rt hjk hЉhщhU hM hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/0")
E = EllipticCurve('37a')
P = plot(E, thickness=3, hue=0.1)
Pru hU1sage_notebook/worksheets/cnta___ec___plot/cells/0rv hU3Graphics object consisting of 2 graphics primitivesrw hډhۉhh$hh&h߉hub(horx }ry (hUP.show()rz hU!r{ hjk hЉhщhUI
r| hMhUfos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/1")
P.show()r} hU1sage_notebook/worksheets/cnta___ec___plot/cells/1r~ hU hډhۉhh$hh&h߉hub(hor }r (hT P = []
k = int(4)
v = [(E.cremona_label(), E) for E in cremona_optimal_curves(range(1,25))]
v.sort()
for lbl, E in v:
P.append(text(lbl[:-1],(0,0)) + \
plot(E,xmin=-1, xmax=3, plot_points=80,thickness=5,rgbcolor=hue(random())))
print lbl,
Q = [[P[k*i+j] for j in range(k)] for i in range(len(P)/k)]
Q.append(P[k*i+j+1:] + [Graphics()]*(k - len(P[k*i+j+1:])))
show(graphics_array(Q), axes=False,figsize=[4,4])r hU!r hjk hЉhщhUI
r hMhT os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/2")
P = []
k = int(4)
v = [(E.cremona_label(), E) for E in cremona_optimal_curves(range(1,25))]
v.sort()
for lbl, E in v:
P.append(text(lbl[:-1],(0,0)) + plot(E,xmin=-1, xmax=3, plot_points=80,thickness=5,rgbcolor=hue(random())))
print lbl,
Q = [[P[k*i+j] for j in range(k)] for i in range(len(P)/k)]
Q.append(P[k*i+j+1:] + [Graphics()]*(k - len(P[k*i+j+1:])))
show(graphics_array(Q), axes=False,figsize=[4,4])r hU1sage_notebook/worksheets/cnta___ec___plot/cells/2r hU'11a1 14a1 15a1 17a1 19a1 20a1 21a1 24a1r hډhۉhh$hh&h߉hub(hor }r (hUE = EllipticCurve('389a')
def f(p):
return plot(E.change_ring(GF(p)),pointsize=40,hue=0.9)
F = [[f(5), f(7)], [f(11),f(997)]]
show(graphics_array(F))r hU!r hjk hЉhщhUI
r hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/3")
E = EllipticCurve('389a')
def f(p):
return plot(E.change_ring(GF(p)),pointsize=40,hue=0.9)
F = [[f(5), f(7)], [f(11),f(997)]]
show(graphics_array(F))r hU1sage_notebook/worksheets/cnta___ec___plot/cells/3r hU hډhۉhh$hUwrapr h߉hub(hor }r (hT[ def sato_tate(E, N):
return [acos(E.ap(p)/(2*sqrt(p))) for p in prime_range(N+1) if N%p != 0]
def dist(E, N, digits=1):
t = verbose('computing st values...')
v = sato_tate(E, N)
t = verbose('finished computing st values:',t)
w = [round(x, digits) for x in v]
w.sort()
vals = {}
for a in w:
if a in vals.keys():
vals[a] += 1
else:
vals[a] = 1
g = vals.items()
g.sort()
verbose('finished post processing:',t)
return g
def graph(label, num=5000):
d = dist(EllipticCurve(label),num,2)
m = max(y for _, y in d)
#m = 2.0 * pi * prime_pi(num)/num
s = Graphics()
for x, y in d:
s += line([(x,0),(x,y)], hue=0.05, thickness=4)
s += plot(lambda x: m*sin(x)^2, 0,pi, plot_points=200,
rgbcolor=(0.3,0.1,0.1), thickness=2)
return sr hU!r hjk hЉhщhU hMhT os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/4")
def sato_tate(E, N):
return [acos(E.ap(p)/(2*sqrt(p))) for p in prime_range(N+1) if N%p != 0]
def dist(E, N, digits=1):
t = verbose('computing st values...')
v = sato_tate(E, N)
t = verbose('finished computing st values:',t)
w = [round(x, digits) for x in v]
w.sort()
vals = {}
for a in w:
if a in vals.keys():
vals[a] += 1
else:
vals[a] = 1
g = vals.items()
g.sort()
verbose('finished post processing:',t)
return g
def graph(label, num=5000):
d = dist(EllipticCurve(label),num,2)
m = max(y for _, y in d)
#m = 2.0 * pi * prime_pi(num)/num
s = Graphics()
for x, y in d:
s += line([(x,0),(x,y)], hue=0.05, thickness=4)
s += plot(lambda x: m*sin(x)^2, 0,pi, plot_points=200,
rgbcolor=(0.3,0.1,0.1), thickness=2)
return sr hU1sage_notebook/worksheets/cnta___ec___plot/cells/4r hU hډhۉhh$hh&h߉hub(hor }r (hUUshow(graphics_array([[graph('37a'), graph('389a')],[ graph('5077a'), graph('32a')]]))r hU!r hjk hЉhщhUJ
r hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/5")
show(graphics_array([[graph('37a'), graph('389a')],[ graph('5077a'), graph('32a')]]))r hU1sage_notebook/worksheets/cnta___ec___plot/cells/5r hU hډhۉhh$hUwrapr h߉hub(hor }r (hU|show(graphics_array([[graph('37a',100), graph('37a',1000)],
[ graph('37a',5000), graph('37a',20000)]]))r hU!r hjk hЉhщhUJ
r hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/7")
show(graphics_array([[graph('37a',100), graph('37a',1000)],
[ graph('37a',5000), graph('37a',20000)]]))r hU1sage_notebook/worksheets/cnta___ec___plot/cells/7r hU hډhۉhh$hUwrapr h߉hub(hor }r (U _Cell__inr U!show(plot(EllipticCurve([-1,0])))r U_Cell__introspect_htmlr U!r U_Cell__worksheetr jk U_Cell__completionsr U_Cell__introspectr U_Cell__out_htmlr UJ
r U _Cell__idr M
U_before_preparser Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/10")
show(plot(EllipticCurve([-1,0])))r U
_Cell__dirr U2sage_notebook/worksheets/cnta___ec___plot/cells/10r U
_Cell__outr U Uhas_new_outputr U_Cell__is_htmlr U _Cell__sager h$U _Cell__typer Uwrapr U _Cell__timer U_Cell__interruptedr ub(hor }r (hUtdef f(x,y):
z = x+I*y
return (sqrt(z^3-z)).imag()
P = plot3dsoya(f, (0,0), 3.0, res=32)
P.show(step=0.04)
r h^U!r hjk hЉh`hU hMU_word_being_completedr UP.shr haUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/8")
def f(x,y):
z = x+I*y
return (sqrt(z^3-z)).imag()
P = plot3dsoya(f, (0,0), 3.0, res=32)
P.show(step=0.04)r hU1sage_notebook/worksheets/cnta___ec___plot/cells/8r hTL [> ] 0%[--------> ] 14%[----------------> ] 27%[--------------------------------> ] 53%[-------------------------------------------------> ] 82%[------------------------------------------------------------>] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - vendor : ATI Technologies Inc.
* - maximum number of lights : 8
* - maximum number of clip planes : 6
* - maximum number of texture units : 8
* - maximum texture size : 2048 pixels
* Soya Pudding * Version: 0.1-0
* Soya * Using 16 bits stencil buffer
* Soya3D * Quit...r hډhdheh$hUwrapr hfhub(hor }r (hiUdef f(x,y):
z = x+I*y
return (sqrt(z^3-z)).real()
P = plot3dsoya(f, (0,0), 3.0, res=32)
P.show(step=0.04,pointer=True)
r j U!r hjjk hkj hlU hmM j Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___plot/cells/9")
def f(x,y):
z = x+I*y
return (sqrt(z^3-z)).real()
P = plot3dsoya(f, (0,0), 3.0, res=32)
P.show(step=0.04,pointer=True)r hnU1sage_notebook/worksheets/cnta___ec___plot/cells/9r hpT [> ] 0%[----------> ] 17%[---------------------> ] 36%[------------------------------------------> ] 70%[------------------------------------------------------------>] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - vendor : ATI Technologies Inc.
* - maximum number of lights : 8
* - maximum number of clip planes : 6
* - maximum number of texture units : 8
* - maximum texture size : 2048 pixels
* Soya Pudding * Version: 0.1-0
* Soya * Using 16 bits stencil buffer
* Soya3D * Quit...r hqj j h$hrUwrapr j hsub(hor }r (j U j jk j j U j M j U2sage_notebook/worksheets/cnta___ec___plot/cells/11r j U j j hj ubeU_Worksheet__synchror KQU_Worksheet__namer Ucnta - ec - plotr U_Worksheet__dirr U)sage_notebook/worksheets/cnta___ec___plotr U_Worksheet__attachedr }r U/home/was/.sage/init.sager J7DsU_Worksheet__queuer ]r U_Worksheet__next_idr M U_Worksheet__comp_is_runningr U_Worksheet__notebookr hU_Worksheet__idr K U_Worksheet__next_block_idr K U_Worksheet__systemr NubUcnta - db - cremonar (hor }r (U_Worksheet__filenamer Ucnta___db___cremonar U_Worksheet__cellsr ]r ((hor }r (U _Cell__inr U"C = CremonaDatabase()
C.allcurves?r U_Cell__introspect_htmlr T C.allcurves
File: /home/was/s/local/lib/python2.4/site-packages/sage/databases/cremona.py
Type: <type 'instancemethod'>
Definition: C.allcurves(N)
Docstring:
Returns the allcurves table of curves of conductor N.
INPUT:
N -- int, the conductor
OUTPUT:
dict -- {id:[ainvs, rank, tor], ...}r U_Cell__worksheetr j U_Cell__completionsr U_Cell__introspectr ]r (j U eU_Cell__out_htmlr U U _Cell__idr M @j UC.r U_before_preparser Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/0")
C = CremonaDatabase()
C.allcurves?r U
_Cell__dirr U4sage_notebook/worksheets/cnta___db___cremona/cells/0r U
_Cell__outr U%Cremona's database of elliptic curvesr Uhas_new_outputr U_Cell__is_htmlr U _Cell__sager h$U _Cell__typer hU _Cell__timer U_Cell__interruptedr ub(hor }r (j UC.number_of_curves()r j U!r j j j j j U j M@j Uuos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/1")
C.number_of_curves()r j U4sage_notebook/worksheets/cnta___db___cremona/cells/1r j U782493r j j j h$j Uwrapr j j ub(hor
}r (j U C.number_of_isogeny_classes()r j U!r
j j j j j U j M @j UC.number_of_isr j Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/11")
C.number_of_isogeny_classes()r j U5sage_notebook/worksheets/cnta___db___cremona/cells/11r j U524169r j j j h$j Uwrapr j j ub(hor }r (j UC.largest_conductor()r j U!r j j j j j U j M@j Uvos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/2")
C.largest_conductor()r j U4sage_notebook/worksheets/cnta___db___cremona/cells/2r j U120000r j j j h$j hj j ub(hor }r (j UC[11]r j U!r j j j j j U j M@j Ufos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/3")
C[11]r j U4sage_notebook/worksheets/cnta___db___cremona/cells/3r j T {'a': {'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5], 'a2': [[0, -1, 1, -7820, -263580], 0, 1]}, 'c': {'a1': ['5', '1.2692093042795534217', '0.25384186085591068434', '1', '1.00000000000000000000'], 'a3': ['1', '6.3460465213977671084', '0.25384186085591068434', '1', '1'], 'a2': ['1', '0.25384186085591068434', '0.25384186085591068434', '1', '1.00000000000000000000']}, 'b': {'a1': 1}, 'd': {}}r j j j h$j Uwrapr! j j ub(hor" }r# (j U#list(cremona_curves([11, 37, 389]))r$ j U!r% j j j j j U j M@j Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/4")
list(cremona_curves([11, 37, 389]))r& j U4sage_notebook/worksheets/cnta___db___cremona/cells/4r' j T` [Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 - x^2 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x +1 over Rational Field, Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field]r( j j j h$j Uwrapr) j j ub(hor* }r+ (j U0E = EllipticCurve([1,2,3,4,5])
E.cremona_label()r, hU!r- j j j hj U j M@hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/5")
E = EllipticCurve([1,2,3,4,5])
E.cremona_label()r. j U4sage_notebook/worksheets/cnta___db___cremona/cells/5r/ j U '10351a1'r0 j hhh$j Uwrapr1 hj ub(hor2 }r3 (hU0E = EllipticCurve([0,2,0,0,6])
E.cremona_label()r4 hU!r5 hj hhhU hM@hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/6")
E = EllipticCurve([0,2,0,0,6])
E.cremona_label()r6 hU4sage_notebook/worksheets/cnta___db___cremona/cells/6r7 hU
'18624bj1'r8 hhhh$hUwrapr9 hhub(hor: }r; (hU hU!r< hj hhhU hM@hUaos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___cremona/cells/8")
r= hU4sage_notebook/worksheets/cnta___db___cremona/cells/8r> hU hhhh$hUhiddenr? hhub(hor@ }rA (hiU hjj hkhlU hmM
@hnU5sage_notebook/worksheets/cnta___db___cremona/cells/10rB hpU hqhrj? hsubeU_Worksheet__synchrorC K!U_Worksheet__namerD Ucnta - db - cremonarE U_Worksheet__dirrF U,sage_notebook/worksheets/cnta___db___cremonarG U_Worksheet__attachedrH }rI U/home/was/.sage/init.sagerJ J7DsU_Worksheet__queuerK ]rL U_Worksheet__next_idrM M @U_Worksheet__comp_is_runningrN U_Worksheet__notebookrO hU_Worksheet__idrP KU_Worksheet__next_block_idrQ K U_Worksheet__systemrR NubUcnta - db - sloanerS (horT }rU (hUcnta___db___sloanerV h]rW ((horX }rY (hUS = SloaneEncyclopedia
SrZ hU!r[ hjT hhhU hM pU_word_being_completedr\ USlor] hUxos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/0")
S = SloaneEncyclopedia
Sr^ hU3sage_notebook/worksheets/cnta___db___sloane/cells/0r_ hU/Sloane Online Encyclopedia of Integer Sequencesr` hhhh$hhhhub(hora }rb (hU v = S.find(prime_range(2,100))rc hU!rd hjT hhhU hMpj\ US.finre hU~os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/1")
v = S.find(prime_range(2,100))rf hU3sage_notebook/worksheets/cnta___db___sloane/cells/1rg hU hhhh$hUwraprh hhub(hori }rj (hUlen(v)rk hU!rl hjT hhhU hMphUfos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/2")
len(v)rm hU3sage_notebook/worksheets/cnta___db___sloane/cells/2rn hU30ro hhhh$hhhhub(horp }rq (hUv[0]rr hU!rs hjT hhhU hMphUdos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/3")
v[0]rt hU3sage_notebook/worksheets/cnta___db___sloane/cells/3ru hT (40, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271])rv hhhh$hhhhub(horw }rx (hUv[1]ry hU!rz hjT hhhU hMphUdos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/4")
v[1]r{ hU3sage_notebook/worksheets/cnta___db___sloane/cells/4r| hT (8578, [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271])r} hhhh$hhhhub(hor~ }r (hUCw = [len(str(numerator(abs(bernoulli(10^n))))) for n in range(5)]
wr hU!r hjT hhhU hMpj\ Ubernr hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/5")
w = [len(str(numerator(abs(bernoulli(10^n))))) for n in range(5)]
wr hU3sage_notebook/worksheets/cnta___db___sloane/cells/5r hU[1, 1, 83, 1779, 27691]r hhhh$hUwrapr hhub(hor }r (hU S.find(w)r hU!r hjT hhhU hMphUios.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/6")
S.find(w)r hU3sage_notebook/worksheets/cnta___db___sloane/cells/6r hU4[(103233, [1, 1, 83, 1779, 27691, 376772, 4767554])]r hhhh$hUwrapr hhub(hor }r (hU h^U!r hjT hh`hU hMphaU`os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___sloane/cells/7")
r hU3sage_notebook/worksheets/cnta___db___sloane/cells/7r hU hhdheh$hh&hfhub(hor }r (hiU hjjT hkhlU hmMphnU3sage_notebook/worksheets/cnta___db___sloane/cells/8r hpU hqhrh&hsubejZ K)j[ j\ }r U/home/was/.sage/init.sager J7Dsj_ U+sage_notebook/worksheets/cnta___db___sloaner ja ]r jc M pjd Ucnta - db - sloaner jf hjg Kjh Kji NubUcnta - db - jonesr (hor }r (j Ucnta___db___jonesr j ]r ((hor }r (j UJ = JonesDatabase()
Jr j U!r j j j j j U j M PU_word_being_completedr UJoner j Utos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___jones/cells/0")
J = JonesDatabase()
Jr j U2sage_notebook/worksheets/cnta___db___jones/cells/0r j UMJohn Jones's table of number fields with bounded ramification and degree <= 6r j j j h$j hj j ub(hor }r (j U0for K in J.unramified_outside([11]):
print Kr j U!r j j j j j U j MPj UJ.unramr j Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___jones/cells/1")
for K in J.unramified_outside([11]):
print Kr j U2sage_notebook/worksheets/cnta___db___jones/cells/1r j UNumber Field in a with defining polynomial x - 1
Number Field in a with defining polynomial x^2 - x + 3
Number Field in a with defining polynomial x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1r j j j h$j Uwrapr j j ub(hor }r (j UV# CAN also get direct access to underlying data, which may be faster.
