# Isogenies of small prime degree.¶

Functions for the computation of isogenies of small primes degree. First: $$l$$ = 2, 3, 5, 7, or 13, where the modular curve $$X_0(l)$$ has genus 0. Second: $$l$$ = 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71, where $$X_0^+(l)$$ has genus 0 and $$X_0(l)$$ is elliptic or hyperelliptic. Also: $$l$$ = 11, 17, 19, 37, 43, 67 or 163 over $$\QQ$$ (the sporadic cases with only finitely many $$j$$-invariants each). All the above only require factorization of a polynomial of degree $$l+1$$. Finally, a generic function which works for arbitrary odd primes $$l$$ (including the characteristic), but requires factorization of the $$l$$-division polynomial, of degree $$(l^2-1)/2$$.

AUTHORS:

• John Cremona and Jenny Cooley: 2009-07..11: the genus 0 cases the sporadic cases over $$\QQ$$.
• Kimi Tsukazaki and John Cremona: 2013-07: The 10 (hyper)-elliptic cases and the generic algorithm. See [KT2013].

REFERENCES:

 [CW2005] Cremona and M. Watkins. Computing isogenies of elliptic curves. preprint, 2005.
 [KT2013] (1, 2, 3, 4) K. Tsukazaki, Explicit Isogenies of Elliptic Curves, PhD thesis, University of Warwick, 2013.
sage.schemes.elliptic_curves.isogeny_small_degree.Fricke_module(l)

Fricke module for l =2,3,5,7,13.

For these primes (and these only) the modular curve $$X_0(l)$$ has genus zero, and its field is generated by a single modular function called the Fricke module (or Hauptmodul), $$t$$. There is a classical choice of such a generator $$t$$ in each case, and the $$j$$-function is a rational function of $$t$$ of degree $$l+1$$ of the form $$P(t)/t$$ where $$P$$ is a polynomial of degree $$l+1$$. Up to scaling, $$t$$ is determined by the condition that the ramification points above $$j=\infty$$ are $$t=0$$ (with ramification degree $$1$$) and $$t=\infty$$ (with degree $$l$$). The ramification above $$j=0$$ and $$j=1728$$ may be seen in the factorizations of $$j(t)$$ and $$k(t)$$ where $$k=j-1728$$.

OUTPUT:

The rational function $$P(t)/t$$.

TESTS:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Fricke_module
sage: Fricke_module(2)
(t^3 + 48*t^2 + 768*t + 4096)/t
sage: Fricke_module(3)
(t^4 + 36*t^3 + 270*t^2 + 756*t + 729)/t
sage: Fricke_module(5)
(t^6 + 30*t^5 + 315*t^4 + 1300*t^3 + 1575*t^2 + 750*t + 125)/t
sage: Fricke_module(7)
(t^8 + 28*t^7 + 322*t^6 + 1904*t^5 + 5915*t^4 + 8624*t^3 + 4018*t^2 + 748*t + 49)/t
sage: Fricke_module(13)
(t^14 + 26*t^13 + 325*t^12 + 2548*t^11 + 13832*t^10 + 54340*t^9 + 157118*t^8 + 333580*t^7 + 509366*t^6 + 534820*t^5 + 354536*t^4 + 124852*t^3 + 15145*t^2 + 746*t + 13)/t

sage.schemes.elliptic_curves.isogeny_small_degree.Fricke_polynomial(l)

Fricke polynomial for l =2,3,5,7,13.

For these primes (and these only) the modular curve $$X_0(l)$$ has genus zero, and its field is generated by a single modular function called the Fricke module (or Hauptmodul), $$t$$. There is a classical choice of such a generator $$t$$ in each case, and the $$j$$-function is a rational function of $$t$$ of degree $$l+1$$ of the form $$P(t)/t$$ where $$P$$ is a polynomial of degree $$l+1$$. Up to scaling, $$t$$ is determined by the condition that the ramification points above $$j=\infty$$ are $$t=0$$ (with ramification degree $$1$$) and $$t=\infty$$ (with degree $$l$$). The ramification above $$j=0$$ and $$j=1728$$ may be seen in the factorizations of $$j(t)$$ and $$k(t)$$ where $$k=j-1728$$.

