€csage.server.notebook.notebook Notebook q)q}q(U_Notebook__worksheetsq}q(U _scratch_q(csage.server.notebook.worksheet Worksheet qoq}q (U_Worksheet__filenameq U _scratch_q U_Worksheet__cellsq ]q ((csage.server.notebook.cell Cell qoq}q(U _Cell__inqU2^3qU_Cell__introspect_htmlqU!
qU_Cell__worksheetqhU_Cell__completionsq‰U_Cell__introspectq‰U_Cell__out_htmlqUU	_Cell__idqKU_before_preparseqU^os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/0")
2^3qU
_Cell__dirqU*sage_notebook/worksheets/_scratch_/cells/0qU
_Cell__outqU8Uhas_new_outputq‰U_Cell__is_htmlq ‰U_Cell__sageq!csage.interfaces.sage0
reduce_load_Sage
q")Rq#U_Cell__typeq$Uwrapq%U_Cell__timeq&‰U_Cell__interruptedq'‰ub(hoq(}q)(hUfactor(2007)q*hU!
q+hhh‰h‰hUhKhUgos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/1")
factor(2007)q,hU*sage_notebook/worksheets/_scratch_/cells/1q-hU	3^2 * 223q.h‰h ‰h!h#h$h%h&‰h'‰ub(hoq/}q0(hUview(factor(2007))q1hU!
q2hhh‰h‰hUhKhUmos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/5")
view(factor(2007))q3hU*sage_notebook/worksheets/_scratch_/cells/5q4hU63^{2} \cdot 223q5h‰h ‰h!h#h$h%h&‰h'‰ub(hoq6}q7(U	_Cell__inq8Ug%hide%html

You can embed HTML

It can even include math: \prod(1-q^n)q9U_Cell__introspect_htmlq:U!
q;U_Cell__worksheetq‰U_Cell__out_htmlq?UU	_Cell__idq@K	U_before_preparseqAUÂos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/9")
%hide%html

You can embed HTML

It can even include math: \prod(1-q^n)qBU
_Cell__dirqCU*sage_notebook/worksheets/_scratch_/cells/9qDU
_Cell__outqEUj
You can embed HTML