D = J.as_dict()r hU!r j j j hj U j MPj\ U J.as_dictr hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___jones/cells/2")
# CAN also get direct access to underlying data, which may be faster.
D = J.as_dict()r j U2sage_notebook/worksheets/cnta___db___jones/cells/2r j U j hhh$j Uwrapr hj ub(hor }r (hU D[(5,31)]r hU!r hj hhhU hMPhUhos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___jones/cells/3")
D[(5,31)]r hU2sage_notebook/worksheets/cnta___db___jones/cells/3r hU[x^2 + 155, x^4 + 2*x^3 + 4*x^2 + 3*x - 9, x^4 + x^3 + 3*x - 1, x^4 + 2*x^3 + 2*x^2 + x + 8, x^4 + 5*x^2 + 45, x^4 + 13*x^2 + 81, x^4 + x^3 - 39*x^2 - 39*x + 281]r hhhh$hhhhub(hor }r (hiU hjj hkhlU hmMPhnU2sage_notebook/worksheets/cnta___db___jones/cells/6r hpU hqhrh&hsubejC K'jD Ucnta - db - jonesr jF U*sage_notebook/worksheets/cnta___db___jonesr jH }r U/home/was/.sage/init.sager J7DsjK ]r jM MPjN jO hjP KjQ KjR NubUcnta - save / loadr (hor }r (U_Worksheet__filenamer Ucnta___save___loadr U_Worksheet__cellsr ]r ((hor }r (hU?R. = PolynomialRing(QQ,2)
f = y^2 + y - x^3 - 17/3*x + 2/3r hU!r hj hhhU hM hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/0")
R. = PolynomialRing(QQ,2)
f = y^2 + y - x^3 - 17/3*x + 2/3r hU3sage_notebook/worksheets/cnta___save___load/cells/0r hU hhhh$hh&hhub(hor }r (hUsave fr hU!r hj hhhU hMhUfos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/1")
save fr hU3sage_notebook/worksheets/cnta___save___load/cells/1r hU hhhh$hh&hhub(hor }r (hUload fr hU!r hj hhhU hMhUfos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/2")
load fr hU3sage_notebook/worksheets/cnta___save___load/cells/2r hU hhhh$hh&hhub(hor }r (hU load('f')r hU!r hj hhhU hMhUios.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/3")
load('f')r hU3sage_notebook/worksheets/cnta___save___load/cells/3r hU 2/3 + y + y^2 - 17/3*x - x^3r hhhh$hh&hhub(hor }r (hU'A = MatrixSpace(QQ,50).random_element()r hU!r hj hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/4")
A = MatrixSpace(QQ,50).random_element()r hU3sage_notebook/worksheets/cnta___save___load/cells/4r hU hhhh$hh&hhub(hor }r (hUA.save('amat')r hU!r hj hhhU hMhUnos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/5")
A.save('amat')r hU3sage_notebook/worksheets/cnta___save___load/cells/5r hU hhhh$hh&hhub(hor }r (hU load('amat')r hU!r hj hhhU hMhUlos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/6")
load('amat')r hU3sage_notebook/worksheets/cnta___save___load/cells/6r hT [-1 -2 2 2 -1 -1 1 1 -2 -2 1 2 1 -1 1 -2 2 2 -1 -2 2 1 1 2 2 -1 1 -2 2 2 1 1 -1 1 -1 2 -1 1 -2 2 2 -2 2 -1 2 -2 -2 2 -1 2]
[-1 -2 -2 -2 2 -1 1 -2 2 1 1 -2 -1 -1 -2 -2 -2 -1 2 -1 -2 -2 -2 -2 -1 -2 -2 -1 -2 1 2 2 -2 -1 2 1 2 -1 -1 1 -1 1 -2 -1 -2 -2 -1 -1 1 1]
[-1 1 -2 -1 -2 2 -2 1 -1 -2 1 -1 -1 2 -2 -1 2 1 -2 2 -2 -1 2 2 -1 -2 1 -2 -2 1 1 -1 -2 -2 1 2 2 -1 2 1 1 -2 2 -2 2 1 2 -2 2 -2]
[ 2 2 -2 -2 1 2 1 2 -2 -1 -1 -2 2 2 -1 2 1 1 1 1 -1 -2 2 2 1 -2 -1 -1 1 1 -1 2 2 -1 -1 2 1 1 -2 1 2 2 -2 -1 -1 -1 -2 -1 -2 -2]
[ 1 -2 1 1 -1 2 1 2 -2 -2 2 1 -2 2 2 2 1 2 -2 2 2 -1 -2 1 -1 2 -2 2 2 2 1 -1 -2 1 2 1 2 -1 1 -2 -2 -1 1 2 1 -2 1 2 1 -1]
[-2 1 -1 -2 2 -1 -2 -2 -2 -1 -2 -1 1 -2 1 -1 1 -1 -2 -1 -2 2 -1 -2 -1 1 1 -1 1 -2 1 1 1 -1 1 -2 -2 -2 2 2 -2 2 -2 -1 -2 -1 -1 -2 1 -1]
[-2 1 -1 -1 2 -1 -2 1 -1 1 -2 -2 2 2 1 1 -2 2 -1 2 -2 1 -2 -1 -2 -2 2 1 1 2 2 -2 -2 -1 -1 -1 -1 2 -2 2 1 2 -1 2 2 -1 2 -2 2 -1]
[ 1 1 2 2 1 -2 2 -1 2 2 1 1 -2 1 1 -1 -1 1 2 1 1 1 -2 1 -2 2 2 2 1 1 1 1 -2 2 -2 1 1 -1 2 -2 2 1 -2 -1 1 -1 1 -2 2 -1]
[-2 2 1 -2 1 1 -2 1 -2 1 -1 1 -2 2 2 -1 2 -1 -1 -2 -2 1 1 2 2 2 2 2 1 1 2 -1 1 -1 -2 1 -2 1 1 -1 2 -2 -2 -2 1 -1 1 1 -2 -2]
[-1 -1 -1 -1 2 1 1 -2 -1 -2 2 -2 -2 1 2 1 2 -2 -1 -1 -2 1 -1 -2 1 1 2 -2 2 2 -1 -2 1 2 -2 1 1 1 2 2 -2 1 -2 2 2 2 1 2 1 -2]
[ 2 -1 2 -2 -1 -2 2 -2 -2 1 1 1 1 1 1 1 -1 1 1 -2 1 -1 2 -1 1 -2 -2 1 2 -1 1 2 2 1 1 -2 -1 1 1 2 2 2 2 1 -1 2 -2 -2 -1 1]
[ 2 1 -2 2 2 1 -2 -2 -2 -1 -1 2 1 -1 2 2 1 2 2 -2 2 -2 -2 2 -2 -2 2 -2 -2 2 2 -2 2 -2 1 2 1 -1 -1 2 -2 -2 1 1 1 1 1 1 -2 1]
[-2 -1 2 2 -2 2 2 -2 1 -2 -2 -1 2 -1 -1 1 -2 2 1 -2 1 -2 1 -2 -1 -1 1 -2 -2 1 -2 -1 -1 1 1 -2 2 -1 1 -1 -1 1 -2 2 -1 1 -1 -1 -2 1]
[-2 -2 -1 1 1 -2 1 1 2 1 -1 1 1 1 -2 -2 -1 2 1 -1 2 1 2 2 2 -1 2 2 -2 -2 2 1 2 2 -2 1 -1 1 -2 -2 2 -2 1 -1 1 2 -2 1 2 1]
[ 2 -2 -1 -2 -2 2 -2 1 -2 2 -1 1 -2 1 -2 2 1 -1 -2 2 2 2 2 1 2 -2 -2 2 -2 -1 2 -1 1 2 2 1 2 -2 -1 1 2 2 1 1 1 -1 2 -1 2 2]
[-2 2 2 2 2 1 -2 1 1 1 -1 2 -2 -2 -2 2 -1 -2 -2 -2 -1 -2 -2 2 1 2 -2 -1 -2 -1 2 1 2 -2 -2 -2 1 1 2 -1 1 -1 -1 -1 2 1 2 1 -1 -2]
[ 1 -1 -2 -2 -2 -1 2 2 -2 2 -2 -1 -2 -2 -2 -1 1 2 -1 -2 -1 -2 2 -2 2 -1 2 -2 -2 -1 2 -2 -1 -2 2 -2 -1 2 2 -1 -1 -1 1 2 1 1 -1 -2 2 -1]
[-1 2 -1 -1 -2 1 -1 -2 -1 -1 -1 -2 -2 -2 -2 -1 2 -1 -2 1 2 -2 -1 1 1 1 -2 -1 1 2 -2 -1 -2 -2 1 2 -2 -1 -2 1 -2 -2 -2 -1 2 -1 2 -2 2 2]
[-2 -1 2 -1 -2 2 -2 1 2 -1 -1 1 2 1 -2 2 1 -1 2 1 1 -1 -1 1 -2 2 -2 2 2 2 -2 -2 1 -2 2 2 1 -2 1 1 1 1 2 -1 -1 1 -2 -1 -2 -2]
[ 2 1 -1 1 -1 2 -1 1 2 -1 -2 -2 -1 -1 -1 2 -2 -2 -1 -1 2 -1 -1 1 2 1 2 -2 -2 1 2 2 2 2 1 -1 2 -2 -2 1 -1 -1 1 -2 -1 -1 2 -2 -2 1]
[-2 2 2 -2 -1 -2 -1 2 -1 2 1 -1 -2 -2 2 -1 -2 2 -2 1 2 1 1 -1 1 1 -1 2 2 -1 -2 -2 2 -2 -2 2 2 -1 2 1 1 1 1 -2 1 -1 -1 -1 -2 -1]
[ 2 2 2 -1 -1 1 -2 2 -2 2 2 -2 -2 1 1 -2 -2 1 -2 -2 2 2 -1 2 -2 1 1 -2 2 -1 2 2 -1 1 1 2 2 2 -1 1 -2 2 1 1 -1 1 -1 2 -2 1]
[ 1 2 2 -2 -1 -1 -2 -2 -1 -2 -2 1 1 -2 1 -1 1 2 -2 -2 2 -2 -1 -1 -2 2 1 -2 -2 -2 1 -2 2 1 -1 -2 -2 -1 -1 1 -1 2 -2 1 1 -1 1 -1 1 2]
[ 1 2 -2 2 1 1 1 2 -1 2 -1 -1 -1 2 -2 1 -2 1 1 -1 -2 -1 -1 2 -2 2 2 2 -1 -1 -2 2 -2 2 -2 2 1 -2 2 1 2 -2 -1 -2 -2 1 1 -1 1 -2]
[-1 1 -2 -2 1 -1 -2 -2 -1 -1 -1 2 -1 1 1 1 1 2 1 -1 -2 2 1 -1 -2 -2 1 1 2 -2 -1 2 -1 2 2 1 2 1 -1 1 -1 -2 -1 2 -2 -1 -2 -1 -1 -2]
[-2 -2 2 2 2 -1 -2 -1 -2 1 1 -1 2 -2 -2 -2 -2 -2 2 2 -2 2 -2 -1 -1 -1 -2 2 -2 1 -2 -2 1 2 -1 -2 -1 -2 1 -1 2 -1 -2 1 2 1 -1 -2 -1 -2]
[-1 1 -2 -1 2 -1 -2 1 1 1 -1 2 -1 -2 1 -1 1 2 -1 -1 -1 1 -2 -2 2 -1 2 2 -1 -2 -2 -1 -1 -1 1 2 -2 2 -1 -1 -2 -2 2 2 -1 1 -2 2 2 -2]
[-2 2 2 2 -2 2 1 -2 1 2 2 -1 -2 1 -2 -2 -1 -2 2 -1 -1 -2 1 2 2 -2 2 -1 -2 -1 -2 2 -2 -1 2 2 2 2 -2 -1 -2 -2 -2 2 -2 2 2 -2 -1 1]
[-2 -1 1 -2 1 -1 -1 1 -2 -2 1 1 -2 1 1 -2 2 2 1 2 -2 -2 -2 -2 1 1 2 2 -1 -1 2 -1 -1 -1 1 1 2 1 -2 1 -2 -2 1 -1 -2 1 1 -2 1 2]
[ 1 1 -2 -2 -1 1 1 -2 -1 -2 1 -1 -1 -2 1 -2 -2 2 -2 2 1 1 2 -1 2 1 2 -1 -1 -1 2 1 1 2 -2 1 -1 -2 1 -1 -1 1 -2 -2 -1 1 -2 2 -2 1]
[ 1 -1 -2 -1 -2 2 1 -1 2 -1 1 1 -1 2 1 -2 2 -1 -2 2 -2 -2 -2 -1 -1 -2 -1 1 -1 -2 -1 1 -2 1 -1 2 -1 -1 2 -1 -1 -1 1 -1 1 2 -2 -2 -2 -1]
[ 2 2 1 -1 -2 2 -2 -1 1 2 -2 -2 -2 1 -2 -2 -2 -1 -1 2 1 -2 1 1 1 -1 1 -1 -2 2 -1 -2 -2 2 2 2 -2 -1 2 -1 -1 -2 -1 -1 1 -1 -1 -1 2 2]
[ 2 2 -1 1 -2 1 2 -1 1 -2 -1 2 2 1 2 -2 -1 -1 1 -2 2 2 -1 1 2 2 2 -1 -1 1 -2 2 -1 -1 1 1 -1 2 -2 -2 -1 -1 -2 1 1 -2 1 2 2 -2]
[-2 -1 -1 2 -1 -1 2 1 2 2 -2 2 -1 1 1 1 -1 -1 2 2 2 -1 -2 2 -2 2 -2 1 1 1 1 2 -1 1 -2 1 1 -1 1 -1 -1 1 1 -2 -2 1 -1 1 2 1]
[-1 -2 1 2 -2 2 -1 2 -1 2 1 2 -1 2 -1 -1 -1 -1 1 -1 -2 2 1 1 -2 1 2 -2 -1 -2 -1 -2 2 1 -2 1 1 -2 2 1 2 -2 -1 1 -2 2 2 2 2 -1]
[ 2 1 1 -2 -1 1 -1 2 1 -2 -2 -1 1 -1 2 2 1 2 -1 1 -1 1 -2 -1 1 1 2 -2 -1 1 -2 1 1 1 -2 2 1 2 2 2 -1 1 -2 1 2 -1 -2 -2 1 2]
[ 1 2 1 1 -2 1 1 1 2 -1 -2 2 1 1 2 -2 -2 1 2 2 1 2 1 1 -2 -2 -1 -2 -2 -1 -1 -2 -1 1 2 1 1 -2 2 1 -2 1 -1 -2 -1 1 1 -2 1 1]
[-2 2 1 -2 -2 2 2 -1 1 -2 -1 2 2 1 2 1 2 -1 -1 1 2 1 2 1 -1 -1 2 1 2 1 1 1 1 -2 -2 -2 -2 -2 -2 -1 2 -1 2 2 1 2 1 -1 1 1]
[ 2 2 -2 2 -1 2 2 2 -2 -2 2 -1 -2 -2 -2 2 2 2 -1 2 -2 1 -1 -1 1 -1 -1 2 2 -2 1 -2 -2 -1 -2 1 -1 2 1 -1 1 2 -2 2 1 1 1 -1 2 1]
[-2 -2 -1 -2 -1 1 -2 -2 1 2 2 2 2 1 -1 -1 -2 2 -2 2 -2 2 2 1 -1 -1 1 2 -2 2 1 1 1 -1 2 -2 1 -1 -1 1 -2 2 -2 -1 1 -1 1 1 -2 -2]
[ 2 -1 -1 -1 -1 2 1 -1 2 2 2 1 -2 -2 -1 -2 1 1 -1 -2 1 -1 -2 2 1 2 2 1 -1 -1 2 2 2 2 -2 -1 1 1 2 -1 -1 2 -2 -2 1 -2 -1 1 -2 -1]
[ 2 1 -1 -2 1 2 1 -1 2 2 -2 -2 -1 -2 1 -1 1 -2 -1 2 1 -1 2 1 2 1 1 1 -2 2 2 -1 -2 1 -1 -2 1 -1 -2 1 2 1 1 1 2 1 -2 -1 2 2]
[ 2 -1 -2 -1 1 -2 2 2 -1 -2 -1 1 -2 -1 2 -2 2 2 2 -2 -1 -1 -2 2 1 -2 -2 -1 -1 2 2 2 2 1 2 2 -1 -2 -2 2 -1 1 -1 -2 -2 2 2 -1 -1 2]
[-1 2 -2 2 2 -2 -2 1 -2 -1 -1 -1 -1 -1 -1 1 2 -2 1 2 1 -1 1 1 1 -2 -1 -2 -1 2 -1 2 1 -1 2 -1 -1 2 2 1 1 -2 1 -2 1 -1 2 2 -2 -1]
[ 2 -1 2 -1 2 -2 2 2 -2 2 2 1 1 -2 -2 -1 -2 -1 -2 2 1 2 1 -2 2 1 -2 1 -1 2 1 -2 2 2 2 2 -2 -1 -1 1 -2 -1 1 -1 -2 1 2 -1 -2 -1]
[-2 -2 -1 1 -1 -1 2 2 2 1 -2 1 -2 -1 -2 2 -1 2 -1 1 -1 -2 2 -2 -1 1 -2 -1 2 -2 -2 -2 1 1 -2 2 2 2 -1 2 -1 -1 1 -1 2 1 -1 2 -2 2]
[ 2 2 2 -2 -1 2 2 2 1 -1 -1 2 2 -2 -2 -2 -1 -2 -1 -1 1 1 2 1 -2 -2 -2 2 -1 -1 2 1 2 -1 -1 2 -1 1 2 1 2 -1 1 -1 -1 -1 2 -2 -2 -2]
[ 1 2 -2 1 2 -1 2 1 2 2 -1 -1 -1 -1 1 1 -2 -1 1 -1 -1 2 -1 2 2 -2 -2 1 -1 2 2 -1 -2 1 -2 -2 -2 1 -2 -1 -1 1 1 1 -1 -1 -2 2 1 1]
[ 2 1 2 -2 -1 2 2 2 1 2 -1 -2 1 -2 -2 -2 1 1 -1 1 -2 2 2 1 2 2 1 2 -2 -2 -2 1 -2 2 -1 -2 -2 1 1 1 -2 2 -2 -2 -2 -1 1 2 1 -2]
[ 2 2 -1 1 -1 1 -1 -2 -2 -2 -1 1 1 -2 1 -1 -1 -2 -1 1 -2 -2 1 -1 -2 2 -2 -2 2 -1 2 -2 -1 2 -2 -1 2 -1 2 -1 1 1 2 2 -1 2 -2 1 1 -2]r hhhh$hh&hhub(hor }r (hU#M = ModularSymbols(Gamma1(13),2); Mr U_Cell__introspect_htmlr U!r hj hU_Cell__introspectr hU hMU_before_preparser Uos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/17")
M = ModularSymbols(Gamma1(13),2); Mr hU4sage_notebook/worksheets/cnta___save___load/cells/17r hUeModular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Fieldr hU_Cell__is_htmlr U _Cell__sager h$hh&U _Cell__timer hub(hor
}r (U _Cell__inr UM == loads(dumps(M))r
j U!r U_Cell__worksheetr j U_Cell__completionsr j U_Cell__out_htmlr U U _Cell__idr Mj Uuos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/20")
M == loads(dumps(M))r U
_Cell__dirr U4sage_notebook/worksheets/cnta___save___load/cells/20r U
_Cell__outr UTruer Uhas_new_outputr j j h$U _Cell__typer h&j U_Cell__interruptedr ub(hor }r (hUD = M.decomposition(2)r hU!r hj hhhU hM-h-UM.decomr hUwos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/45")
D = M.decomposition(2)r h U4sage_notebook/worksheets/cnta___save___load/cells/45r! h U h h!h"h$h%h&h'h(ub(hor" }r# (hUDhU!r$ hj hhhU hM.hUbos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/46")
Dr% h U4sage_notebook/worksheets/cnta___save___load/cells/46r& h T [
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field
]r' h h!h"h$h%h&h'h(ub(hor( }r) (hUsave Dr* hU!r+ hj hhhU hM/hUgos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/47")
save Dr, h U4sage_notebook/worksheets/cnta___save___load/cells/47r- h U h h!h"h$h%h&h'h(ub(hor. }r/ (hU load('D')r0 hU!r1 hj hhhU hM0hUjos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/48")
load('D')r2 h U4sage_notebook/worksheets/cnta___save___load/cells/48r3 h T [
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field
]r4 h h!h"h$h%h&h'h(ub(hor5 }r6 (hU h^U!r7 hj hh`hU hM1haUaos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/49")
r8 h U4sage_notebook/worksheets/cnta___save___load/cells/49r9 h U h hdheh$h%h&hfh(ub(hor: }r; (hiU hjj hkhlU hmM2hnU4sage_notebook/worksheets/cnta___save___load/cells/50r< hpU hqhrh&hsubeU_Worksheet__synchror= MU_Worksheet__comp_is_runningr> U_Worksheet__dirr? U+sage_notebook/worksheets/cnta___save___loadr@ U_Worksheet__attachedrA }rB U/home/was/.sage/init.sagerC J7DsU_Worksheet__queuerD ]rE U_Worksheet__next_idrF M3U_Worksheet__namerG Ucnta - save / loadrH U_Worksheet__notebookrI hU_Worksheet__idrJ K U_Worksheet__next_block_idrK KU_Worksheet__systemrL NubUcnta - combinatorial geomrM (horN }rO (h
Ucnta___combinatorial_geomrP h ]rQ ((horR }rS (hUSP. = ProjectiveSpace(3,QQ)
C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
CrT hU!rU hjN hhhU hM hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/0")
P. = ProjectiveSpace(3,QQ)
C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
CrV h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/0rW h U}Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
y^2 - x*z
z^2 - y*w
-1*y*z + x*wrX h h!h"h$h%h&h'h(ub(horY }rZ (hU1len(C.irreducible_components()) # twisted cubicr[ hU!r\ hjN hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/1")
len(C.irreducible_components()) # twisted cubicr] h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/1r^ h U1h h!h"h$h%h&h'h(ub(hor_ }r` (hUJ = C.defining_ideal()
Jra hU!rb hjN hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/2")
J = C.defining_ideal()
Jrc h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/2rd h U_Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational Fieldre h h!h"h$h%h&h'h(ub(horf }rg (hUG = J.groebner_fan()
Grh hU!ri hjN hhhU hMhU}os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/3")
G = J.groebner_fan()
Grj h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/3rk h U{Groebner fan of the ideal:
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational Fieldrl h h!h"h$h%h&h'h(ub(horm }rn (hUG.reduced_groebner_bases()ro hU!rp hjN hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/4")