OUTPUT:

The polynomial $$P(t)$$ as an element of $$\ZZ[t]$$.

TESTS:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Fricke_polynomial
sage: Fricke_polynomial(2)
t^3 + 48*t^2 + 768*t + 4096
sage: Fricke_polynomial(3)
t^4 + 36*t^3 + 270*t^2 + 756*t + 729
sage: Fricke_polynomial(5)
t^6 + 30*t^5 + 315*t^4 + 1300*t^3 + 1575*t^2 + 750*t + 125
sage: Fricke_polynomial(7)
t^8 + 28*t^7 + 322*t^6 + 1904*t^5 + 5915*t^4 + 8624*t^3 + 4018*t^2 + 748*t + 49
sage: Fricke_polynomial(13)
t^14 + 26*t^13 + 325*t^12 + 2548*t^11 + 13832*t^10 + 54340*t^9 + 157118*t^8 + 333580*t^7 + 509366*t^6 + 534820*t^5 + 354536*t^4 + 124852*t^3 + 15145*t^2 + 746*t + 13

sage.schemes.elliptic_curves.isogeny_small_degree.Psi(l, use_stored=True)

Generic kernel polynomial for genus zero primes.

For each of the primes $$l$$ for which $$X_0(l)$$ has genus zero (namely $$l=2,3,5,7,13$$), we may define an elliptic curve $$E_t$$ over $$\QQ(t)$$, with coefficients in $$\ZZ[t]$$, which has good reduction except at $$t=0$$ and $$t=\infty$$ (which lie above $$j=\infty$$) and at certain other values of $$t$$ above $$j=0$$ when $$l=3$$ (one value) or $$l\equiv1\pmod{3}$$ (two values) and above $$j=1728$$ when $$l=2$$ (one value) or $$l\equiv1 \pmod{4}$$ (two values). (These exceptional values correspond to endomorphisms of $$E_t$$ of degree $$l$$.) The $$l$$-division polynomial of $$E_t$$ has a unique factor of degree $$(l-1)/2$$ (or 1 when $$l=2$$), with coefficients in $$\ZZ[t]$$, which we call the Generic Kernel Polynomial for $$l$$. These are used, by specialising $$t$$, in the function isogenies_prime_degree_genus_0(), which also has to take into account the twisting factor between $$E_t$$ for a specific value of $$t$$ and the short Weierstrass form of an elliptic curve with $$j$$-invariant $$j(t)$$. This enables the computation of the kernel polynomials of isogenies without having to compute and factor division polynomials.

All of this data is quickly computed from the Fricke modules, except that for $$l=13$$ the factorization of the Generic Division Polynomial takes a long time, so the value have been precomputed and cached; by default the cached values are used, but the code here will recompute them when use_stored is False, as in the doctests.

INPUT:

• l – either 2, 3, 5, 7, or 13.
• use_stored (boolean, default True) – If True, use precomputed values, otherwise compute them on the fly.

TESTS:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Fricke_module, Psi
sage: assert Psi(2, use_stored=True) == Psi(2, use_stored=False)
sage: assert Psi(3, use_stored=True) == Psi(3, use_stored=False)
sage: assert Psi(5, use_stored=True) == Psi(5, use_stored=False)
sage: assert Psi(7, use_stored=True) == Psi(7, use_stored=False)
sage: assert Psi(13, use_stored=True) == Psi(13, use_stored=False) # not tested (very long time)

sage.schemes.elliptic_curves.isogeny_small_degree.Psi2(l)

Returns the generic kernel polynomial for hyperelliptic $$l$$-isogenies.

INPUT:

• l – either 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.

OUTPUT:

The generic $$l$$-kernel polynomial.