It can even include math: \prod(1-q^n)qFUhas_new_outputqG‰U_Cell__sageqHh")RqIU_Cell__is_htmlqJˆU_Cell__typeqKUwrapqLU_Cell__timeqM‰U_Cell__interruptedqN‰ub(hoqO}qP(h8UE = EllipticCurve('37a'); EqQh:U!
qRh‰h?Uh@KhAUvos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/8")
E = EllipticCurve('37a'); EqShCU*sage_notebook/worksheets/_scratch_/cells/8qThEU?Elliptic Curve defined by y^2 + y = x^3 - x over Rational FieldqUhG‰hHhIhJ‰hKhLhM‰hN‰ub(hoqV}qW(h8Uv%latex
Cells can be written in latex, which can refer to SAGE objects.
For example, consider $E$ given by $\sage{E}$. qXh:U!
qYh‰h?UEqZh@KhAUÑos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/11")
%latex
Cells can be written in latex, which can refer to SAGE objects.
For example, consider $E$ given by $\sage{E}$.q[hCU+sage_notebook/worksheets/_scratch_/cells/11q\hEUhG‰hHhIhJ‰hKhLhM‰hN‰ub(hoq]}q^(h8U[L = E.Lseries_dokchitser(20)
show(plot(L,-1,2, thickness=2, hue=0.6), ymin=-0.25, ymax=0.5)q_h:U!
q`h‰h?UBqah@KU_word_being_completedqbUE.Lseries_doqchAU¶os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/7")
L = E.Lseries_dokchitser(20)
show(plot(L,-1,2, thickness=2, hue=0.6), ymin=-0.25, ymax=0.5)qdhCU*sage_notebook/worksheets/_scratch_/cells/7qehEUhG‰hHhIhJ‰hKhLhM‰hN‰ub(hoqf}qg(hU)ModularSymbols(43,sign=1).decomposition()qhhU!
qihhh‰h‰hUhKhU„os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/2")
ModularSymbols(43,sign=1).decomposition()qjhU*sage_notebook/worksheets/_scratch_/cells/2qkhT­[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over Rational Field
]qlh‰h ‰h!h#h$h%h&‰h'‰ub(hoqm}qn(hUtime n=factorial(10^6)qohU!
qphhh‰h‰hUhKhU—os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/3")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
n=factorial(10^6)qqhU*sage_notebook/worksheets/_scratch_/cells/3qrhU$CPU time: 3.61 s,  Wall time: 3.63 sqsh‰h ‰h!h#h$h%h&ˆh'‰ub(hoqt}qu(hU-show(maxima('x*sin(x)*cos(x)^2').integrate())qvhU!
qwhhh‰h‰hUhKhUˆos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/_scratch_/cells/4")
show(maxima('x*sin(x)*cos(x)^2').integrate())qxhU*sage_notebook/worksheets/_scratch_/cells/4qyhUu
\frac{\sin \left(3 x\right)-3 x \cos \left(3 x\right)+9 \sin x-9 x   \cos x}{36}
qzh‰h ‰h!h#h$h%h&‰h'‰ub(hoq{}q|(hUhhh‰hUhKhU*sage_notebook/worksheets/_scratch_/cells/6q}hUh‰h$h%h'‰ubeU_Worksheet__synchroq~K"U_Worksheet__nameqU	_scratch_q€U_Worksheet__attachedq}q‚U_Worksheet__dirqƒU"sage_notebook/worksheets/_scratch_q„U_Worksheet__queueq…]q†U_Worksheet__next_idq‡KU_Worksheet__sageqˆhIU_Worksheet__comp_is_runningq‰‰U_Worksheet__variablesqŠ]q‹(UNE-sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_fieldqŒU'L-sage.lfunctions.dokchitser.DokchitserqU
f-functionqŽeU_Worksheet__notebookqhU_Worksheet__idqKU_Worksheet__next_block_idq‘KU_Worksheet__systemq’NubUcnta - using other systemsq“(hoq”}q•(U_Worksheet__filenameq–Ucnta___using_other_systemsq—U_Worksheet__cellsq˜]q™((hoqš}q›(h8U)E = magma.EllipticCurve('[1,2,3,4,5]'); Eqœh:U!
qh‰h?Uh@M°hAU–os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/cnta___using_other_systems/cells/14")
E = magma.EllipticCurve('[1,2,3,4,5]'); EqžhCUq¦h‰h?Uh@M°hAU‹os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/cnta___using_other_systems/cells/15")
E.MordellWeilGroup(Bound = 10)q§hCUq­h‰h?Uh@M°hAUtos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/cnta___using_other_systems/cells/16")
type(E)q®hCUq°hG‰hHh¡hJ‰hKhLhM‰hN‰ub(hoq±}q²(h8Umathematica('BernoulliB[100]')q³h:U!
q´h‰h?Uh@M°hAU‹os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/cnta___using_other_systems/cells/17")
mathematica('BernoulliB[100]')qµhCUq½U_Cell__worksheetq¾h”U_Cell__completionsq¿‰U_Cell__introspectqÀ‰U_Cell__out_htmlqÁUU	_Cell__idqÂM°U_before_preparseqÃU’os.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)qÄU
_Cell__dirqÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/0qÆU
_Cell__outqÇU-1 * 3^2 * 223
-1 * 3^2 * 223qÈUhas_new_outputqɉU_Cell__is_htmlqʉU_Cell__sageqËh#U_Cell__typeqÌUwrapqÍU_Cell__timeqΉU_Cell__interruptedqωub(hoqÐ}qÑ(hºUn.factor(algorithm="kash")qÒh¼U!
qÓh¾h”h¿‰hÀ‰hÁUhÂM°hÃU‚os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/1")
n.factor(algorithm="kash")qÔhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/1qÕhÇU-1 * 3^2 * 223qÖhɉhʉhËh#hÌUwrapq×hΉhωub(hoqØ}qÙ(hºUgap(n).FactorsInt()qÚh¼U!
qÛh¾h”h¿‰hÀ‰hÁUhÂM°hÃU{os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/2")
gap(n).FactorsInt()qÜhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/2qÝhÇU[ -3, 3, 223 ]qÞhɉhʉhËh#hÌhÍhΉhωub(hoqß}qà(hºUpari(n).factor()qáh¼U!
qâh¾h”h¿‰hÀ‰hÁUhÂM°hÃUxos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/3")
pari(n).factor()qãhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/3qähÇU[-1, 1; 3, 2; 223, 1]qåhɉhʉhËh#hÌhÍhΉhωub(hoqæ}qç(hºUgp(n).factor()qèh¼U!
qéh¾h”h¿‰hÀ‰hÁUhÂM
°hÃUwos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/10")
gp(n).factor()qêhÅUqðh¾h”h¿‰hÀ‰hÁUhÂM°hÃUzos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/4")
maxima(n).factor()qñhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/4qòhÇU-3^2*223qóhɉhʉhËh#hÌhÍhΉhωub(hoqô}qõ(hºUkash(n).Factorization()qöh¼U!
q÷h¾h”h¿‰hÀ‰hÁUhÂM°hÃUos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/5")
kash(n).Factorization()qøhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/5qùhÇUE[ <3, 2>, <223, 1> ], extended by:
  ext1 := -1,
  ext2 := UnassignqúhɉhʉhËh#hÌhÍhΉhωub(hoqû}qü(hºU!magma(n).Factorization(nvals = 2)qýh¼U!
qþh¾h”h¿‰hÀ‰hÁUhÂM°hÃU‰os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/6")
magma(n).Factorization(nvals = 2)qÿhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/6rhÇU([ <3, 2>, <223, 1> ], -1)rhɉhʉhËh#hÌhÍhΉhωub(hor}r(hºUmaple(n).ifactor()rh¼U!
rh¾h”h¿‰hÀ‰hÁUhÂM°hÃUzos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/7")
maple(n).ifactor()rhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/7rhÇU-``(3)^2*``(223)rhɉhʉhËh#hÌhÍhΉhωub(hor	}r
(hºUmathematica(n).FactorInteger()rh¼U!
rh¾h”h¿‰hÀ‰hÁUhÂM°hÃU†os.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/8")
mathematica(n).FactorInteger()r
hÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/8rhÇU{{-1, 1}, {3, 2}, {223, 1}}rhɉhʉhËh#hÌhÍhΉhωub(hor}r(hºU<%magma