G.reduced_groebner_bases()rq h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/4rr h Ts [[-1*z^2 + y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2 - y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2 - y*w, y*z - x*w, -1*y^2 + x*z, -1*y^3 + x^2*w], [z^2 - y*w, y*z - x*w, y^3 - x^2*w, -1*y^2 + x*z], [z^2 - y*w, y*z - x*w, y^2 - x*z], [-1*z^2 + y*w, y^2 - x*z, -1*y*z + x*w], [-1*z^2 + y*w, y*z - x*w, y^2 - x*z, -1*z^3 + x*w^2], [z^3 - x*w^2, -1*z^2 + y*w, y*z - x*w, y^2 - x*z]]rs h h!h"h$h%h&h'h(ub(hort }ru (hU G.fvector()rv hU!rw hjN hhhU hMhUros.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/6")
G.fvector()rx h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/6ry h U (1, 8, 8)rz h h!h"h$h%h&h'h(ub(hor{ }r| (hUf = prod(J.gens())r} hU!r~ hjN hhhU hMhUyos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/5")
f = prod(J.gens())r h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/5r h U h h!h"h$h%h&h'h(ub(hor }r (hU@NP = polymake.convex_hull(f.exponents()) # -- newton polytoper hU!r hjN hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/7")
NP = polymake.convex_hull(f.exponents()) # -- newton polytoper h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/7r h U h h!h"h$h%h&h'h(ub(hor }r (hU NP.facets()r hU!r hjN hhhU hMhUros.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/8")
NP.facets()r h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/8r h U_[(3/2, 5/2, -1, 0), (3, 1, -1, 0), (1, 0, 0, 0), (-3/2, 2, 1, 0), (3, -1, 4, 0), (-3, 1, 5, 0)]r h h!h"h$h%h&h'h(ub(hor }r (hU h^U!r hjN hh`hU hM haUgos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___combinatorial_geom/cells/9")
r h U:sage_notebook/worksheets/cnta___combinatorial_geom/cells/9r h U h hdheh$h%h&hfh(ub(hor }r (hiU hjjN hkhlU hmM hnU;sage_notebook/worksheets/cnta___combinatorial_geom/cells/11r hpU hqhrh&hsubehtK)huUcnta - combinatorial geomr hwU2sage_notebook/worksheets/cnta___combinatorial_geomr hy}r U/home/was/.sage/init.sager J7Dsh|]r h~M hhhhNhKhKubUcnta - ec - lserr (hor }r (jm Ucnta___ec___lserr jo ]r ((hor }r (hUE = EllipticCurve('37a')
Er hU!r hj hЉhщhU hM hUE.Lserier hUxos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/0")
E = EllipticCurve('37a')
Er hU1sage_notebook/worksheets/cnta___ec___lser/cells/0r hU?Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr hډhۉhh$hh&h߉hub(hor }r (hUtime v = E.anlist(100000)r hU!r hj hhhU hMjE UE.anlir hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/21")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
v = E.anlist(100000)r hU2sage_notebook/worksheets/cnta___ec___lser/cells/21r hU$CPU time: 1.43 s, Wall time: 1.49 sr hhhh$hh&hhub(hor }r (hU(time v = E.anlist(100000,pari_ints=True)r hU!r hj hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/22")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
v = E.anlist(100000,pari_ints=True)r hU2sage_notebook/worksheets/cnta___ec___lser/cells/22r hU$CPU time: 0.31 s, Wall time: 0.30 sr hhhh$hUwrapr hhub(hor }r (hUCE.Lseries_at1() # directly and with proved error bound if nonzeror hU!r hj hЉhщhU hMhU E.Lseriesr hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/1")
E.Lseries_at1() # directly and with proved error bound if nonzeror hU1sage_notebook/worksheets/cnta___ec___lser/cells/1r hU0hډhۉhh$hh&h߉hub(hor }r (hU E.Lseries(1) # via PARIr hU!r hj hЉhщhU hMhUE.Lserr hUzos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/2")
E.Lseries(1) # via PARIr hU1sage_notebook/worksheets/cnta___ec___lser/cells/2r hU0.00000000000000000r hډhۉhh$hh&h߉hub(hor }r (hUGE.Lseries_sympow(2,16) # Watkins C program -- symmetric square L-valuer hU!r hj hЉhщhU hMhU E.Lseries_syr hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/3")
E.Lseries_sympow(2,16) # Watkins C program -- symmetric square L-valuer hU1sage_notebook/worksheets/cnta___ec___lser/cells/3r hU'2.492262044273650E+00'r hډhۉhh$hh&h߉hub(hor }r (hU:L = E.Lseries_dokchitser() # Tim Dokchitser's GP packager hU!r hj hЉhщhU hMhU
E.Lseries_dokr hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/4")
L = E.Lseries_dokchitser() # Tim Dokchitser's GP packager hU1sage_notebook/worksheets/cnta___ec___lser/cells/4r hU hډhۉhh$hh&h߉hub(hor }r (hUL(1)r hU!r hj hЉhщhU hMhUbos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/5")
L(1)r hU1sage_notebook/worksheets/cnta___ec___lser/cells/5r hU0hډhۉhh$hh&h߉hub(hor }r (hUL.derivative(1)r hU!r hj hЉhщhU hMhUL.derr hUmos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/7")
L.derivative(1)r hU1sage_notebook/worksheets/cnta___ec___lser/cells/7r hU0.30599977383405230r hډhۉhh$hh&h߉hub(hor }r (hU>L(1+I) # works very well for arbitrary inputr hU!r hj hЉhщhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/8")
L(1+I) # works very well for arbitrary inputr hU1sage_notebook/worksheets/cnta___ec___lser/cells/8r hU,-0.15892526330137721 + 0.45791106676511545*Ir hډhۉhh$hh&h߉hub(hor }r (hUr hj hЉhщhU hM hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/9")
set_verbose(-2)
v = [tuple(L(1 + t/8*I)) for t in range(75)]r hU1sage_notebook/worksheets/cnta___ec___lser/cells/9r hU hډhۉhh$hUwrapr h߉hub(hor }r (hUshow(line(v, hue=0.8))r hU!r hj hЉhщhUJ
r hM
hUuos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/10")
show(line(v, hue=0.8))r hU2sage_notebook/worksheets/cnta___ec___lser/cells/10r hU hډhۉhh$hh&h߉hub(hor }r (U _Cell__inr UVfor P in E.Lseries_zeros_in_interval(0,10,0.3): # Mike Rubinstein's
print P[0]r hU!r U_Cell__worksheetr j U_Cell__completionsr hU_Cell__out_htmlr U U _Cell__idr MjE UE.Lseries_zeros_ir hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/19")
for P in E.Lseries_zeros_in_interval(0,10,0.3): # Mike Rubinstein's
print P[0]r U
_Cell__dirr U2sage_notebook/worksheets/cnta___ec___lser/cells/19r U
_Cell__outr U65.0031700134
6.8703912161
8.0143308081
9.9330983534r Uhas_new_outputr hhh$U _Cell__typer h&hU_Cell__interruptedr
ub(hor }r (hU}%time
# Christophe Doche and Sylvain Duquesne implemented SEA, which is
# included with SAGE
print E.sea(next_prime(10^50))r
hU!r hj hhhU hMjE UE.ser hT os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/20")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
# Christophe Doche and Sylvain Duquesne implemented SEA, which is
# included with SAGE
print E.sea(next_prime(10^50))r hU2sage_notebook/worksheets/cnta___ec___lser/cells/20r hU(52
CPU time: 0.00 s, Wall time: 0.13 sr hhhh$hh&hhub(hor }r (hU h^U!r hj hh`hU hMhaU_os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___ec___lser/cells/26")
r hU2sage_notebook/worksheets/cnta___ec___lser/cells/26r hU hhdheh$hh&hfhub(hor }r (hiU hjj hkhlU hmMhnU2sage_notebook/worksheets/cnta___ec___lser/cells/27r hpU hqhrh&hsubej Kj Ucnta - ec - lserr j U)sage_notebook/worksheets/cnta___ec___lserr j }r U/home/was/.sage/init.sager J7Dsj ]r j M j j hj K
j Kj NubUcnta - modformr (hor! }r" (U_Worksheet__filenamer# Ucnta___modformr$ U_Worksheet__cellsr% ]r& ((hor' }r( (hiU!M = ModularForms(1, 12)
M.basis()r) h^U!r* hjj! hkh`hlU hmM haU}os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/0")
M = ModularForms(1, 12)
M.basis()r+ hnU/sage_notebook/worksheets/cnta___modform/cells/0r, hpU[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6),
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + 3199218815520/691*q^5 + O(q^6)
]r- hqhdheh$hrUwrapr. hfhsub(hor/ }r0 (hiU$S = M.cuspidal_submodule()
S.basis()r1 h^U!r2 hjj! hkh`hlU hmMU_word_being_completedr3 UM.cuspidal_submor4 haUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/1")
S = M.cuspidal_submodule()
S.basis()r5 hnU/sage_notebook/worksheets/cnta___modform/cells/1r6 hpU9[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]r7 hqhdheh$hrUwrapr8 hfhsub(hor9 }r: (hiU(M = ModularForms(Gamma0(23),2)
M.basis()r; h^U!r< hjj! hkh`hlU hmMhaUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/2")
M = ModularForms(Gamma0(23),2)
M.basis()r= hnU/sage_notebook/worksheets/cnta___modform/cells/2r> hpU[
q - q^3 - q^4 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6),
1 + 12/11*q + 36/11*q^2 + 48/11*q^3 + 84/11*q^4 + 72/11*q^5 + O(q^6)
]r? hqhdheh$hrUwrapr@ hfhsub(horA }rB (hiU0M = ModularForms(Gamma1(13),2,prec=20)
M.basis()rC h^U!rD hjj! hkh`hlU hmMhaUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/3")
M = ModularForms(Gamma1(13),2,prec=20)
M.basis()rE hnU/sage_notebook/worksheets/cnta___modform/cells/3rF hpTt [
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 + 10*q^16 - 3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 + 5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + O(q^20),
1 + 21060/19*q^11 - 36504/19*q^12 - 10270/19*q^13 - 388440/19*q^14 + 595140/19*q^15 + 89856/19*q^16 - 323856/19*q^17 - 939120/19*q^18 + 617760/19*q^19 + O(q^20),
q + 11709/19*q^11 - 20687/19*q^12 - 17570/57*q^13 - 219842/19*q^14 + 336765/19*q^15 + 50783/19*q^16 - 183429/19*q^17 - 532306/19*q^18 + 350304/19*q^19 + O(q^20),
q^2 + 262*q^11 - 467*q^12 - 386/3*q^13 - 4779*q^14 + 7306*q^15 + 1097*q^16 - 3974*q^17 - 11550*q^18 + 7608*q^19 + O(q^20),
q^3 + 918/19*q^11 - 1215/19*q^12 - 1115/57*q^13 - 15203/19*q^14 + 23529/19*q^15 + 3666/19*q^16 - 12783/19*q^17 - 36661/19*q^18 + 23964/19*q^19 + O(q^20),
q^4 - 882/19*q^11 + 2095/19*q^12 + 1607/57*q^13 + 18130/19*q^14 - 27230/19*q^15 - 4014/19*q^16 + 14916/19*q^17 + 43745/19*q^18 - 29064/19*q^19 + O(q^20),
q^5 - 1287/19*q^11 + 2607/19*q^12 + 2114/57*q^13 + 25201/19*q^14 - 38200/19*q^15 - 5761/19*q^16 + 20878/19*q^17 + 60950/19*q^18 - 40260/19*q^19 + O(q^20),
q^6 - 1080/19*q^11 + 2024/19*q^12 + 1637/57*q^13 + 20053/19*q^14 - 30520/19*q^15 - 4608/19*q^16 + 16608/19*q^17 + 48445/19*q^18 - 31908/19*q^19 + O(q^20),
q^7 - 675/19*q^11 + 1056/19*q^12 + 940/57*q^13 + 12355/19*q^14 - 19075/19*q^15 - 2880/19*q^16 + 10380/19*q^17 + 29872/19*q^18 - 19553/19*q^19 + O(q^20),
q^8 - 360/19*q^11 + 453/19*q^12 + 419/57*q^13 + 5975/19*q^14 - 9350/19*q^15 - 1403/19*q^16 + 5042/19*q^17 + 14489/19*q^18 - 9420/19*q^19 + O(q^20),
q^9 - 153/19*q^11 + 98/19*q^12 + 44/19*q^13 + 2290/19*q^14 - 3665/19*q^15 - 592/19*q^16 + 2007/19*q^17 + 5575/19*q^18 - 3576/19*q^19 + O(q^20),
q^10 - 54/19*q^11 - 9/19*q^12 + 22/57*q^13 + 597/19*q^14 - 994/19*q^15 - 143/19*q^16 + 534/19*q^17 + 1420/19*q^18 - 900/19*q^19 + O(q^20)
]rG hqhdheh$hrUnowraprH hfhsub(horI }rJ (hiU#S = M.cuspidal_subspace()
S.basis()rK h^U!rL hjj! hkh`hlU hmMhaUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/4")
S = M.cuspidal_subspace()
S.basis()rM hnU/sage_notebook/worksheets/cnta___modform/cells/4rN hpU[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 + 10*q^16 - 3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 + 5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + O(q^20)
]rO hqhdheh$hrUnowraprP hfhsub(horQ }rR (hiU%E = M.eisenstein_subspace()
E.basis()rS h^U!rT hjj! hkh`hlU hmMj3 UM.eisenstein_subsprU haUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/5")
E = M.eisenstein_subspace()
E.basis()rV hnU/sage_notebook/worksheets/cnta___modform/cells/5rW hpT [
1 + 21060/19*q^11 - 36504/19*q^12 - 10270/19*q^13 - 388440/19*q^14 + 595140/19*q^15 + 89856/19*q^16 - 323856/19*q^17 - 939120/19*q^18 + 617760/19*q^19 + O(q^20),
q + 11709/19*q^11 - 20687/19*q^12 - 17570/57*q^13 - 219842/19*q^14 + 336765/19*q^15 + 50783/19*q^16 - 183429/19*q^17 - 532306/19*q^18 + 350304/19*q^19 + O(q^20),
q^2 + 262*q^11 - 467*q^12 - 386/3*q^13 - 4779*q^14 + 7306*q^15 + 1097*q^16 - 3974*q^17 - 11550*q^18 + 7608*q^19 + O(q^20),
q^3 + 918/19*q^11 - 1215/19*q^12 - 1115/57*q^13 - 15203/19*q^14 + 23529/19*q^15 + 3666/19*q^16 - 12783/19*q^17 - 36661/19*q^18 + 23964/19*q^19 + O(q^20),
q^4 - 882/19*q^11 + 2095/19*q^12 + 1607/57*q^13 + 18130/19*q^14 - 27230/19*q^15 - 4014/19*q^16 + 14916/19*q^17 + 43745/19*q^18 - 29064/19*q^19 + O(q^20),
q^5 - 1287/19*q^11 + 2607/19*q^12 + 2114/57*q^13 + 25201/19*q^14 - 38200/19*q^15 - 5761/19*q^16 + 20878/19*q^17 + 60950/19*q^18 - 40260/19*q^19 + O(q^20),
q^6 - 1080/19*q^11 + 2024/19*q^12 + 1637/57*q^13 + 20053/19*q^14 - 30520/19*q^15 - 4608/19*q^16 + 16608/19*q^17 + 48445/19*q^18 - 31908/19*q^19 + O(q^20),
q^7 - 675/19*q^11 + 1056/19*q^12 + 940/57*q^13 + 12355/19*q^14 - 19075/19*q^15 - 2880/19*q^16 + 10380/19*q^17 + 29872/19*q^18 - 19553/19*q^19 + O(q^20),
q^8 - 360/19*q^11 + 453/19*q^12 + 419/57*q^13 + 5975/19*q^14 - 9350/19*q^15 - 1403/19*q^16 + 5042/19*q^17 + 14489/19*q^18 - 9420/19*q^19 + O(q^20),
q^9 - 153/19*q^11 + 98/19*q^12 + 44/19*q^13 + 2290/19*q^14 - 3665/19*q^15 - 592/19*q^16 + 2007/19*q^17 + 5575/19*q^18 - 3576/19*q^19 + O(q^20),
q^10 - 54/19*q^11 - 9/19*q^12 + 22/57*q^13 + 597/19*q^14 - 994/19*q^15 - 143/19*q^16 + 534/19*q^17 + 1420/19*q^18 - 900/19*q^19 + O(q^20)