TESTS:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Psi2
sage: Psi2(11)
x^5 - 55*x^4*u + 994*x^3*u^2 - 8774*x^2*u^3 + 41453*x*u^4 - 928945/11*u^5 + 33*x^4 + 276*x^3*u - 7794*x^2*u^2 + 4452*x*u^3 + 1319331/11*u^4 + 216*x^3*v - 4536*x^2*u*v + 31752*x*u^2*v - 842616/11*u^3*v + 162*x^3 + 38718*x^2*u - 610578*x*u^2 + 33434694/11*u^3 - 4536*x^2*v + 73872*x*u*v - 2745576/11*u^2*v - 16470*x^2 + 580068*x*u - 67821354/11*u^2 - 185976*x*v + 14143896/11*u*v + 7533*x - 20437029/11*u - 12389112/11*v + 19964151/11

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_13_0(E)

Returns list of all 13-isogenies from E when the j-invariant is 0.

OUTPUT:

(list) 13-isogenies with codomain E. In general these are normalised; but if $$-3$$ is a square then there are two endomorphisms of degree $$13$$, for which the codomain is the same as the domain.

Note

This implementation requires that the characteristic is not 2, 3 or 13.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(13).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_13_0


Endomorphisms of degree 13 will exist when -3 is a square:

sage: K.<r> = QuadraticField(-3)
sage: E = EllipticCurve(K, [0, r]); E
Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3
sage: isogenies_13_0(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3]
sage: isogenies_13_0(E)[0].rational_maps()
(((7/338*r + 23/338)*x^13 + (-164/13*r - 420/13)*x^10 + (720/13*r + 3168/13)*x^7 + (3840/13*r - 576/13)*x^4 + (4608/13*r + 2304/13)*x)/(x^12 + (4*r + 36)*x^9 + (1080/13*r + 3816/13)*x^6 + (2112/13*r - 5184/13)*x^3 + (-17280/169*r - 1152/169)), ((18/2197*r + 35/2197)*x^18*y + (23142/2197*r + 35478/2197)*x^15*y + (-1127520/2197*r - 1559664/2197)*x^12*y + (-87744/2197*r + 5992704/2197)*x^9*y + (-6625152/2197*r - 9085824/2197)*x^6*y + (-28919808/2197*r - 2239488/2197)*x^3*y + (-1990656/2197*r - 3870720/2197)*y)/(x^18 + (6*r + 54)*x^15 + (3024/13*r + 11808/13)*x^12 + (31296/13*r + 51840/13)*x^9 + (487296/169*r - 2070144/169)*x^6 + (-940032/169*r + 248832/169)*x^3 + (1990656/2197*r + 3870720/2197)))


An example of endomorphisms over a finite field:

sage: K = GF(19^2,'a')
sage: E = EllipticCurve(j=K(0)); E
Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2
sage: isogenies_13_0(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2]
sage: isogenies_13_0(E)[0].rational_maps()
((6*x^13 - 6*x^10 - 3*x^7 + 6*x^4 + x)/(x^12 - 5*x^9 - 9*x^6 - 7*x^3 + 5), (-8*x^18*y - 9*x^15*y + 9*x^12*y - 5*x^9*y + 5*x^6*y - 7*x^3*y + 7*y)/(x^18 + 2*x^15 + 3*x^12 - x^9 + 8*x^6 - 9*x^3 + 7))


A previous implementation did not work in some characteristics:

sage: K = GF(29)
sage: E = EllipticCurve(j=K(0))
sage: isogenies_13_0(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 29 to Elliptic Curve defined by y^2 = x^3 + 26*x + 12 over Finite Field of size 29, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 29 to Elliptic Curve defined by y^2 = x^3 + 16*x + 28 over Finite Field of size 29]

sage: K = GF(101)
sage: E = EllipticCurve(j=K(0)); E.ainvs()
(0, 0, 0, 0, 1)
sage: [phi.codomain().ainvs() for phi in isogenies_13_0(E)]
[(0, 0, 0, 64, 36), (0, 0, 0, 42, 66)]