n := -2007;
F, s := Factorization(-2007);
print F, srh¼U!
rh¾h”h¿‰hÀ‰hÁUhÂM	°hÃU¤os.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, srhÅU;sage_notebook/worksheets/cnta___using_other_systems/cells/9rhÇU[ <3, 2>, <223, 1> ]
-1rhɉhʉhËh#hÌhÍhΉhωub(hor}r(hºU
magma('F')rh¼U!
rh¾h”h¿‰hÀ‰hÁUhÂM°hÃUsos.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/11")
magma('F')rhÅU, <223, 1> ]rhɉhʉhËh#hÌhÍhΉhωub(hor}r(hºUU_Cell__introspect_htmlr U!
r!h¾h”h¿‰U_Cell__introspectr"‰hÁUhÂM°U_before_preparser#Uios.chdir("/home/was/talks/2006-07-09-cnta/sage_notebook/worksheets/cnta___using_other_systems/cells/12")
r$hÅU]r?U_Worksheet__next_idr@M°U_default_filenamerAUp/home/was/talks/2006-08-14-ccr-sage/sage_notebook/tmp/cnta___using_other_systems/cnta___using_other_systems.sobjrBU_Worksheet__comp_is_runningrC‰U_Worksheet__next_block_idrDKU_Worksheet__notebookrEhU_Worksheet__systemrFNhˆh¡U_Worksheet__idrGKubUcoveragerH(horI}rJ(h
UcoveragerKh]rL((horM}rN(hUbuzzard_tpslopes(2,1,50)rOhU!
rPhjIh‰h‰hUhM U_word_being_completedrQUbuzzard_rRhUros.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/0")
buzzard_tpslopes(2,1,50)rShU)sage_notebook/worksheets/coverage/cells/0rThT%[[], [], [], [], [], [], [], [], [], [], [], [], [3], [], [], [], [3], [], [4], [], [3], [], [5], [], [3, 7], [], [4], [], [3, 8], [], [6, 6], [], [3, 7], [], [4, 8], [], [3, 9, 9], [], [5, 8], [], [3, 7, 11], [], [4, 9, 12], [], [3, 8, 11], [], [6, 6, 13], [], [3, 7, 12, 15], [], [4, 8, 13]]rUh‰h ‰h!h#h$h%h&‰h'‰ub(horV}rW(hU@factor(genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5).conductor)rXhU!
rYhjIh‰h‰hUhM jQUgenurZhUšos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/1")
factor(genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5).conductor)r[hU)sage_notebook/worksheets/coverage/cells/1r\hU
5^4 * 2267r]h‰h ‰h!h#h$h%h&‰h'‰ub(hor^}r_(hU=ecm = ECM(); 
ecm.factor(next_prime(10^10)*next_prime(10^30))r`hU!
rahjIh‰h‰hUhM jQUECM.rbhU—os.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/2")
ecm = ECM(); 
ecm.factor(next_prime(10^10)*next_prime(10^30))rchU)sage_notebook/worksheets/coverage/cells/2rdhU.[10000000019, 1000000000000000000000000000057]reh‰h ‰h!h#h$h%h&‰h'‰ub(horf}rg(hU4factor(ModularSymbols(389,2,sign=1).T(2).charpoly())rhhU!
rihjIh‰h‰hUhM hUŽos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/3")
factor(ModularSymbols(389,2,sign=1).T(2).charpoly())rjhU)sage_notebook/worksheets/coverage/cells/3rkhTE(x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * (x^6 + 3*x^5 - 2*x^4 - 8*x^3 + 2*x^2 + 4*x - 1) * (x^20 - 3*x^19 - 29*x^18 + 91*x^17 + 338*x^16 - 1130*x^15 - 2023*x^14 + 7432*x^13 + 6558*x^12 - 28021*x^11 - 10909*x^10 + 61267*x^9 + 6954*x^8 - 74752*x^7 + 1407*x^6 + 46330*x^5 - 1087*x^4 - 12558*x^3 - 942*x^2 + 960*x + 148)rlh‰h ‰h!h#h$h%h&‰h'‰ub(horm}rn(hU)A. = QuaternionAlgebra(QQ,-1,-1)
ArohU!
rphjIh‰h‰hUhM jQU
QuaternionAlgrqhUƒos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/4")
A. = QuaternionAlgebra(QQ,-1,-1)
ArrhU)sage_notebook/worksheets/coverage/cells/4rshU@Quaternion algebra with generators (i, j, k) over Rational Fieldrth‰h ‰h!h#h$h%h&‰h'‰ub(horu}rv(hUi^2 + jrwhU!
rxhjIh‰h‰hUhM hUaos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/5")
i^2 + jryhU)sage_notebook/worksheets/coverage/cells/5rzhU-1 + jr{h‰h ‰h!h#h$h%h&‰h'‰ub(hor|}r}(hU%SupersingularModule(37).T(2).matrix()r~hU!
rhjIh‰h‰hUhM hUos.chdir("/home/was/talks/2006-08-14-ccr-sage/sage_notebook/worksheets/coverage/cells/6")
SupersingularModule(37).T(2).matrix()r€hU)sage_notebook/worksheets/coverage/cells/6rhUbSupersingular Module -- work in progress; use at own risk. (2006-08-08)
[1 1 1]
[1 0 2]
[1 2 0]r‚h‰h ‰h!h#h$h%h&‰h'‰ub(horƒ}r„(hUhjIh‰hUhM hU)sage_notebook/worksheets/coverage/cells/7r…hUh‰h$h%h'‰ubeh~KhUcoverager†h}r‡hƒU!sage_notebook/worksheets/coveragerˆh…]r‰h‡M h‰‰hhhKh‘Kh’NubuU_Notebook__historyrŠ]r‹(U9# Worksheet '_scratch_' (2006-08-15 at 00:38)
sage: 2^3
8rŒUL# Worksheet '_scratch_' (2006-08-15 at 00:38)
sage: factor(2006)
2 * 17 * 59rUJ# Worksheet '_scratch_' (2006-08-15 at 00:38)
sage: factor(2007)
3^2 * 223rŽTh# Worksheet '_scratch_' (2006-08-15 at 00:39)
sage: ModularSymbols(43).decomposition()
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0 over ...rTo# Worksheet '_scratch_' (2006-08-15 at 00:39)
sage: ModularSymbols(43,sign=1).decomposition()
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over Rational Field,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over ...rUo# Worksheet '_scratch_' (2006-08-15 at 00:39)
sage: time n=factorial(10^6)
CPU time: 3.61 s,  Wall time: 3.63 sr‘U{# Worksheet '_scratch_' (2006-08-15 at 00:40)
sage: show(factor(2007))