]rX hqhdheh$hrUnowraprY hfhsub(horZ }r[ (hiUM.eisenstein_series()r\ h^U!r] hjj! hkh`hlU hmMj3 UM.eisenstein_ser^ haUqos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/6")
M.eisenstein_series()r_ hnU/sage_notebook/worksheets/cnta___modform/cells/6r` hpT [
1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + 18*q^10 + 12*q^11 + 28*q^12 + q^13 + 24*q^14 + 24*q^15 + 31*q^16 + 18*q^17 + 39*q^18 + 20*q^19 + O(q^20),
-7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^4 + -4*q^5 + O(q^6),
q + (zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (3*zeta6 + 3)*q^4 + 4*q^5 + O(q^6),
-zeta6 + q + (2*zeta6 - 1)*q^2 + (3*zeta6 - 2)*q^3 + (-2*zeta6 - 1)*q^4 + 6*q^5 + O(q^6),
q + (zeta6 + 1)*q^2 + (zeta6 + 2)*q^3 + (zeta6 + 2)*q^4 + 6*q^5 + O(q^6),
-1 + q - q^2 + 4*q^3 + 3*q^4 - 4*q^5 - 4*q^6 - 6*q^7 - 5*q^8 + 13*q^9 + 4*q^10 - 10*q^11 + 12*q^12 + 6*q^14 - 16*q^15 + 11*q^16 + 18*q^17 - 13*q^18 - 18*q^19 + O(q^20),
q + q^2 + 4*q^3 + 3*q^4 + 4*q^5 + 4*q^6 + 6*q^7 + 5*q^8 + 13*q^9 + 4*q^10 + 10*q^11 + 12*q^12 + 6*q^14 + 16*q^15 + 11*q^16 + 18*q^17 + 13*q^18 + 18*q^19 + O(q^20),
zeta6 - 1 + q + (-2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (2*zeta6 - 3)*q^4 + 6*q^5 + O(q^6),
q + (-zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (-zeta6 + 3)*q^4 + 6*q^5 + O(q^6),
7/13*zeta6 - 18/13 + q + (-2*zeta6 + 3)*q^2 + (3*zeta6 - 2)*q^3 + (-6*zeta6 + 3)*q^4 + -4*q^5 + O(q^6),
q + (-zeta6 + 3)*q^2 + (zeta6 + 2)*q^3 + (-3*zeta6 + 6)*q^4 + 4*q^5 + O(q^6)
]ra hqhdheh$hrUnowraprb hfhsub(horc }rd (hiU:M = ModularForms(Gamma0(11),6)
S = M.cuspidal_subspace()
Sre h^U!rf hjj! hkh`hlU hmM haUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/9")
M = ModularForms(Gamma0(11),6)
S = M.cuspidal_subspace()
Srg hnU/sage_notebook/worksheets/cnta___modform/cells/9rh hpUCuspidal subspace of dimension 4 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(11) of weight 6 over Rational Fieldri hqhdheh$hrUwraprj hfhsub(hork }rl (hiU t2 = S.hecke_operator(2); t2rm h^U!rn hjj! hkh`hlU hmM
j3 U
S.hecke_opro haUyos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/10")
t2 = S.hecke_operator(2); t2rp hnU0sage_notebook/worksheets/cnta___modform/cells/10rq hpUHecke operator T_2 on Cuspidal subspace of dimension 4 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(11) of weight 6 over Rational Fieldrr hqhdheh$hrUwraprs hfhsub(hort }ru (hiU t2.matrix()rv h^U!rw hjj! hkh`hlU hmM haUhos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/11")
t2.matrix()rx hnU0sage_notebook/worksheets/cnta___modform/cells/11ry hpUJ[ 0 32 10 4]
[ 1 0 3 38]
[ 0 0 -2 -12]
[ 0 1 -2 -2]rz hqhdheh$hrUwrapr{ hfhsub(hor| }r} (hiU factor(charpoly(t2.matrix()))r~ h^U!r hjj! hkh`hlU hmM haUzos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/12")
factor(charpoly(t2.matrix()))r hnU0sage_notebook/worksheets/cnta___modform/cells/12r hpU (x + 4) * (x^3 - 90*x + 188)r hqhdheh$hrUwrapr hfhsub(hor }r (hiUJModularSymbols(Gamma0(11),6, sign=1).cuspidal_submodule().decomposition(2)r h^U!r hjj! hkh`hlU hmMhaUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___modform/cells/14")
ModularSymbols(Gamma0(11),6, sign=1).cuspidal_submodule().decomposition(2)r hnU0sage_notebook/worksheets/cnta___modform/cells/14r hpT [
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 6 with sign 1 over Rational Field,
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 6 with sign 1 over Rational Field
]r hqhdheh$hrUwrapr hfhsub(hor }r (hiU hjj! hkhlU hmMhnU0sage_notebook/worksheets/cnta___modform/cells/15r hpU hqhrh&hsubeU_Worksheet__synchror K!U_Worksheet__namer Ucnta - modformr U_Worksheet__dirr U'sage_notebook/worksheets/cnta___modformr U_Worksheet__attachedr }r U/home/was/.sage/init.sager J7DsU_Worksheet__queuer ]r U_Worksheet__next_idr MU_Worksheet__comp_is_runningr U_Worksheet__notebookr hU_Worksheet__systemr NU_Worksheet__next_block_idr K"U_Worksheet__idr KubUcnta - db - stein-watkinsr (hor }r (hUcnta___db___stein_watkinsr h]r ((hor }r (hU D = SteinWatkinsPrimeData(0)
Dr hU!r hj hhhU hM `j\ USteinWatkinsPrimr hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___stein_watkins/cells/0")
D = SteinWatkinsPrimeData(0)
Dr hU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/0r hU3Stein-Watkins Prime Conductor Database p.0 Iteratorr hhhh$hUwrapr hhub(hor }r (hUC = D.next()
Cr hU!r hj hhhU hM`hUuos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___stein_watkins/cells/1")
C = D.next()
Cr hU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/1r hU+Stein-Watkins isogeny class of conductor 11r hhhh$hUwrapr hhub(hor }r (hUlist(C)r hU!r hj hhhU hM`hUnos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___stein_watkins/cells/5")
list(C)r hU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/5r hU}[[[0, -1, 1, 0, 0], '(1)', '1', '5'], [[0, -1, 1, -10, -20], '(5)', '1', '5'], [[0, -1, 1, -7820, -263580], '(1)', '1', '1']]r hhhh$hhhhub(hor }r (hUD.next()r hU!r hj hhhU hM`hUoos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___stein_watkins/cells/2")
D.next()r hU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/2r hU+Stein-Watkins isogeny class of conductor 17r hhhh$hUwrapr hhub(hor }r (hUD.next()r hU!r hj hhhU hM`hUoos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___stein_watkins/cells/3")
D.next()r hU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/3r hU+Stein-Watkins isogeny class of conductor 19r hhhh$hh&hhub(hor }r (hU h^U!r hj hh`hU hM`haUgos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___db___stein_watkins/cells/4")
r hU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/4r hU hhdheh$hh&hfhub(hor }r (hiU hjj hkhlU hmM`hnU:sage_notebook/worksheets/cnta___db___stein_watkins/cells/7r hpU hqhrh&hsubejZ K%j[ j\ }r U/home/was/.sage/init.sager J7Dsj_ U2sage_notebook/worksheets/cnta___db___stein_watkinsr ja ]r jc M`jd Ucnta - db - stein-watkinsr jf hjg Kjh K ji NubU _scratch_r (hor }r (U_Worksheet__filenamer U _scratch_r U_Worksheet__cellsr ]r ((hor }r (hiU2+3r h^U!r hjj hkh`hlU hmKhaUZos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/_scratch_/cells/4")
2+3r hnU*sage_notebook/worksheets/_scratch_/cells/4r hpU5hqhdheh$hrh&hfhsub(hor }r (hiUGshow(plot(EllipticCurve('37a'), rgbcolor=(1,0,1), thickness=3),dpi=100)r h^U!r hjj hkh`hlUB
r hmKhaUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/_scratch_/cells/5")
show(plot(EllipticCurve('37a'), rgbcolor=(1,0,1), thickness=3),dpi=100)r hnU*sage_notebook/worksheets/_scratch_/cells/5r hpU hqhdheh$hrh&hfhsub(hor }r (hiU hjj hkhlU hmKhnU*sage_notebook/worksheets/_scratch_/cells/7r hpU hqhrh&hsubeU_Worksheet__synchror K U_Worksheet__namer U _scratch_r U_Worksheet__dirr U"sage_notebook/worksheets/_scratch_r U_Worksheet__attachedr }r U/home/was/.sage/init.sager J7DsU_Worksheet__queuer ]r U_Worksheet__next_idr KU_Worksheet__comp_is_runningr U_Worksheet__notebookr hU_Worksheet__idr K U_Worksheet__next_block_idr K U_Worksheet__systemr NubUcnta - using other systemsr (hor }r (j Ucnta___using_other_systemsr j ]r ((hor }r (hU*n = -2007
print n.factor()
print factor(n)r hU!r hj hhhU hM hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/0")
n = -2007
print n.factor()
print factor(n)r hU;sage_notebook/worksheets/cnta___using_other_systems/cells/0r hU -1 * 3^2 * 223
-1 * 3^2 * 223r
hhhh$hh&hhub(hor }r (hUn.factor(algorithm="kash")r
hU!r hj hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/1")
n.factor(algorithm="kash")r hU;sage_notebook/worksheets/cnta___using_other_systems/cells/1r hU-1 * 3^2 * 223r hhhh$hUwrapr hhub(hor }r (hUgap(n).FactorsInt()r hU!r hj hhhU hMhU{os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/2")
gap(n).FactorsInt()r hU;sage_notebook/worksheets/cnta___using_other_systems/cells/2r hU[ -3, 3, 223 ]r hhhh$hh&hhub(hor }r (hUpari(n).factor()r hU!r hj hhhU hMhUxos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/3")
pari(n).factor()r hU;sage_notebook/worksheets/cnta___using_other_systems/cells/3r hU[-1, 1; 3, 2; 223, 1]r hhhh$hh&hhub(hor! }r" (hUgp(n).factor()r# hU!r$ hj hhhU hM
hUwos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/10")
gp(n).factor()r% hUr+ hj hhhU hMhUzos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/4")
maxima(n).factor()r, hU;sage_notebook/worksheets/cnta___using_other_systems/cells/4r- hU-3^2*223r. hhhh$hh&hhub(hor/ }r0 (hUkash(n).Factorization()r1 hU!r2 hj hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/5")
kash(n).Factorization()r3 hU;sage_notebook/worksheets/cnta___using_other_systems/cells/5r4 hUE[ <3, 2>, <223, 1> ], extended by:
ext1 := -1,
ext2 := Unassignr5 hhhh$hh&hhub(hor6 }r7 (hU!magma(n).Factorization(nvals = 2)r8 hU!r9 hj hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/6")
magma(n).Factorization(nvals = 2)r: hU;sage_notebook/worksheets/cnta___using_other_systems/cells/6r; hU([ <3, 2>, <223, 1> ], -1)r< hhhh$hh&hhub(hor= }r> (hUmaple(n).ifactor()r? hU!r@ hj hhhU hMhUzos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/7")
maple(n).ifactor()rA hU;sage_notebook/worksheets/cnta___using_other_systems/cells/7rB hU-``(3)^2*``(223)rC hhhh$hh&hhub(horD }rE (hU mathematica(n).FactorInteger()rF hU!rG hj hhhU hMhUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/8")
mathematica(n).FactorInteger()rH hU;sage_notebook/worksheets/cnta___using_other_systems/cells/8rI hU{{-1, 1}, {3, 2}, {223, 1}}rJ hhhh$hh&hhub(horK }rL (hU<%magma
n := -2007;
F, s := Factorization(-2007);
print F, srM hU!rN hj hhhU hM hUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/9")
%magma
n := -2007;
F, s := Factorization(-2007);
print F, srO hU;sage_notebook/worksheets/cnta___using_other_systems/cells/9rP hU[ <3, 2>, <223, 1> ]
-1rQ hhhh$hh&hhub(horR }rS (hU
magma('F')rT hU!rU hj hhhU hM hUsos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/11")
magma('F')rV hU, <223, 1> ]rX hhhh$hh&hhub(horY }rZ (hU h^U!r[ hj hh`hU hM haUios.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/12")
r\ hU jA }ra U/home/was/.sage/init.sagerb J7Dsj? U3sage_notebook/worksheets/cnta___using_other_systemsrc jD ]rd jF MjG Ucnta - using other systemsre jI hjL NjK K
jJ K ubuU_Notebook__historyrf ]rg (UG# Worksheet 'cnta - save / load' (2006-07-12 at 07:33)
sage: A.lksajd
5rh TD # Worksheet 'cnta - save / load' (2006-07-12 at 07:35)
sage: search_sage('helper for pickle')
--------------------------------------------------------
| SAGE Version 1.3.5.2, Build Date: 2006-07-08-0825 |
| Distributed under the GNU General Public License V2. |
--------------------------------------------------------ri T= # Worksheet 'cnta - save / load' (2006-07-12 at 07:35)
sage: search_sage('for pickle')
--------------------------------------------------------
| SAGE Version 1.3.5.2, Build Date: 2006-07-08-0825 |
| Distributed under the GNU General Public License V2. |
--------------------------------------------------------rj Ud# Worksheet 'cnta - save / load' (2006-07-12 at 07:39)
sage: A = MatrixSpace(QQ,2).random_element()
rk T # Worksheet 'cnta - save / load' (2006-07-12 at 07:39)
sage: loads(dumps(A))
Traceback (most recent call last):
loads(dumps(A))
...
cPickle.PicklingError: Second element of tuple returned by must be a tuplerl Ud# Worksheet 'cnta - save / load' (2006-07-12 at 07:40)
sage: A = MatrixSpace(QQ,2).random_element()
rm T # Worksheet 'cnta - save / load' (2006-07-12 at 07:40)
sage: loads(dumps(A))
Traceback (most recent call last):
loads(dumps(A))
...
cPickle.PicklingError: Second element of tuple returned by must be a tuplern Ud# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: A = MatrixSpace(QQ,2).random_element()
ro U# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: loads(dumps(A))
Traceback (most recent call last):
loads(dumps(A))
...
NotImplementedErrorrp UQ# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: b = loads(dumps(A))
rq UH# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: b.nrows()
2rr Ux# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.lkasjdf = 5
rs UQ# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: b = loads(dumps(A))
rt U# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: b.lkasjdf
Traceback (most recent call last):
b.lkasjdf
...
AttributeError: 'matrix_pyx.Matrix' object has no attribute 'lkasjdf'ru Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
rv UQ# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: b = loads(dumps(A))
rw U# Worksheet 'cnta - save / load' (2006-07-12 at 07:41)
sage: b.l
Traceback (most recent call last):
b.l
...
AttributeError: 'matrix_pyx.Matrix' object has no attribute 'l'rx Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:44)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
ry U# Worksheet 'cnta - save / load' (2006-07-12 at 07:44)
sage: b = loads(dumps(A))
Traceback (most recent call last):
b = loads(dumps(A))
...
IOError: invalid data streamrz TW # Worksheet 'cnta - save / load' (2006-07-12 at 07:45)
sage: dumps(A)
'x\x9cuP\xcbN\xc30\x10\x14\x01ZH)\xef\x97\xc4\x01\x89S\xb8\xe43\xb8T\xf8P\xe4\xf3\xcaq\x96\
\xc8R\x9at\xbd\x8e\x08\x87Jp\xe4\xafq\x14S\nJ}\xf1\xeejgvf>"\xcd\xaa\xc0t\xa1\x9c5m\xf8`\x\
f9\xde\xc6\x00\x16\xf3F#\xc0\xf3\xef\x90v\x92\xa1}^*\x8dq\xbf\xf7\xd2\xd5\xf0j\xb0\xcc...r{ U# Worksheet 'cnta - save / load' (2006-07-12 at 07:45)
sage: loads(dumps(A))
Traceback (most recent call last):
loads(dumps(A))
...
IOError: invalid data streamr| Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r} U# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: loads(dumps(A))
Traceback (most recent call last):
loads(dumps(A))
...
NotImplementedErrorr~ UQ# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: B = loads(dumps(A))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: B.l
Traceback (most recent call last):
B.l
...
AttributeError: 'matrix_pyx.Matrix' object has no attribute 'l'r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: B.nrows
r UH# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: B.nrows()
2r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: B[1,1]
Traceback (most recent call last):
B[1,1]
...
NotImplementedErrorr U_# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: type(B)
r Uw# Worksheet 'cnta - save / load' (2006-07-12 at 07:46)
sage: type(A)
r UM# Worksheet 'cnta - save / load' (2006-07-12 at 07:47)
sage: c = A.__class__
r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:47)
sage: new(c)
Traceback (most recent call last):
new(c)
...
NameError: name 'new' is not definedr U# Worksheet 'cnta - save / load' (2006-07-12 at 07:47)
sage: c.__new__()
Traceback (most recent call last):
c.__new__()
...
TypeError: matrix_pyx.Matrix.__new__(): not enough argumentsr U# Worksheet 'cnta - save / load' (2006-07-12 at 07:47)
sage: c.__new__(A.parent())
Traceback (most recent call last):
c.__new__(A.parent())
...
TypeError: matrix_pyx.Matrix.__new__(X): X is not a type object (MatrixSpace_field)r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:47)
sage: c.__new__(type(A))
Traceback (most recent call last):
c.__new__(type(A))
...
TypeError: int() argument must be a string or a numberr U# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: c.__new__(type(A),2,3)
Traceback (most recent call last):
c.__new__(type(A),2,3)
...
TypeError: int() argument must be a string or a numberr UX# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: d = c.__new__(type(A),2,3)
r Uw# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: type(d)
r Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r UM# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: c = A.__class__
r Uq# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: c
r UX# Worksheet 'cnta - save / load' (2006-07-12 at 07:48)
sage: d = c.__new__(type(A),2,3)
r Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:50)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:50)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:50)
sage: B = loads(dumps(A))
Traceback (most recent call last):
B = loads(dumps(A))
...
AttributeError: 'Matrix_dense_rational' object has no attribute '__type__'r UM# Worksheet 'cnta - save / load' (2006-07-12 at 07:50)
sage: c = A.__class__
r U\# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: d = c.__new__(A.__class__,2,3)
r Uw# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: type(d)
r UX# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: d = c.__new__(A.__class__)
r Uw# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: type(d)
r Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r UQ# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: B = loads(dumps(A))
r UG# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: B.nrows()
r Uw# Worksheet 'cnta - save / load' (2006-07-12 at 07:51)
sage: type(B)
r UG# Worksheet 'cnta - save / load' (2006-07-12 at 07:52)
sage: B.nrows()
r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:52)
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Fieldr Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:54)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:54)
sage: B = loads(dumps(A))
Traceback (most recent call last):
B = loads(dumps(A))
...
cPickle.PicklingError: Can't pickle : import of module mutability_pyx failedr Ur# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: A = MatrixSpace(QQ,2).random_element()
sage: A.l = 5
r UQ# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: B = loads(dumps(A))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Fieldr UH# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: B.nrows()
2r UK# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: B
[1 2]
[1 1]r UV# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: A == loads(dumps(A))
Truer UW# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: A is loads(dumps(A))
Falser U{# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: A = MatrixSpace(Integers(8),2).random_element()
sage: A.l = 5
r UW# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: A is loads(dumps(A))
Falser UU# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: A ==loads(dumps(A))
Truer UY# Worksheet 'cnta - save / load' (2006-07-12 at 07:55)
sage: loads(dumps(A))
[7 6]
[2 1]r U{# Worksheet 'cnta - save / load' (2006-07-12 at 08:00)
sage: A = MatrixSpace(Integers(8),2).random_element()
sage: A.l = 5
r U{# Worksheet 'cnta - save / load' (2006-07-12 at 08:01)
sage: A = MatrixSpace(Integers(8),2).random_element()
sage: A.l = 5
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:01)
sage: A ==loads(dumps(A))
Traceback (most recent call last):
A ==loads(dumps(A))
...
NameError: sager U{# Worksheet 'cnta - save / load' (2006-07-12 at 08:01)
sage: A = MatrixSpace(Integers(8),2).random_element()
sage: A.l = 5
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:01)
sage: A ==loads(dumps(A))
Traceback (most recent call last):
A ==loads(dumps(A))
...
cPickle.PicklingError: Can't pickle : import of module mutability_pyx failedr U{# Worksheet 'cnta - save / load' (2006-07-12 at 08:03)
sage: A = MatrixSpace(Integers(8),2).random_element()
sage: A.l = 5
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:03)
sage: A ==loads(dumps(A))
Traceback (most recent call last):
A ==loads(dumps(A))
...
NameError: sager U{# Worksheet 'cnta - save / load' (2006-07-12 at 08:04)
sage: A = MatrixSpace(Integers(8),2).random_element()
sage: A.l = 5
r UU# Worksheet 'cnta - save / load' (2006-07-12 at 08:04)
sage: A ==loads(dumps(A))
Truer UB# Worksheet 'cnta - save / load' (2006-07-12 at 08:04)
sage: A.l
5r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:04)
sage: A.charpoly()
Traceback (most recent call last):
A.charpoly()
...
NotImplementedErrorr Uf# Worksheet 'cnta - save / load' (2006-07-12 at 08:04)
sage: A = MatrixSpace(QQ,100).random_element()
r Ue# Worksheet 'cnta - save / load' (2006-07-12 at 08:05)
sage: A = MatrixSpace(QQ,50).random_element()
r UL# Worksheet 'cnta - save / load' (2006-07-12 at 08:05)
sage: A.save('amat')
r T[ # Worksheet 'cnta - save / load' (2006-07-12 at 08:05)
sage: load('amat')
[ 2 -1 2 2 2 -2 -1 -2 2 2 -2 1 -1 -1 2 -2 1 -1 -1 -1 1 -1 1 -1 2 1 1 -1 -1 2
-1 -1 -2 1 -2 -1 1 2 -2 2 2 -1 -2 1 -1 1 -2 2 2 -1]
[ 2 -1 1 2 2 -2 2 1 2 -2 1 1 -1 -1 -1 -2 -1 2 1 2 1 -2 -1 1 1 1 1 -2 2 1
-2 1 -1 1 1 -1 -2 -2 1...r UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:05)
sage: f = A.charpoly()
r TY # Worksheet 'cnta - save / load' (2006-07-12 at 08:05)
sage: f.factor()
(x^50 + 4*x^49 - 17*x^48 + 223*x^47 - 3175*x^46 + 39419*x^45 + 178933*x^44 + 6721023*x^43
+ 62309197*x^42 + 686595910*x^41 + 9869795996*x^40 - 115422131577*x^39 - 180040202922*x^38
+ 2496681744386*x^37 + 137615943938023*x^36 + 945174540623537*x^35 - 8597542011686976*x^3...r Tg # Worksheet 'cnta - save / load' (2006-07-12 at 08:05)
sage: f = A.charpoly()
sage: f
x^50 + 4*x^49 - 17*x^48 + 223*x^47 - 3175*x^46 + 39419*x^45 + 178933*x^44 + 6721023*x^43 +
62309197*x^42 + 686595910*x^41 + 9869795996*x^40 - 115422131577*x^39 - 180040202922*x^38 +
2496681744386*x^37 + 137615943938023*x^36 + 945174540623537*x^35 - 8597542011686976*x^34...r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:06)
sage: M = ModularSymbols(Gamma1(13),2); M
Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over
Rational Fieldr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:07)
sage: A = Matrix(Integer(8),3,range(9))
Traceback (most recent call last):
A = Matrix(Integer(8),3,range(9))
...
TypeError: R (=8) must be a ring.r U`# Worksheet 'cnta - save / load' (2006-07-12 at 08:07)
sage: A = Matrix(Integers(8),3,range(9))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:07)
sage: A = Matrix(Integer(8),3,range(9))
sage: loads(dumps(A)) == A
Traceback (most recent call last):
A = Matrix(Integer(8),3,range(9))
...
TypeError: R (=8) must be a ring.r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:07)
sage: A = Matrix(Integers(8),3,range(9))
sage: loads(dumps(A)) == A
Truer UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:08)
sage: A.determinant()
0r UH# Worksheet 'cnta - save / load' (2006-07-12 at 08:08)
sage: A[0,0] = 5
r UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:08)
sage: A.determinant()
7r UH# Worksheet 'cnta - save / load' (2006-07-12 at 08:08)
sage: A[0,0] = 5
r UO# Worksheet 'cnta - save / load' (2006-07-12 at 08:08)
sage: A.set_immutable()
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:08)
sage: A[0,0] = 5
Traceback (most recent call last):
A[0,0] = 5
...
ValueError: object is immutable; please change a copy instead.r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:11)
sage: latex(A)
Traceback (most recent call last):
...
NameError: name 'A' is not definedr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:11)
sage: A = Matrix(Integers(8),3,range(9))
sage: loads(dumps(A)) == A
Truer UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:11)
sage: A.determinant()
0r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:11)
sage: latex(A)
\left(\begin{array}{rrr}
0&1&2\\
3&4&5\\
6&7&0
\end{array}\right)r Uc# Worksheet 'cnta - save / load' (2006-07-12 at 08:14)
sage: M=ntl.mat_ZZ(3,3,[1,2,3,4,5,6,7,8,9])
r UL# Worksheet 'cnta - save / load' (2006-07-12 at 08:14)
sage: M.LLL()
(2, 54)r US# Worksheet 'cnta - save / load' (2006-07-12 at 08:15)
sage: 2/3 == pari(2/3)
Falser UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:15)
sage: 1 == pari(1)
Truer U# Worksheet 'cnta - save / load' (2006-07-12 at 08:15)
sage: QQ._coerce_(pari(2/3))
Traceback (most recent call last):
QQ._coerce_(pari(2/3))
...
TypeErrorr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:15)
sage: pari.__coerce__(2/3)
Traceback (most recent call last):
pari.__coerce__(2/3)
...
AttributeError: 'gen.PariInstance' object has no attribute '__coerce__'r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:15)
sage: pari._coerce_(2/3)
Traceback (most recent call last):
pari._coerce_(2/3)
...
TypeError: x must be a PARI objectr U_# Worksheet 'cnta - save / load' (2006-07-12 at 08:16)
sage: type(pari(1) + 1)
r Ue# Worksheet 'cnta - save / load' (2006-07-12 at 08:16)
sage: type(1+pari(1))
r Ue# Worksheet 'cnta - save / load' (2006-07-12 at 08:16)
sage: type(2/3 + 1)
r Uc# Worksheet 'cnta - save / load' (2006-07-12 at 08:16)
sage: type(1+2/3)
r Ua# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: canonical_coercion(1,pari(1))
(1, 1)r U_# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: a,b=canonical_coercion(1,pari(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(a), type(b)
(, )r U_# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: a,b=canonical_coercion(pari(1),1)
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(a), type(b)
(, )r UB# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: a+b
2r U_# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(a+b)
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(pari(1),1)
Traceback (most recent call last):
type(pari(1),1)
...
TypeError: type() takes 1 or 3 argumentsr U]# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(pari(1)+1)
r Ue# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(1+pari(1))
r U]# Worksheet 'cnta - save / load' (2006-07-12 at 08:17)
sage: type(pari(1)+1)
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:20)
sage: type(pari(1)+1)
Traceback (most recent call last):
os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/34")
...
NameError: name 'os' is not definedr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:20)
sage: type(pari(1)+1)
Traceback (most recent call last):
os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___save___load/cells/34")
...
NameError: name 'os' is not definedr Ue# Worksheet 'cnta - save / load' (2006-07-12 at 08:21)
sage: type(pari(1)+1)
r Ue# Worksheet 'cnta - save / load' (2006-07-12 at 08:21)
sage: type(1+pari(1))
r T # Worksheet 'cnta - save / load' (2006-07-12 at 08:21)
sage: type(1+pari(1/3))
Traceback (most recent call last):
type(1+pari(1/3))
...
TypeError: unable to find a common parent for 1 (parent: Integer Ring) and 1/3 (parent: Interface to the PARI C library)r UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:21)
sage: pari(1) + gp(1)
2r Uy# Worksheet 'cnta - save / load' (2006-07-12 at 08:21)
sage: type(pari(1) + gp(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:22)
sage: print type(pari(1) + gp(1)), type(gp(1) + pari(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:24)
sage: print type(pari(1) + gp(1)), type(gp(1) + pari(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:24)
sage: print type(pari(1) + gp(1)), type(gp(1) + pari(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:25)
sage: pari(gp(1))
Traceback (most recent call last):
pari(gp(1))
...
AttributeErrorr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:26)
sage: print type(pari(1) + gp(1)), type(gp(1) + pari(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:26)
sage: pari(gp(1))
Traceback (most recent call last):
pari(gp(1))
...
AttributeErrorr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:26)
sage: pari(gp(1))
Traceback (most recent call last):
...
AttributeErrorr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:27)
sage: pari._coerce_(gp(1))
Traceback (most recent call last):
pari._coerce_(gp(1))
...
TypeError: no canonical coercion of 1 into PARIr U# Worksheet 'cnta - save / load' (2006-07-12 at 08:27)
sage: print type(pari(1) + gp(1)), type(gp(1) + pari(1))
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:27)
sage: pari(gp(1))
Traceback (most recent call last):
pari(gp(1))
...
AttributeErrorr UJ# Worksheet 'cnta - save / load' (2006-07-12 at 08:29)
sage: pari(gp(1))
1r TQ # Worksheet 'cnta - save / load' (2006-07-12 at 08:29)
sage: print type(pari(1) + gp(1)), type(gp(1) + pari(1))
Traceback (most recent call last):
print type(pari(1) + gp(1)), type(gp(1) + pari(1))
...
TypeError: unable to find an unambiguous parent for 1 (parent: Interface to the PARI C library) and 1 (parent: GP/PARI interpreter)r TE # Worksheet 'cnta - save / load' (2006-07-12 at 08:29)
sage: print type(pari(1) + gap(1)), type(gap(1) + pari(1))
Traceback (most recent call last):
print type(pari(1) + gap(1)), type(gap(1) + pari(1))
...
TypeError: unable to find an unambiguous parent for 1 (parent: Interface to the PARI C library) and 1 (parent: Gap)r T1 # Worksheet 'cnta - save / load' (2006-07-12 at 08:29)
sage: print type(gp(1) + gap(1)), type(gap(1) + gp(1))
Traceback (most recent call last):
print type(gp(1) + gap(1)), type(gap(1) + gp(1))
...
TypeError: unable to find an unambiguous parent for 1 (parent: GP/PARI interpreter) and 1 (parent: Gap)r UJ# Worksheet 'cnta - save / load' (2006-07-12 at 08:29)
sage: pari(gp(1))
1r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:29)
sage: pari(ModularSymbols(37))
Traceback (most recent call last):
pari(ModularSymbols(37))
...
gen.PariError: unknown function or error in formal parameters (3)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:31)
sage: pari(ModularSymbols(37))
Traceback (most recent call last):
...
gen.PariError: unknown function or error in formal parameters (3)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:31)
sage: pari(ModularSymbols(37))
Traceback (most recent call last):
...
gen.PariError: unknown function or error in formal parameters (3)r Tg # Worksheet 'cnta - save / load' (2006-07-12 at 08:32)
sage: pari(ModularSymbols(37))
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
...r TI # Worksheet 'cnta - save / load' (2006-07-12 at 08:32)
sage: pari(ModularSymbols(37))
1
Traceback (most recent call last):
pari(ModularSymbols(37))
File "/home/was/talks/2006-07-09-cnta/", line 1, in ?
File "gen.pyx", line 5645, in gen._pari_trap
gen.PariError: unknown function or error in formal parameters (3)r TM # Worksheet 'cnta - save / load' (2006-07-12 at 08:32)
sage: pari('ModularSymbols(37)')
1
Traceback (most recent call last):
pari('ModularSymbols(37)')
File "/home/was/talks/2006-07-09-cnta/", line 1, in ?
File "gen.pyx", line 5645, in gen._pari_trap
gen.PariError: unknown function or error in formal parameters (3)r T # Worksheet 'cnta - save / load' (2006-07-12 at 08:32)
sage: pari('2/0')
1
Traceback (most recent call last):
pari('2/0')
File "/home/was/talks/2006-07-09-cnta/", line 1, in ?
File "gen.pyx", line 5645, in gen._pari_trap
gen.PariError: division by zero (46)r UJ# Worksheet 'cnta - save / load' (2006-07-12 at 08:35)
sage: pari(gp(1))
1r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:35)
sage: pari('2/0')
Traceback (most recent call last):
pari('2/0')
...
gen.PariError: division by zero (46)r T1 # Worksheet 'cnta - save / load' (2006-07-12 at 08:36)
sage: print type(gp(1) + gap(1)), type(gap(1) + gp(1))
Traceback (most recent call last):
print type(gp(1) + gap(1)), type(gap(1) + gp(1))
...