3^{2} \cdot 223
r’U}# Worksheet '_scratch_' (2006-08-15 at 00:40)
sage: view(factor(2007))
3^{2} \cdot 223r“Tj# Worksheet '_scratch_' (2006-08-15 at 00:40)
sage: view(maxima('sin(x^2)').integrate())
\frac{\sqrt{\pi} \left(\left(\sqrt{2} i+\sqrt{2}\right)   \mathrm{erf}\left(\frac{\left(\sqrt{2} i+\sqrt{2}\right) x}{2}  \right)+\left(\sqrt{2} i-\sqrt{2}\right) \mathrm{erf}\left(\frac{  \left(\sqrt{2} i-\sqrt{2}\right) x}{2}\right)\right)}{8}...r”Tj# Worksheet '_scratch_' (2006-08-15 at 00:40)
sage: view(maxima('cos(x^2)').integrate())
-\frac{\sqrt{\pi} \left(\left(\sqrt{2} i-\sqrt{2}\right)   \mathrm{erf}\left(\frac{\left(\sqrt{2} i+\sqrt{2}\right) x}{2}  \right)+\left(\sqrt{2} i+\sqrt{2}\right) \mathrm{erf}\left(\frac{  \left(\sqrt{2} i-\sqrt{2}\right) x}{2}\right)\right)}{8...r•U•# Worksheet '_scratch_' (2006-08-15 at 00:40)
sage: view(maxima('tan(x^2)').integrate())
\int {\tan x^2}{\;dx}r–U‰# Worksheet '_scratch_' (2006-08-15 at 00:41)
sage: view(maxima('tan(x)').integrate())
\log \sec xr—Uª# Worksheet '_scratch_' (2006-08-15 at 00:41)
sage: view(maxima('tan(1/x)').integrate())
\int {\tan \left(\frac{1}{x}\right)}{\;dx}r˜UÖ# Worksheet '_scratch_' (2006-08-15 at 00:41)
sage: view(maxima('sin(1/x)').integrate())
\sin \left(\frac{1}{x}\right) x+\int {\frac{\cos \left(\frac{1}{x}  \right)}{x}}{\;dx}r™UÙ# Worksheet '_scratch_' (2006-08-15 at 00:41)
sage: view(maxima('x*sin(x)*cos(x)^2').integrate())
\frac{\sin \left(3 x\right)-3 x \cos \left(3 x\right)+9 \sin x-9 x   \cos x}{36}ršU×# Worksheet '_scratch_' (2006-08-15 at 00:41)
sage: show(maxima('x*sin(x)*cos(x)^2').integrate())