TypeError: unable to find an unambiguous parent for 1 (parent: GP/PARI interpreter) and 1 (parent: Gap)r TQ # Worksheet 'cnta - save / load' (2006-07-12 at 08:36)
sage: print type(gp(1) + pari(1)), type(pari(1) + gp(1))
Traceback (most recent call last):
print type(gp(1) + pari(1)), type(pari(1) + gp(1))
...
TypeError: unable to find an unambiguous parent for 1 (parent: GP/PARI interpreter) and 1 (parent: Interface to the PARI C library)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:36)
sage: pari(1) + 1
Traceback (most recent call last):
pari(1) + 1
...
TypeError: unable to find an unambiguous parent for 1 (parent: Interface to the PARI C library) and 1 (parent: Integer Ring)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:36)
sage: 1 + pari(1)
Traceback (most recent call last):
1 + pari(1)
...
TypeError: unable to find an unambiguous parent for 1 (parent: Integer Ring) and 1 (parent: Interface to the PARI C library)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:36)
sage: pari(GF(3)(2))
Traceback (most recent call last):
pari(GF(3)(2))
...
TypeError: instance exception may not have a separate valuer U# Worksheet 'cnta - save / load' (2006-07-12 at 08:36)
sage: pari(1) + GF(3)(2)
Traceback (most recent call last):
pari(1) + GF(3)(2)
...
TypeError: instance exception may not have a separate valuer U# Worksheet 'cnta - save / load' (2006-07-12 at 08:37)
sage: pari(1) + GF(3)(2)
Traceback (most recent call last):
...
TypeError: no way to coerce 1 into this ring.r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:37)
sage: pari(1) + GF(3)(2)
Traceback (most recent call last):
...
TypeError: no way to coerce 1 of type into Finite Field of size 3.r
U# Worksheet 'cnta - save / load' (2006-07-12 at 08:38)
sage: GF(3)(int(1))
Traceback (most recent call last):
GF(3)(int(1))
...
TypeError: no way to coerce 1 of type into Finite Field of size 3.r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:38)
sage: GF(3)(int(1))
Traceback (most recent call last):
...
TypeError:
no way to coerce 1 of type into Finite Field of size 3.r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:38)
sage: GF(3)(1)
Traceback (most recent call last):
GF(3)(1)
...
TypeError:
no way to coerce 1 of type into Finite Field of size 3.r
U# Worksheet 'cnta - save / load' (2006-07-12 at 08:38)
sage: GF(3)(1)
Traceback (most recent call last):
GF(3)(1)
...
TypeError:
no way to coerce 1 of type into Finite Field of size 3.r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:39)
sage: GF(3)(1)
Traceback (most recent call last):
...
TypeError: no way to coerce 1 of type into Finite Field of size 3.r Uz# Worksheet 'cnta - save / load' (2006-07-12 at 08:39)
sage: GF(3)(1)
Traceback (most recent call last):
...
RuntimeErrorr UG# Worksheet 'cnta - save / load' (2006-07-12 at 08:40)
sage: GF(3)(1)
1r UY# Worksheet 'cnta - save / load' (2006-07-12 at 08:40)
sage: pari(1) + GF(3)(2)
Mod(0, 3)r UU# Worksheet 'cnta - save / load' (2006-07-12 at 08:40)
sage: pari(GF(3)(1))
Mod(1, 3)r UY# Worksheet 'cnta - save / load' (2006-07-12 at 08:41)
sage: GF(3)(2) + pari(2)
Mod(1, 3)r T # Worksheet 'cnta - save / load' (2006-07-12 at 08:41)
sage: gap(1) + gp(1) + magma(1)
Traceback (most recent call last):
gap(1) + gp(1) + magma(1)
...
TypeError: unable to find an unambiguous parent for 1 (parent: Gap) and 1 (parent: GP/PARI interpreter)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:42)
sage: R. = PolynomialRing(QQ,2)
sage: f = y^2 + y - x^3 - 17/3*x + 2/3
r UD# Worksheet 'cnta - save / load' (2006-07-12 at 08:42)
sage: save f
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:42)
sage: A = Matrix(Integers(8),3,range(9))
sage: loads(dumps(A)) == A
True
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:42)
sage: M.LLL()
Traceback (most recent call last):
M.LLL()
...
NameError: name 'M' is not definedr U8# Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: view(A)
\left(\begin{array}{rrr}
0&1&2\\
3&4&5\\
6&7&0
\end{array}\right)r U# Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: A = Matrix(Integers(8),3,range(9))
sage: view(A)
sage: loads(dumps(A)) == A
Truer U# Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: A = Matrix(Integers(8),3,range(9))
sage: view(A)
sage: loads(dumps(A)) == A
Truer U# Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: A = Matrix(Integers(8),3,range(9))
sage: show(A)
sage: loads(dumps(A)) == A
\left(\begin{array}{rrr}
0&1&2\\
3&4&5\\
6&7&0
\end{array}\right)
Truer UN# Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: A.determinant()
0r Ta # Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: view(load('amat'))
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr}
2&-1&2&2&2&-2&-1&-2&2&2&-2&1&-1&-1&2&-2&1&-1&-1&-1&1&-1&1&-1&2&1&1&-1&-1&2&-1&-1&-2&1&-2&-\
1&1&2&-2&2&2&-1&-2&1&-1&1&-2&2&2&-1\\
2&-1&1&2&2&-2&2&1&2&-2&1&1&-1&-1&-1&-2&-1...r T[ # Worksheet 'cnta - save / load' (2006-07-12 at 08:43)
sage: load('amat')
[ 2 -1 2 2 2 -2 -1 -2 2 2 -2 1 -1 -1 2 -2 1 -1 -1 -1 1 -1 1 -1 2 1 1 -1 -1 2
-1 -1 -2 1 -2 -1 1 2 -2 2 2 -1 -2 1 -1 1 -2 2 2 -1]
[ 2 -1 1 2 2 -2 2 1 2 -2 1 1 -1 -1 -1 -2 -1 2 1 2 1 -2 -1 1 1 1 1 -2 2 1
-2 1 -1 1 1 -1 -2 -2 1...r! U# Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: M = ModularSymbols(Gamma1(13),2); M
Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over
Rational Fieldr" TW # Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: dumps(M)
'x\x9c\xad\x9bw|\x14\xd5\xd7\xc6C&u\x13D\x10\x0b\xa2\x12\x1b\x065\xc8&\xa1Y\xb1\xa2.D\x01\\
xd7\x82\xe2\xba\tK6\x97\x14fv#\xa0\xae\x8aJ\x16\xb0\xa3`\x05A\x14{\xef\xbd\xf7\xde{\xef\xb\
d\xf7\xf6{\xce\xb9[\xce\xd9]\xd0\xf7\xf3y\xfd\x83\x87y\xe6;\xcf\xdc\xb9g\xe6\xce\x9d\x...r# U# Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: loads(dumps(M))
Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over
Rational Fieldr$ UV# Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: M == loads(dumps(M))
Truer% UV# Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: M == loads(dumps(M))
Truer& UT# Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: D = M.decomposition(2)
r' TP # Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: D
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0...r( UD# Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: save D
r) TX # Worksheet 'cnta - save / load' (2006-07-12 at 08:44)
sage: load('D')
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0...r* U# Worksheet 'cnta - galois' (2006-07-12 at 08:51)
sage: K = NumberField(x^7 - 3*x + 2)
Traceback (most recent call last):
...
ValueError: defining polynomial (x^7 - 3*x + 2) must be irreducibler+ UW# Worksheet 'cnta - galois' (2006-07-12 at 08:51)
sage: K = NumberField(x^7 - 3*x + 1)
r, UM# Worksheet 'cnta - galois' (2006-07-12 at 08:51)
sage: G = K.galois_group()
r- U_# Worksheet 'cnta - galois' (2006-07-12 at 08:51)
sage: G
Transitive group number 7 of degree 7r. T, # Worksheet 'cnta - galois' (2006-07-12 at 08:51)
sage: G.conjugacy_classes_representatives()
[(), (1,2), (1,2)(3,4), (1,2)(3,4)(5,6), (1,2,3), (1,2,3)(4,5), (1,2,3)(4,5)(6,7),
(1,2,3)(4,5,6), (1,2,3,4), (1,2,3,4)(5,6), (1,2,3,4)(5,6,7), (1,2,3,4,5),
(1,2,3,4,5)(6,7), (1,2,3,4,5,6), (1,2,3,4,5,6,7)]r/ UD# Worksheet 'cnta - galois' (2006-07-12 at 08:51)
sage: gg = gap(G)
r0 Uq# Worksheet 'cnta - galois' (2006-07-12 at 08:52)
sage: gg.NormalSubgroups()
[ Group(()), Alt( [ 1 .. 7 ] ), S7 ]r1 UF# Worksheet 'cnta - galois' (2006-07-12 at 08:53)
sage: G.order()
5040r2 U[# Worksheet 'cnta - galois' (2006-07-12 at 08:53)
sage: factor(G.order())
2^4 * 3^2 * 5 * 7r3 Un# Worksheet 'cnta - galois' (2006-07-12 at 08:53)
sage: G.order(), factor(G.order())
(5040, 2^4 * 3^2 * 5 * 7)r4 U# Worksheet 'cnta - galois' (2006-07-12 at 08:53)
sage: G.order(), factor(G.order()), factorial(7)
(5040, 2^4 * 3^2 * 5 * 7, 5040)r5 U# Worksheet 'cnta - galois' (2006-07-12 at 08:54)
sage: G.order(), factor(G.order()), factorial(7) # It's just S_7
(5040, 2^4 * 3^2 * 5 * 7, 5040)r6 T # Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: K = NumberField(x^2+17).compositum(NumberField(x^2-19))
Traceback (most recent call last):
K = NumberField(x^2+17).compositum(NumberField(x^2-19))
...
AttributeError: 'NumberField_quadratic' object has no attribute 'compositum'r7 Uv# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: K = NumberField(x^2+17).composite_fields(NumberField(x^2-19))
r8 U# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: V = NumberField(x^2+17).composite_fields(NumberField(x^2-19))
sage: print V
[Number Field in a with defining polynomial x^4 - 4*x^2 + 1296]r9 U# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: V = NumberField(x^2+17).composite_fields(NumberField(x^2-19))
sage: print V
sage: K = V[0]
[Number Field in a with defining polynomial x^4 - 4*x^2 + 1296]r: UM# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: G = K.galois_group()
r; U_# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: G
Transitive group number 2 of degree 4r< U# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: G.order(), factor(G.order()), factorial(7) # It's just S_7
(4, 2^2, 5040)r= Uk# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: G.order(), factor(G.order()) # It's just
(4, 2^2)r> U# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: G.order(), factor(G.order()) # It's just Z/2 x Z/2 of course
(4, 2^2)r? U# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: G.conjugacy_classes_representatives()
[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)]r@ UD# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: gg = gap(G)
rA U# Worksheet 'cnta - galois' (2006-07-12 at 08:56)
sage: gg.NormalSubgroups()
[ Group(()), Group([ (1,3)(2,4) ]), Group([ (1,2)(3,4) ]),
Group([ (1,4)(2,3) ]), E(4) = 2[x]2 ]rB U# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: V = NumberField(x^2+17).composite_fields(NumberField(x^3-19))
sage: print V
sage: K = V[0]
[Number Field in a with defining polynomial x^6 + 51*x^4 - 38*x^3 + 867*x^2 + 1938*x +
5274]rC U# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: V = NumberField(x^2+17).composite_fields(NumberField(x^3-2))
sage: print V
sage: K = V[0]
[Number Field in a with defining polynomial x^6 + 51*x^4 - 4*x^3 + 867*x^2 + 204*x + 4917]rD UM# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G = K.galois_group()
rE U_# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G
Transitive group number 3 of degree 6rF U# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.order(), factor(G.order()) # It's just Z/2 x Z/2 of course
(12, 2^2 * 3)rG Ub# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.order(), factor(G.order())
(12, 2^2 * 3)rH U# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.conjugacy_classes_representatives()
[(), (2,6)(3,5), (1,2)(3,6)(4,5), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6)]rI UG# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.identity()
()rJ UF# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.id()
[12, 4]rK Ub# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.order(), factor(G.order())
(12, 2^2 * 3)rL U# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.conjugacy_classes_representatives()
[(), (2,6)(3,5), (1,2)(3,6)(4,5), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6)]rM UD# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.order()
12rN U# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: G.conjugacy_classes_representatives()
[(), (2,6)(3,5), (1,2)(3,6)(4,5), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6)]rO UD# Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: gg = gap(G)
rP T+ # Worksheet 'cnta - galois' (2006-07-12 at 08:57)
sage: gg.NormalSubgroups()
[ Group(()), Group([ (1,4)(2,5)(3,6) ]), Group([ (1,3,5)(2,4,6) ]),
Group([ (1,3,5)(2,4,6), (1,2)(3,6)(4,5) ]),
Group([ (1,3,5)(2,4,6), (2,6)(3,5) ]),
Group([ (1,2,3,4,5,6), (1,3,5)(2,4,6) ]), D(6) = S(3)[x]2 ]rQ Uu# Worksheet 'cnta - galois' (2006-07-12 at 08:58)
sage: K.class_group()
Multiplicative Abelian Group isomorphic to C4rR U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: P. = ProjectiveSpace(3,QQ)
sage: C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
rS U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: P. = ProjectiveSpace(3,QQ)
sage: C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
rT T! # Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: P. = ProjectiveSpace(3,QQ)
sage: C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
sage: C
Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
y^2 - x*z
z^2 - y*w
-1*y*z + x*wrU Uw# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: len(C.irreducible_components()) # twisted cubic
1rV U[# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: J = C.defining_ideal()
rW U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: J = C.defining_ideal()
sage: J
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational
FieldrX UY# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: G = J.groebner_fan()
rY U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: G = J.groebner_fan()
sage: G
Groebner fan of the ideal:
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational
FieldrZ Tp # Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: G.reduced_groebner_bases()
[[-1*z^2 + y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2 - y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2
- y*w, y*z - x*w, -1*y^2 + x*z, -1*y^3 + x^2*w], [z^2 - y*w, y*z - x*w, y^3 - x^2*w,
-1*y^2 + x*z], [z^2 - y*w, y*z - x*w, y^2 - x*z], [-1*z^2 + y*w, y^2 - x*z, -1*y*z + x*w],
[-1...r[ U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: K.class_group()
Multiplicative Abelian Group isomorphic to C4
Traceback (most recent call last):
sage: K.class_group()
...
NameError: name 'K' is not definedr\ UY# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:00)
sage: G.fvector()
(1, 8, 8)r] Ur# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: f = prod(J.gens()) # \/-- newton polytope
r^ Ur# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: f = prod(J.gens()) # \/-- newton polytope
r_ Ur# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: f = prod(J.gens()) # \/-- newton polytope
r` Um# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: NP = polymake.convex_hull(f.exponents())
ra U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: NP.facets()
[(3/2, 5/2, -1, 0), (3, 1, -1, 0), (1, 0, 0, 0), (-3/2, 2, 1, 0), (3, -1, 4, 0), (-3, 1,
5, 0)]rb U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: NP = polymake.convex_hull(f.exponents()) # -- newton polytope
rc U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:01)
sage: NP.facets()
[(3/2, 5/2, -1, 0), (3, 1, -1, 0), (1, 0, 0, 0), (-3/2, 2, 1, 0), (3, -1, 4, 0), (-3, 1,
5, 0)]rd UW# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:02)
sage: f = prod(J.gens())
re U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: NP.facets()
Traceback (most recent call last):
...
NameError: name 'NP' is not definedrf T! # Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: P. = ProjectiveSpace(3,QQ)
sage: C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
sage: C
Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
y^2 - x*z
z^2 - y*w
-1*y*z + x*wrg Uw# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: len(C.irreducible_components()) # twisted cubic
1rh U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: J = C.defining_ideal()
sage: J
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational
Fieldri U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: G = J.groebner_fan()
sage: G
Groebner fan of the ideal:
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational
Fieldrj Tp # Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: G.reduced_groebner_bases()
[[-1*z^2 + y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2 - y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2
- y*w, y*z - x*w, -1*y^2 + x*z, -1*y^3 + x^2*w], [z^2 - y*w, y*z - x*w, y^3 - x^2*w,
-1*y^2 + x*z], [z^2 - y*w, y*z - x*w, y^2 - x*z], [-1*z^2 + y*w, y^2 - x*z, -1*y*z + x*w],
[-1...rk UY# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: G.fvector()
(1, 8, 8)rl UW# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: f = prod(J.gens())
rm U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: NP = polymake.convex_hull(f.exponents()) # -- newton polytope
rn U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: NP.facets()
[(3/2, 5/2, -1, 0), (3, 1, -1, 0), (1, 0, 0, 0), (-3/2, 2, 1, 0), (3, -1, 4, 0), (-3, 1,
5, 0)]ro U?# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
rp UW# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: f = prod(J.gens())
rq UW# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:04)
sage: f = prod(J.gens())
rr UQ# Worksheet '_scratch_' (2006-07-12 at 09:10)
sage: M = ModularForms(Gamma0(23))
rs U# Worksheet '_scratch_' (2006-07-12 at 09:10)
sage: M
Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(23) of weight 2 over
Rational Fieldrt U# Worksheet '_scratch_' (2006-07-12 at 09:10)
sage: M.basis()
[
q - q^3 - q^4 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6),
1 + 12/11*q + 36/11*q^2 + 48/11*q^3 + 84/11*q^4 + 72/11*q^5 + O(q^6)
]ru T # Worksheet 'cnta - modform' (2006-07-12 at 09:10)
sage: M = ModularForms(1, 12)
sage: M.basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6),
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 +
3199218815520/691*q^5 + O(q^6)
]rv U# Worksheet 'cnta - modform' (2006-07-12 at 09:10)
sage: S = M.cuspidal_submodule()
sage: S.basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]rw U# Worksheet 'cnta - modform' (2006-07-12 at 09:10)
sage: M = ModularForms(Gamma0(23),2)
sage: M.basis()
[
q - q^3 - q^4 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6),
1 + 12/11*q + 36/11*q^2 + 48/11*q^3 + 84/11*q^4 + 72/11*q^5 + O(q^6)
]rx Ud# Worksheet 'cnta - modform' (2006-07-12 at 09:11)
sage: M = ModularForms(DirichletGroup(13).0^2,2)
ry U# Worksheet 'cnta - modform' (2006-07-12 at 09:11)
sage: M = ModularForms(DirichletGroup(13).0^2,2)
sage: M.basis()
Traceback (most recent call last):
M.basis()
...
NotImplementedError: must restrict scalars down correctly.rz T6 # Worksheet 'cnta - modform' (2006-07-12 at 09:11)
sage: M = ModularForms(Gamma1(13),2)
sage: M.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6),
1 + O(q^6),
q + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6),
q^5 + O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6)
]r{ T # Worksheet 'cnta - modform' (2006-07-12 at 09:11)
sage: M = ModularForms(Gamma1(13),2,prec=10)
sage: M.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 + O(q^10),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 + O(q^10),
1 + O(q^10),
q + O(q^10),
q^2 + O(q^10),
q^3 + O(q^10),
q^4 + O(q^10),
q^5 + O(q^10),
q^6 + O(q^10),
q^7 + O(q^10),
q^8 + O(q^10),
q^9 + O(q...r| T # Worksheet 'cnta - modform' (2006-07-12 at 09:11)
sage: M = ModularForms(Gamma1(13),2,prec=20)
sage: M.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 + 10*q^16 -
3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 + 5*q^16 -
3*q^17 - 2*q^18 - 2*q^19 + O(q^20),
1 + 21060/19*q^11 - 36...r} T[ # Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: S = M.cuspidal_subspace()
sage: S.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 + 10*q^16 -
3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 + 5*q^16 -
3*q^17 - 2*q^18 - 2*q^19 + O(q^20)
]r~ Tv # Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: E = M.eisenstein_subspace()
sage: E.basis()
[
1 + 21060/19*q^11 - 36504/19*q^12 - 10270/19*q^13 - 388440/19*q^14 + 595140/19*q^15 +
89856/19*q^16 - 323856/19*q^17 - 939120/19*q^18 + 617760/19*q^19 + O(q^20),
q + 11709/19*q^11 - 20687/19*q^12 - 17570/57*q^13 - 219842/19*q^14 + 336765/19*q^15 +
50783/19*q^16 - 18...r T` # Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: M.eisenstein_series()
[
1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + 18*q^10 +
12*q^11 + 28*q^12 + q^13 + 24*q^14 + 24*q^15 + 31*q^16 + 18*q^17 + 39*q^18 + 20*q^19 +
O(q^20),
-7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^...r U# Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: t2 = M.T(2)
sage: t2
Hecke operator T_2 on Modular Forms space of dimension 13 for Congruence Subgroup
Gamma1(13) of weight 2 over Rational Fieldr U# Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: t2.matrix()
Traceback (most recent call last):
t2.matrix()
...
NotImplementedErrorr U# Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: t2 = S.T(2)
sage: t2
Hecke operator T_2 on Cuspidal subspace of dimension 2 of Modular Forms space of dimension
13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Fieldr U# Worksheet 'cnta - modform' (2006-07-12 at 09:12)
sage: t2.matrix()
Traceback (most recent call last):
t2.matrix()
...
NotImplementedErrorr Ux# Worksheet 'cnta - modform' (2006-07-12 at 09:13)
sage: M = ModularForms(Gamma0(11),6)
sage: S = M.cuspidal_subspace()
r T # Worksheet 'cnta - modform' (2006-07-12 at 09:13)
sage: M = ModularForms(Gamma0(11),6)
sage: S = M.cuspidal_subspace()
sage: S
Cuspidal subspace of dimension 4 of Modular Forms space of dimension 6 for Congruence
Subgroup Gamma0(11) of weight 6 over Rational Fieldr U# Worksheet 'cnta - modform' (2006-07-12 at 09:13)
sage: t2 = S.hecke_operator(2); t2
Hecke operator T_2 on Cuspidal subspace of dimension 4 of Modular Forms space of dimension
6 for Congruence Subgroup Gamma0(11) of weight 6 over Rational Fieldr U# Worksheet 'cnta - modform' (2006-07-12 at 09:13)
sage: t2.matrix()
[ 0 32 10 4]
[ 1 0 3 38]
[ 0 0 -2 -12]
[ 0 1 -2 -2]r Us# Worksheet 'cnta - modform' (2006-07-12 at 09:13)
sage: factor(charpoly(t2.matrix()))
(x + 4) * (x^3 - 90*x + 188)r U# Worksheet 'cnta - modform' (2006-07-12 at 09:14)
sage: S.decomposition()
Traceback (most recent call last):
S.decomposition()
...
NotImplementedErrorr Us# Worksheet 'cnta - modform' (2006-07-12 at 09:14)
sage: factor(charpoly(t2.matrix()))
(x + 4) * (x^3 - 90*x + 188)r Tx # Worksheet 'cnta - modform' (2006-07-12 at 09:14)
sage: ModularSymbols(Gamma0(11),6).decomposition(2)
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for
Gamma_0(11) of weight 6 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for
Gamma_0(11) of weight 6 with sign 0 ove...r T # Worksheet 'cnta - modform' (2006-07-12 at 09:14)
sage: ModularSymbols(Gamma0(11),6).cuspidal_submodule().decomposition(2)
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for
Gamma_0(11) of weight 6 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 10 for
Gamma_0(11) of weight 6 with sign 0 ove...r T # Worksheet 'cnta - modform' (2006-07-12 at 09:14)
sage: ModularSymbols(Gamma0(11),6, sign=1).cuspidal_submodule().decomposition(2)
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 6 for
Gamma_0(11) of weight 6 with sign 1 over Rational Field,
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 6 for
Gamma_0(11) of weight 6 with sign 1 over ...r U9# Worksheet '_scratch_' (2006-07-12 at 09:16)
sage: 2+3
5r U# Worksheet '_scratch_' (2006-07-12 at 09:16)
sage: plot(EllipticCurve('11a'))
Graphics object consisting of 1 graphics primitiver UU# Worksheet '_scratch_' (2006-07-12 at 09:16)
sage: show(plot(EllipticCurve('11a')))
r U]# Worksheet '_scratch_' (2006-07-12 at 09:16)
sage: show(plot(EllipticCurve('37a')),dpi=100)
r Uo# Worksheet '_scratch_' (2006-07-12 at 09:17)
sage: show(plot(EllipticCurve('37a'), rgbcolor=(1,0,1)),dpi=100)
r U|# Worksheet '_scratch_' (2006-07-12 at 09:17)
sage: show(plot(EllipticCurve('37a'), rgbcolor=(1,0,1), thickness=3),dpi=100)
r U9# Worksheet '_scratch_' (2006-07-12 at 09:19)
sage: 2+3
5r U|# Worksheet '_scratch_' (2006-07-12 at 09:19)
sage: show(plot(EllipticCurve('37a'), rgbcolor=(1,0,1), thickness=3),dpi=100)
r U/# Worksheet '_scratch_' (2006-07-12 at 09:20)
r U# Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: C = CremonaDatabase()
sage: C
Cremona's database of elliptic curvesr UY# Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: C.number_of_curves()
782493r UZ# Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: C.largest_conductor()
120000r TU # Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: C[11]
{'a': {'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5], 'a2': [[0, -1,
1, -7820, -263580], 0, 1]}, 'c': {'a1': ['5', '1.2692093042795534217',
'0.25384186085591068434', '1', '1.00000000000000000000'], 'a3': ['1',
'6.3460465213977671084', '0.25384186085...r Ts # Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: list(cremona_curves([11, 37, 389]))
[Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field, Elliptic
Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field, Elliptic Curve
defined by y^2 + y = x^3 - x^2 over Rational Field, Elliptic Curve defined by y^2 + y =
x^...r U~# Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.cremona_label()
'10351a1'r U# Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
sage: E = EllipticCurve([0,2,0,0,6])
sage: E.cremona_label()
'18624bj1'r U9# Worksheet 'cnta - db - cremona' (2006-07-12 at 09:20)
r U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:20)
sage: J = JonesDatabase()
sage: J
Traceback (most recent call last):
...
TypeError: _load() takes exactly 3 arguments (1 given)r U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:20)
sage: for K in J.unramified_outside([11]):
... print K
Traceback (most recent call last):
for K in J.unramified_outside([11]):
...
NameError: name 'J' is not definedr U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:20)
sage: # CAN also get direct access to underlying data, which may be faster.
sage: D = J.as_dict()
Traceback (most recent call last):
D = J.as_dict()
...
NameError: name 'J' is not definedr U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:20)
sage: D[(5,31)]
Traceback (most recent call last):
D[(5,31)]
...
NameError: name 'D' is not definedr U7# Worksheet 'cnta - db - jones' (2006-07-12 at 09:20)
r U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
sage: J = JonesDatabase()
sage: J
John Jones's table of number fields with bounded ramification and degree <= 6r U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
sage: J = JonesDatabase()
sage: J
John Jones's table of number fields with bounded ramification and degree <= 6r T( # Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
sage: for K in J.unramified_outside([11]):
... print K
Number Field in a with defining polynomial x - 1
Number Field in a with defining polynomial x^2 - x + 3
Number Field in a with defining polynomial x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1r U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
sage: # CAN also get direct access to underlying data, which may be faster.
sage: D = J.as_dict()
r U# Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
sage: D[(5,31)]
[x^2 + 155, x^4 + 2*x^3 + 4*x^2 + 3*x - 9, x^4 + x^3 + 3*x - 1, x^4 + 2*x^3 + 2*x^2 + x +
8, x^4 + 5*x^2 + 45, x^4 + 13*x^2 + 81, x^4 + x^3 - 39*x^2 - 39*x + 281]r U7# Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
r U7# Worksheet 'cnta - db - jones' (2006-07-12 at 09:21)
r U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 09:21)
sage: D = SteinWatkinsPrimeData(0)
sage: D
Stein-Watkins Prime Conductor Database p.0 Iteratorr U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 09:21)
sage: C = D.next()
sage: C
Stein-Watkins isogeny class of conductor 11r U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 09:21)
sage: list(C)
[[[0, -1, 1, 0, 0], '(1)', '1', '5'], [[0, -1, 1, -10, -20], '(5)', '1', '5'], [[0, -1, 1,
-7820, -263580], '(1)', '1', '1']]r Ux# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 09:21)
sage: D.next()
Stein-Watkins isogeny class of conductor 17r Ux# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 09:21)
sage: D.next()
Stein-Watkins isogeny class of conductor 19r U?# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 09:21)
r U# Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: S = SloaneEncyclopedia
sage: S
Sloane Online Encyclopedia of Integer Sequencesr U\# Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: v = S.find(prime_range(2,100))
r UF# Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: len(v)
30r TM # Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: v[0]
(40, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271])r TS # Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: v[1]
(8578, [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271...r U# Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: w = [len(str(numerator(abs(bernoulli(10^n))))) for n in range(5)]
sage: w
[1, 1, 83, 1779, 27691]r U{# Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
sage: S.find(w)
[(103233, [1, 1, 83, 1779, 27691, 376772, 4767554])]r U8# Worksheet 'cnta - db - sloane' (2006-07-12 at 09:21)
r U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:21)
sage: E = EllipticCurve('681b')
sage: E
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 1154*x - 15345 over Rational Fieldr T\ # Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:21)
sage: print E.mwrank()
Curve [1,1,0,-1154,-15345] :
3 points of order 2:
[-18:9:1], [-22:11:1], [310:-155:8]
****************************
* Using 2-isogeny number 1 *
****************************
Using 2-isogenous curve [0,422,0,59049,0]
-------------------------------------------...r UM# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:21)
sage: E.analytic_rank()
0r UF# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:21)
sage: E.sha_an()
9r UY# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.Lseries(1)
1.8448152061268208r Uc# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.omega()
0.81991786938969809641267899116r UP# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.tamagawa_number(3)
2r U_# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.non_surjective()
[(2, '2-torsion')]r U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.heegner_discriminants_list(10)
[-8, -20, -35, -56, -68, -83, -95, -107, -119, -143]r Ut# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: n = E.heegner_index(-20); n
[8.99999215656, 9.00000831615]r UR# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.shabound_kato()
[2, 3]r U\# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.shabound_kolyvagin()
([2, 3], 9)r UT# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: F = EllipticCurve('681c')
r U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: Eap = E.aplist(100)
sage: Fap = F.aplist(100)
sage: [(Eap[n][1] - Fap[n][1])%3 for n in range(len(Eap))]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]r Tt # Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.isogeny_class() # written by Cremona
([Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 1154*x - 15345 over Rational Field,
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 19*x - 42812 over Rational Field,
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2369*x + 20862 over Rational Field,
Elliptic ...r U{# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: f = E.modular_form()
sage: f
q + q^2 - q^3 - q^4 + 2*q^5 + O(q^6)r U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: f.parent()
Modular Forms space of dimension 78 for Congruence Subgroup Gamma0(681) of weight 2 over
Rational Fieldr UT# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E = EllipticCurve('389a')
r T # Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
sage: E.padic_regulator(5) # makes a call to MAGMA for part of the algorithm (for now)
1 + 2*5 + 2*5^2 + 4*5^3 + 3*5^4 + 4*5^5 + 3*5^6 + 5^7 + 2*5^10 + 4*5^11 + 5^12 + 4*5^13 +
4*5^14 + 5^15 + 5^17 + 4*5^18 + O(5^20)r U5# Worksheet 'cnta - ec - bsd' (2006-07-12 at 09:22)
r U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: E = EllipticCurve('37a')
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: time v = E.anlist(100000,pari_ints=True)
CPU time: 0.31 s, Wall time: 0.30 sr Uy# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: time v = E.anlist(100000)
CPU time: 1.43 s, Wall time: 1.49 sr U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: E.Lseries_at1() # directly and with proved error bound if nonzero
0r Uk# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: E.Lseries(1) # via PARI
0.00000000000000000r U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: E.Lseries_sympow(2,16) # Watkins C program -- symmetric square L-value
'2.492262044273650E+00'r Uv# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: L = E.Lseries_dokchitser() # Tim Dokchitser's GP package
r UA# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: L(1)
0r U^# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: L.derivative(1)
0.30599977383405230r U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: L(1+I) # works very well for arbitrary input
-0.15892526330137721 + 0.45791106676511545*Ir U~# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: set_verbose(-2)
sage: v = [tuple(L(1 + t/8*I)) for t in range(75)]
r UR# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: show(line(v, hue=0.8))
r U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
sage: for P in E.Lseries_zeros_in_interval(0,10,0.3): # Mike Rubinstein's
... print P[0]
5.0031700134
6.8703912161
8.0143308081
9.9330983534r U# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
time> # Christophe Doche and Sylvain Duquesne implemented SEA, which is
time> # included with SAGE
time> print E.sea(next_prime(10^50))
52
CPU time: 0.00 s, Wall time: 0.13 sr U6# Worksheet 'cnta - ec - lser' (2006-07-12 at 09:22)
r U# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: E = EllipticCurve('37a')
sage: P = plot(E, thickness=3, hue=0.1)
sage: P
Graphics object consisting of 2 graphics primitivesr UD# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: P.show()
r T? # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: P = []
sage: k = int(4)
sage: v = [(E.cremona_label(), E) for E in cremona_optimal_curves(range(1,25))]
sage: v.sort()
sage: for lbl, E in v:
... P.append(text(lbl[:-1],(0,0)) + \
... plot(E,xmin=-1, xmax=3, plot_points=80,thickness=5,rgbcolor=hue(random())))
... print lbl,
...
sage: Q = [[P[k*i+j] for j in range(k)] for i in range(len(P)/k)]
sage: Q.append(P[k*i+j+1:] + [Graphics()]*(k - len(P[k*i+j+1:])))
sage: show(graphics_array(Q), axes=False,figsize=[4,4])
11a1 14a1 15a1 17a1 19a1 20a1 21a1 24a1r U# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: E = EllipticCurve('389a')
sage: def f(p):
... return plot(E.change_ring(GF(p)),pointsize=40,hue=0.9)
...
sage: F = [[f(5), f(7)], [f(11),f(997)]]
sage: show(graphics_array(F))
r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: def sato_tate(E, N):
... return [acos(E.ap(p)/(2*sqrt(p))) for p in prime_range(N+1) if N%p != 0]
...
sage: def dist(E, N, digits=1):
... t = verbose('computing st values...')