\frac{\sin \left(3 x\right)-3 x \cos \left(3 x\right)+9 \sin x-9 x   \cos x}{36}
r›U\# Worksheet 'interfaces' (2006-08-15 at 00:42)
sage: E = magma.EllipticCurve('[1,2,3,4,5]')
rœUŒ# Worksheet 'interfaces' (2006-08-15 at 00:42)
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational FieldrU†# Worksheet 'interfaces' (2006-08-15 at 00:43)
sage: E.MordellWeilGroup()
Abelian Group isomorphic to Z
Defined on 1 generator (free)ržTe# Worksheet 'interfaces' (2006-08-15 at 00:43)
sage: E.MordellWeilGroup(bound = 10)
...
TypeError: Error evaluation Magma code.
IN:_sage_[18] := MordellWeilGroup(_sage_[17] : bound:=10);
OUT:
>> _sage_[18] := MordellWeilGroup(_sage_[17] : bound:=10);
                                 ^
Runtime error in 'MordellWeilGroup': Parameter 'bound' is not d...rŸU# Worksheet 'interfaces' (2006-08-15 at 00:43)
sage: E.MordellWeilGroup(Bound = 10)
Abelian Group isomorphic to Z
Defined on 1 generator (free)r Ui# Worksheet 'interfaces' (2006-08-15 at 00:43)
sage: type(E)
r¡UD# Worksheet 'interfaces' (2006-08-15 at 00:43)
sage: parent(E)
Magmar¢UI# Worksheet 'interfaces' (2006-08-15 at 00:43)
sage: E.name()
'_sage_[2]'r£T]# Worksheet 'coverage' (2006-08-15 at 01:05)
sage: buzzard_tpslopes(2,1,50)
[[], [], [], [], [], [], [], [], [], [], [], [], [3], [], [], [], [3], [], [4], [], [3], [], [5], [], [3, 7], [], [4], [], [3, 8], [], [6, 6], [], [3, 7], [], [4, 8], [], [3, 9, 9], [], [5, 8], [], [3, 7, 11], [], [4, 9, 12], [], [3, 8, 11], [], [6, 6, 13], [], [3, 7, 1...r¤U~# Worksheet 'coverage' (2006-08-15 at 01:06)
sage: factor(genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5).conductor)
5^4 * 2267r¥U¼# Worksheet 'coverage' (2006-08-15 at 01:07)
sage: ECM.factor(2007)
...
TypeError: unbound method factor() must be called with ECM instance as first argument (got Integer instance instead)r¦U€# Worksheet 'coverage' (2006-08-15 at 01:07)
sage: ECM().factor(next_prime(10^10)*next_prime(10^11))
[10000000019, 100000000003]r§U“# Worksheet 'coverage' (2006-08-15 at 01:07)
sage: ECM().factor(next_prime(10^10)*next_prime(10^30))
[10000000019, 1000000000000000000000000000057]r¨Uà# Worksheet 'coverage' (2006-08-15 at 01:07)
sage: ecm = ECM(); print ecm
sage: ecm.factor(next_prime(10^10)*next_prime(10^30))

[10000000019, 1000000000000000000000000000057]r©U¥# Worksheet 'coverage' (2006-08-15 at 01:07)
sage: ecm = ECM(); 
sage: ecm.factor(next_prime(10^10)*next_prime(10^30))
[10000000019, 1000000000000000000000000000057]rªTy# Worksheet 'coverage' (2006-08-15 at 01:08)
sage: factor(ModularSymbols(389,2,sign=1).T(2).charpoly())
(x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * (x^6 + 3*x^5 - 2*x^4 - 8*x^3 + 2*x^2 + 4*x - 1) * (x^20 - 3*x^19 - 29*x^18 + 91*x^17 + 338*x^16 - 1130*x^15 - 2023*x^14 + 7432*x^13 + 6558*x^12 - 28021*x^11 - 10909*x^10 + 61267*x^9 + 6954*x^8 - 74752*x^7 + 1407*x^6 + 46...r«U[# Worksheet 'coverage' (2006-08-15 at 01:09)
sage: A. = QuaternionAlgebra(QQ,-1,-1)
r¬UA# Worksheet 'coverage' (2006-08-15 at 01:09)
sage: i^2 + j
-1 + jr­U£# Worksheet 'coverage' (2006-08-15 at 01:09)
sage: A. = QuaternionAlgebra(QQ,-1,-1)
sage: A
Quaternion algebra with generators (i, j, k) over Rational Fieldr®Uî# Worksheet 'coverage' (2006-08-15 at 01:09)
sage: SupersingularModule(37).T(2)
Supersingular Module -- work in progress; use at own risk. (2006-08-08)
Hecke operator T_2 on Module of supersingular points on X_0(1)/F_37 over Integer Ringr¯U»# Worksheet 'coverage' (2006-08-15 at 01:09)
sage: SupersingularModule(37).T(2).matrix()
Supersingular Module -- work in progress; use at own risk. (2006-08-08)
[1 1 1]
[1 0 2]
[1 2 0]r°U# Worksheet '_scratch_' (2006-08-15 at 08:56)
sage: def f(x):
...    return dimension_cusp_forms(Gamma0(int(x)))
...
sage: show(plot(f,1,100))
r±U# Worksheet '_scratch_' (2006-08-15 at 08:57)
sage: E = EllipticCurve('37a'); E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Fieldr²U‹# Worksheet '_scratch_' (2006-08-15 at 08:58)
sage: L = E.Lseries_dokchitser(5)
sage: show(plot(L,-1,1, plot_points=50, plot_division=50))
r³Ui# Worksheet '_scratch_' (2006-08-15 at 08:59)
sage: L = E.Lseries_dokchitser(5)
sage: show(plot(L,-1,2))
r´Uj# Worksheet '_scratch_' (2006-08-15 at 08:59)
sage: L = E.Lseries_dokchitser(20)
sage: show(plot(L,-1,2))
rµU€# Worksheet '_scratch_' (2006-08-15 at 09:00)
sage: L = E.Lseries_dokchitser(20)
sage: show(plot(L,-1,2, thickness=2, hue=0.6))
r¶U–# Worksheet '_scratch_' (2006-08-15 at 09:00)
sage: L = E.Lseries_dokchitser(20)
sage: show(plot(L,-1,2, thickness=2, hue=0.6, ymin=-0.25, ymax=0.5))
r·U–# Worksheet '_scratch_' (2006-08-15 at 09:01)
sage: L = E.Lseries_dokchitser(20)
sage: show(plot(L,-1,2, thickness=2, hue=0.6), ymin=-0.25, ymax=0.5)
r¸U\# Worksheet 'interfaces' (2006-08-15 at 09:01)
sage: E = magma.EllipticCurve('[1,2,3,4,5]')
r¹T# Worksheet '_scratch_' (2006-08-15 at 09:03)
html> 
You can embed HTML