... v = sato_tate(E, N)
... t = verbose('finished computing st values:',t)
... w = [round(x, digits) for x in v]
... w.sort()
... vals = {}
... for a in w:
... if a in vals.keys():
... vals[a] += 1
... else:
... vals[a] = 1
... g = vals.items()
... g.sort()
... verbose('finished post processing:',t)
... return g
...
sage: def graph(label, num=5000):
... d = dist(EllipticCurve(label),num,2)
... m = max(y for _, y in d)
... #m = 2.0 * pi * prime_pi(num)/num
... s = Graphics()
... for x, y in d:
... s += line([(x,0),(x,y)], hue=0.05, thickness=4)
... s += plot(lambda x: m*sin(x)^2, 0,pi, plot_points=200,
... rgbcolor=(0.3,0.1,0.1), thickness=2)
... return s
r U# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: show(graphics_array([[graph('37a'), graph('389a')],[ graph('5077a'), graph('32a')]]))
r U# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
sage: show(graphics_array([[graph('37a',100), graph('37a',1000)],
... [ graph('37a',5000), graph('37a',20000)]]))
r U6# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:22)
r U# Worksheet 'cnta - using other systems' (2006-07-12 at 09:22)
sage: n = -2007
sage: print n.factor()
sage: print factor(n)
-1 * 3^2 * 223
-1 * 3^2 * 223r Un# Worksheet 'cnta - using other systems' (2006-07-12 at 09:22)
sage: n.factor(algorithm="kash")
-1 * 3^2 * 223r Ug# Worksheet 'cnta - using other systems' (2006-07-12 at 09:22)
sage: gap(n).FactorsInt()
[ -3, 3, 223 ]r Uk# Worksheet 'cnta - using other systems' (2006-07-12 at 09:22)
sage: pari(n).factor()
[-1, 1; 3, 2; 223, 1]r Ui# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: gp(n).factor()
[-1, 1; 3, 2; 223, 1]r U`# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: maxima(n).factor()
-3^2*223r U# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: kash(n).Factorization()
[ <3, 2>, <223, 1> ], extended by:
ext1 := -1,
ext2 := Unassignr U# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: magma(n).Factorization(nvals = 2)
([ <3, 2>, <223, 1> ], -1)r Uh# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: maple(n).ifactor()
-``(3)^2*``(223)r U# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: mathematica(n).FactorInteger()
{{-1, 1}, {3, 2}, {223, 1}}r U# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
magma> n := -2007;
magma> F, s := Factorization(-2007);
magma> print F, s
[ <3, 2>, <223, 1> ]
-1r Ud# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
sage: magma('F')
[ <3, 2>, <223, 1> ]r U@# Worksheet 'cnta - using other systems' (2006-07-12 at 09:23)
r U# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: R. = PolynomialRing(QQ,2)
sage: f = y^2 + y - x^3 - 17/3*x + 2/3
r UD# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: save f
r UD# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: load f
r Uc# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: load('f')
2/3 + y + y^2 - 17/3*x - x^3r Ue# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: A = MatrixSpace(QQ,50).random_element()
r UL# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: A.save('amat')
r T[ # Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: load('amat')
[-1 -2 2 2 -1 -1 1 1 -2 -2 1 2 1 -1 1 -2 2 2 -1 -2 2 1 1 2 2 -1 1 -2 2 2
1 1 -1 1 -1 2 -1 1 -2 2 2 -2 2 -1 2 -2 -2 2 -1 2]
[-1 -2 -2 -2 2 -1 1 -2 2 1 1 -2 -1 -1 -2 -2 -2 -1 2 -1 -2 -2 -2 -2 -1 -2 -2 -1 -2 1
2 2 -2 -1 2 1 2 -1 -1 ...r U# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: M = ModularSymbols(Gamma1(13),2); M
Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over
Rational Fieldr UV# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: M == loads(dumps(M))
Truer UT# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: D = M.decomposition(2)
r TP # Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: D
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0...r UD# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: save D
r TX # Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
sage: load('D')
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 15 for
Gamma_1(13) of weight 2 with sign 0...r U8# Worksheet 'cnta - save / load' (2006-07-12 at 09:23)
r U# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: V = NumberField(x^2+17).composite_fields(NumberField(x^3-2))
sage: print V
sage: K = V[0]
[Number Field in a with defining polynomial x^6 + 51*x^4 - 4*x^3 + 867*x^2 + 204*x + 4917]r UM# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: G = K.galois_group()
r U_# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: G
Transitive group number 3 of degree 6r UD# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: G.order()
12r U# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: G.conjugacy_classes_representatives()
[(), (2,6)(3,5), (1,2)(3,6)(4,5), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6)]r UD# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: gg = gap(G)
r T+ # Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: gg.NormalSubgroups()
[ Group(()), Group([ (1,4)(2,5)(3,6) ]), Group([ (1,3,5)(2,4,6) ]),
Group([ (1,3,5)(2,4,6), (1,2)(3,6)(4,5) ]),
Group([ (1,3,5)(2,4,6), (2,6)(3,5) ]),
Group([ (1,2,3,4,5,6), (1,3,5)(2,4,6) ]), D(6) = S(3)[x]2 ]r Uu# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
sage: K.class_group()
Multiplicative Abelian Group isomorphic to C4r U3# Worksheet 'cnta - galois' (2006-07-12 at 09:23)
r T! # Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: P. = ProjectiveSpace(3,QQ)
sage: C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z])
sage: C
Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
y^2 - x*z
z^2 - y*w
-1*y*z + x*wr Uw# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: len(C.irreducible_components()) # twisted cubic
1r
U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: J = C.defining_ideal()
sage: J
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational
Fieldr U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: G = J.groebner_fan()
sage: G
Groebner fan of the ideal:
Ideal (z^2 - y*w, -1*y*z + x*w, y^2 - x*z) of Polynomial Ring in x, y, z, w over Rational
Fieldr Tp # Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: G.reduced_groebner_bases()
[[-1*z^2 + y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2 - y*w, -1*y*z + x*w, -1*y^2 + x*z], [z^2
- y*w, y*z - x*w, -1*y^2 + x*z, -1*y^3 + x^2*w], [z^2 - y*w, y*z - x*w, y^3 - x^2*w,
-1*y^2 + x*z], [z^2 - y*w, y*z - x*w, y^2 - x*z], [-1*z^2 + y*w, y^2 - x*z, -1*y*z + x*w],
[-1...r
UY# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: G.fvector()
(1, 8, 8)r UW# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: f = prod(J.gens())
r U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: NP = polymake.convex_hull(f.exponents()) # -- newton polytope
r U# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
sage: NP.facets()
[(3/2, 5/2, -1, 0), (3, 1, -1, 0), (1, 0, 0, 0), (-3/2, 2, 1, 0), (3, -1, 4, 0), (-3, 1,
5, 0)]r U?# Worksheet 'cnta - combinatorial geom' (2006-07-12 at 09:23)
r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:49)
sage: plot3dsoya(lambda x,y : y^2 - x^3 - x, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T/ # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:49)
sage: plot3dsoya(lambda x,y : y^2 - x^2 - 2 (0,0), 2.0).show()
Traceback (most recent call last):
plot3dsoya(lambda x,y : y^2 - x^2 - 2 (0,0), 2.0).show()
...
TypeError: p (=2.0000000000000000) must be a point, i.e., a 2-tuple or list of length 2r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:49)
sage: plot3dsoya(lambda x,y : y^2 - x^2 - 2, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:50)
sage: plot3dsoya(lambda x,y : y^2 - x^3 - 2, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:50)
sage: plot3dsoya(lambda x,y : sqrt(x^3 - 2), (2,2), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:51)
sage: def f(x,y):
... z = x+I*y
... return abs(sqrt(x^3 - x))
...
sage: plot3dsoya(f, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:51)
sage: def f(x,y):
... z = x+I*y
... return abs(sqrt(z^3-z))
...
sage: plot3dsoya(f, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:51)
sage: def f(x,y):
... z = x+I*y
... return abs(sqrt(z^3-z))
...
sage: plot3dsoya(f, (0,0), 1.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r U]# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:51)
sage: show(plot(EllipticCurve([-1,0])))
r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:52)
sage: def f(x,y):
... z = x+I*y
... return abs(sqrt(z^3-z))
...
sage: plot3dsoya(f, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:52)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: plot3dsoya(f, (0,0), 2.0).show()
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r U# Worksheet 'cnta - ec - plot' (2006-07-12 at 09:53)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 2.0)
r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:53)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 2.0)
sage: P.show(step=0.01)
[> ]
1%[-----------------------------------------------------------> ] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* Using OpenGL 2.0.5755 (8.24.8)
* - renderer : Generic
* - ve...r TT # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:53)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 2.0, rez=16)
sage: P.show(step=0.01)
Traceback (most recent call last):
P = plot3dsoya(f, (0,0), 2.0, rez=16)
...
TypeError: plot3dsoya() got an unexpected keyword argument 'rez'r T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:53)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 2.0, res=16)
sage: P.show(step=0.01)
[> ]
0%[---------------------------------------> ]
66%[------------------------------------------------------------>] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* ...r! T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:55)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=16)
sage: P.show(step=0.04)
[> ]
0%[-----------------------------------> ]
59%[------------------------------------------------------------>] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* ...r" T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:56)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).imag()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=16)
sage: P.show(step=0.04)
[> ]
0%[---------------------------------------> ]
65%[------------------------------------------------------------>] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* ...r# T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:56)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=16)
sage: P.show(step=0.04)
[> ]
0%[-------------------------------------> ]
62%[------------------------------------------------------------>] 100%
* Soya * Using 8 bits stencil buffer
* Soya * version 0.11.2
* ...r$ T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:57)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).imag()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=32)
sage: P.show(step=0.04)
[> ] 0%[--------->
] 16%[-------------------> ]
32%[-----------------------------> ]
49%[---------------------------------------...r% T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:57)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).imag()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=32)
sage: P.show(step=0.04)
[> ] 0%[---------->
] 17%[--------------------> ]
34%[----------------------------------> ]
57%[---------------------------------------...r& T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:58)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=32)
sage: P.show(step=0.04)
[> ] 0%[---------->
] 17%[----------------------> ]
37%[----------------------------------------> ]
67%[---------------------------------------...r' T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:58)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).real()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=32)
sage: P.show(step=0.04,pointer=True)
[> ] 0%[---------->
] 17%[---------------------> ]
36%[------------------------------------------> ]
70%[---------------------------------------...r( T # Worksheet 'cnta - ec - plot' (2006-07-12 at 09:58)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).imag()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=32)
sage: P.show(step=0.04)
[> ] 0%[---------->
] 17%[--------------------> ]
34%[------------------------------> ]
50%[---------------------------------------...r) T # Worksheet 'cnta - ec - plot' (2006-07-12 at 10:01)
sage: def f(x,y):
... z = x+I*y
... return (sqrt(z^3-z)).imag()
...
sage: P = plot3dsoya(f, (0,0), 3.0, res=32)
sage: P.show(step=0.04)
[> ] 0%[-------->
] 14%[----------------> ]
27%[--------------------------------> ]
53%[---------------------------------------...r* U# Worksheet 'cnta - db - cremona' (2006-07-12 at 10:09)
sage: C = CremonaDatabase()
sage: C
Cremona's database of elliptic curvesr+ UY# Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: C.number_of_curves()
782493r, Ub# Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: C.number_of_isogeny_classes()
524169r- UZ# Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: C.largest_conductor()
120000r. TU # Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: C[11]
{'a': {'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5], 'a2': [[0, -1,
1, -7820, -263580], 0, 1]}, 'c': {'a1': ['5', '1.2692093042795534217',
'0.25384186085591068434', '1', '1.00000000000000000000'], 'a3': ['1',
'6.3460465213977671084', '0.25384186085...r/ Ts # Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: list(cremona_curves([11, 37, 389]))
[Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field, Elliptic
Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field, Elliptic Curve
defined by y^2 + y = x^3 - x^2 over Rational Field, Elliptic Curve defined by y^2 + y =
x^...r0 U~# Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.cremona_label()
'10351a1'r1 U# Worksheet 'cnta - db - cremona' (2006-07-12 at 10:10)
sage: E = EllipticCurve([0,2,0,0,6])
sage: E.cremona_label()
'18624bj1'r2 U# Worksheet 'cnta - db - jones' (2006-07-12 at 10:10)
sage: J = JonesDatabase()
sage: J
John Jones's table of number fields with bounded ramification and degree <= 6r3 T( # Worksheet 'cnta - db - jones' (2006-07-12 at 10:10)
sage: for K in J.unramified_outside([11]):
... print K
Number Field in a with defining polynomial x - 1
Number Field in a with defining polynomial x^2 - x + 3
Number Field in a with defining polynomial x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1r4 U# Worksheet 'cnta - db - jones' (2006-07-12 at 10:10)
sage: # CAN also get direct access to underlying data, which may be faster.
sage: D = J.as_dict()
r5 U# Worksheet 'cnta - db - jones' (2006-07-12 at 10:10)
sage: D[(5,31)]
[x^2 + 155, x^4 + 2*x^3 + 4*x^2 + 3*x - 9, x^4 + x^3 + 3*x - 1, x^4 + 2*x^3 + 2*x^2 + x +
8, x^4 + 5*x^2 + 45, x^4 + 13*x^2 + 81, x^4 + x^3 - 39*x^2 - 39*x + 281]r6 U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: D = SteinWatkinsPrimeData(0)
sage: D
Stein-Watkins Prime Conductor Database p.0 Iteratorr7 U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: C = D.next()
sage: C
Stein-Watkins isogeny class of conductor 11r8 U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: list(C)
[[[0, -1, 1, 0, 0], '(1)', '1', '5'], [[0, -1, 1, -10, -20], '(5)', '1', '5'], [[0, -1, 1,
-7820, -263580], '(1)', '1', '1']]r9 U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: D = SteinWatkinsPrimeData(1)
sage: D
Stein-Watkins Prime Conductor Database p.1 Iteratorr: U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: C = D.next()
sage: C
Traceback (most recent call last):
C = D.next()
...
IOError: The Stein-Watkins data file /home/was/s/data/stein-watkins-ecdb/p.01.bz2 must be installed.r; U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: D = SteinWatkinsPrimeData(0)
sage: D
Stein-Watkins Prime Conductor Database p.0 Iteratorr< U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: C = D.next()
sage: C
Stein-Watkins isogeny class of conductor 11r= U# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: list(C)
[[[0, -1, 1, 0, 0], '(1)', '1', '5'], [[0, -1, 1, -10, -20], '(5)', '1', '5'], [[0, -1, 1,
-7820, -263580], '(1)', '1', '1']]r> Ux# Worksheet 'cnta - db - stein-watkins' (2006-07-12 at 10:10)
sage: D.next()
Stein-Watkins isogeny class of conductor 17r? U# Worksheet 'cnta - db - sloane' (2006-07-12 at 10:10)
sage: S = SloaneEncyclopedia
sage: S
Sloane Online Encyclopedia of Integer Sequencesr@ U\# Worksheet 'cnta - db - sloane' (2006-07-12 at 10:11)
sage: v = S.find(prime_range(2,100))
rA UF# Worksheet 'cnta - db - sloane' (2006-07-12 at 10:11)
sage: len(v)
30rB TM # Worksheet 'cnta - db - sloane' (2006-07-12 at 10:11)
sage: v[0]
(40, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271])rC TS # Worksheet 'cnta - db - sloane' (2006-07-12 at 10:11)
sage: v[1]
(8578, [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271...rD U# Worksheet 'cnta - db - sloane' (2006-07-12 at 10:11)
sage: w = [len(str(numerator(abs(bernoulli(10^n))))) for n in range(5)]
sage: w
[1, 1, 83, 1779, 27691]rE U{# Worksheet 'cnta - db - sloane' (2006-07-12 at 10:11)
sage: S.find(w)
[(103233, [1, 1, 83, 1779, 27691, 376772, 4767554])]rF U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:11)
sage: E = EllipticCurve('681b')
sage: E
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 1154*x - 15345 over Rational FieldrG T\ # Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:11)
sage: print E.mwrank()
Curve [1,1,0,-1154,-15345] :
3 points of order 2:
[-18:9:1], [-22:11:1], [310:-155:8]
****************************
* Using 2-isogeny number 1 *
****************************
Using 2-isogenous curve [0,422,0,59049,0]
-------------------------------------------...rH T[ # Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:11)
sage: time E.mwrank()
"\nCurve [1,1,0,-1154,-15345] :\t\n3 points of order 2:\n[-18:9:1], [-22:11:1],
[310:-155:8]\n\n****************************\n* Using 2-isogeny number 1
*\n****************************\n\nUsing 2-isogenous curve
[0,422,0,59049,0]\n---------------------------------------...rI T\ # Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:11)
sage: print E.mwrank()
Curve [1,1,0,-1154,-15345] :
3 points of order 2:
[-18:9:1], [-22:11:1], [310:-155:8]
****************************
* Using 2-isogeny number 1 *
****************************
Using 2-isogenous curve [0,422,0,59049,0]
-------------------------------------------...rJ UM# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:11)
sage: E.analytic_rank()
0rK UF# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:11)
sage: E.sha_an()
9rL UY# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.Lseries(1)
1.8448152061268208rM Uc# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.omega()
0.81991786938969809641267899116rN UP# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.tamagawa_number(3)
2rO U_# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.non_surjective()
[(2, '2-torsion')]rP U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.heegner_discriminants_list(10)
[-8, -20, -35, -56, -68, -83, -95, -107, -119, -143]rQ Ut# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: n = E.heegner_index(-20); n
[8.99999215656, 9.00000831615]rR UR# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.shabound_kato()
[2, 3]rS U\# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.shabound_kolyvagin()
([2, 3], 9)rT UT# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: F = EllipticCurve('681c')
rU U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: Eap = E.aplist(100)
sage: Fap = F.aplist(100)
sage: [(Eap[n][1] - Fap[n][1])%3 for n in range(len(Eap))]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]rV Tt # Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.isogeny_class() # written by Cremona
([Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 1154*x - 15345 over Rational Field,
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 19*x - 42812 over Rational Field,
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2369*x + 20862 over Rational Field,
Elliptic ...rW U{# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: f = E.modular_form()
sage: f
q + q^2 - q^3 - q^4 + 2*q^5 + O(q^6)rX U# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: f.parent()
Modular Forms space of dimension 78 for Congruence Subgroup Gamma0(681) of weight 2 over
Rational FieldrY UT# Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E = EllipticCurve('389a')
rZ T # Worksheet 'cnta - ec - bsd' (2006-07-12 at 10:12)
sage: E.padic_regulator(5) # makes a call to MAGMA for part of the algorithm (for now)
1 + 2*5 + 2*5^2 + 4*5^3 + 3*5^4 + 4*5^5 + 3*5^6 + 5^7 + 2*5^10 + 4*5^11 + 5^12 + 4*5^13 +
4*5^14 + 5^15 + 5^17 + 4*5^18 + O(5^20)r[ eU_Notebook__defaultsr\ }r] (Ucell_output_colorr^ U#0000EEr_ Umax_history_lengthr` MUcell_input_colorra U#0000000rb Uword_wrap_colsrc KduU_Notebook__worksheet_dirrd Usage_notebook/worksheetsre U_Notebook__filenamerf Usage_notebook/nb.sobjrg U _Notebook__default_worksheetrh j U _Notebook__next_worksheet_idri KU_default_filenamerj U5/home/was/talks/2006-07-09-cnta/sage_notebook/nb.sobjrk U_Notebook__systemrl NU_Notebook__dirrm U
sage_notebookrn U_Notebook__authro U:U_Notebook__colorrp NU_Notebook__object_dirrq Usage_notebook/objectsrr ub.