html> It can even include math: \prod(1-q^n)

You can embed HTML

It can even include math: \prod(1-q^n)rºUÊ# Worksheet '_scratch_' (2006-08-15 at 09:03)
html%hide> 
You can embed HTML

html%hide> It can even include math: \prod(1-q^n)
...
NameError: name 'hide' is not definedr»T# Worksheet '_scratch_' (2006-08-15 at 09:03)
hide%html> 
You can embed HTML

hide%html> It can even include math: \prod(1-q^n)

You can embed HTML

It can even include math: \prod(1-q^n)r¼U«# Worksheet '_scratch_' (2006-08-15 at 09:04)
latex> Cells can be written in latex, which can refer to SAGE objects.
latex> For example, consider $E$ given by $\sage{E}$.
r½UA# Worksheet 'interfaces' (2006-08-15 at 09:04)
sage: gap('2^3')
8r¾U´# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: E = magma.EllipticCurve('[1,2,3,4,5]'); E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Fieldr¿UF# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: singular('2^3')
8rÀU@# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: gp('2^3')
8rÁU×# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: gap('Factorization(2007)')
...
TypeError: Gap produced error output
Function: number of arguments must be 2 (not 1)

   executing $sage1:=Factorization(2007);;rÂUÉ# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: gap('Factorize(2007)')
...
TypeError: Gap produced error output
Variable: 'Factorize' must have a value


   executing $sage2:=Factorize(2007);;rÃUÀ# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: gap('Factor(2007)')
...
TypeError: Gap produced error output
Variable: 'Factor' must have a value


   executing $sage3:=Factor(2007);;rÄUA# Worksheet 'interfaces' (2006-08-15 at 09:05)
sage: gap('2^3')
8rÅUÄ# Worksheet 'cnta - using other systems' (2006-08-15 at 09:08)
sage: E = magma.EllipticCurve('[1,2,3,4,5]'); E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational FieldrÆU # Worksheet 'cnta - using other systems' (2006-08-15 at 09:08)
sage: E.MordellWeilGroup(Bound = 10)
Abelian Group isomorphic to Z
Defined on 1 generator (free)rÇUy# Worksheet 'cnta - using other systems' (2006-08-15 at 09:08)
sage: type(E)
rÈU¾# Worksheet 'cnta - using other systems' (2006-08-15 at 09:09)
sage: mathematica('BernoulliB[100]')
-94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330rÉeU_Notebook__defaultsrÊ}rË(Ucell_output_colorrÌU#0000EErÍUmax_history_lengthrÎMôUcell_input_colorrÏU#0000000rÐUword_wrap_colsrÑKduU_Notebook__worksheet_dirrÒUsage_notebook/worksheetsrÓU_Notebook__filenamerÔUsage_notebook/nb.sobjrÕU_Notebook__default_worksheetrÖhU_Notebook__next_worksheet_idr×KU_default_filenamerØU9/home/was/talks/2006-08-14-ccr-sage/sage_notebook/nb.sobjrÙU_Notebook__systemrÚNU_Notebook__show_debugrÛ‰U_Notebook__dirrÜU
sage_notebookrÝU_Notebook__authrÞU:U_Notebook__colorrßNU_Notebook__object_dirràUsage_notebook/objectsráub.