sage: x = polygen(QQ)
sage: f = x^12 + 78624*x^9 - 130308048*x^6 + 2270840832*x^3 - 54500179968
sage: K.<a> = NumberField(f)
sage: E = EllipticCurve(j=K(0)); E.ainvs()
(0, 0, 0, 0, 1)
sage: [phi.codomain().ainvs() for phi in isogenies_13_0(E)]
[(0, 0, 0, -739946459/23857162861049856*a^11 - 2591641747/1062017577504*a^8 + 16583647773233/4248070310016*a^5 - 14310911337/378211388*a^2, 26146225/4248070310016*a^9 + 7327668845/14750244132*a^6 + 174618431365/756422776*a^3 - 378332499709/94552847), (0, 0, 0, 3501275/5964290715262464*a^11 + 24721025/531008788752*a^8 - 47974903745/1062017577504*a^5 - 6773483100/94552847*a^2, 6699581/4248070310016*a^9 + 1826193509/14750244132*a^6 - 182763866047/756422776*a^3 - 321460597/94552847)]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_13_1728(E)

Returns list of all 13-isogenies from E when the j-invariant is 1728.

OUTPUT:

(list) 13-isogenies with codomain E. In general these are normalised; but if $$-1$$ is a square then there are two endomorphisms of degree $$13$$, for which the codomain is the same as the domain; and over $$\QQ$$, the codomain is a minimal model.

Note

This implementation requires that the characteristic is not 2, 3 or 13.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(13).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_13_1728

sage: E = EllipticCurve([0,0,0,i,0]); E.ainvs()
(0, 0, 0, i, 0)
sage: isogenies_13_1728(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1]

sage: K = GF(83)
sage: E = EllipticCurve(K, [0,0,0,5,0]); E.ainvs()
(0, 0, 0, 5, 0)
sage: isogenies_13_1728(E)
[]
sage: K = GF(89)
sage: E = EllipticCurve(K, [0,0,0,5,0]); E.ainvs()
(0, 0, 0, 5, 0)
sage: isogenies_13_1728(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89 to Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89 to Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89]

sage: K = GF(23)
sage: E = EllipticCurve(K, [1,0])
sage: isogenies_13_1728(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 23 to Elliptic Curve defined by y^2 = x^3 + 16 over Finite Field of size 23, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 23 to Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 23]

sage: x = polygen(QQ)
sage: f = x^12 + 1092*x^10 - 432432*x^8 + 6641024*x^6 - 282896640*x^4 - 149879808*x^2 - 349360128
sage: K.<a> = NumberField(f)
sage: E = EllipticCurve(K, [1,0])
sage: [phi.codomain().ainvs() for phi in isogenies_13_1728(E)]
[(0,
0,
0,
11090413835/20943727039698624*a^10 + 32280103535965/55849938772529664*a^8 - 355655987835845/1551387188125824*a^6 + 19216954517530195/5235931759924656*a^4 - 1079766118721735/5936430566808*a^2 + 156413528482727/8080141604822,
214217013065/82065216155553792*a^11 + 1217882637605/427423000810176*a^9 - 214645003230565/189965778137856*a^7 + 22973355421236025/1282269002430528*a^5 - 2059145797340695/2544184528632*a^3 - 23198483147321/989405094468*a),
(0,
0,
0,
11090413835/20943727039698624*a^10 + 32280103535965/55849938772529664*a^8 - 355655987835845/1551387188125824*a^6 + 19216954517530195/5235931759924656*a^4 - 1079766118721735/5936430566808*a^2 + 156413528482727/8080141604822,
-214217013065/82065216155553792*a^11 - 1217882637605/427423000810176*a^9 + 214645003230565/189965778137856*a^7 - 22973355421236025/1282269002430528*a^5 + 2059145797340695/2544184528632*a^3 + 23198483147321/989405094468*a)]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_2(E)

Returns a list of all 2-isogenies with domain E.

INPUT:

• E – an elliptic curve.

OUTPUT:

(list) 2-isogenies with domain E. In general these are normalised, but over $$\QQ$$ the codomain is a minimal model.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_2
sage: E = EllipticCurve('14a1'); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field
sage: [phi.codomain().ainvs() for phi in isogenies_2(E)]
[(1, 0, 1, -36, -70)]

sage: E = 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 Field
sage: [phi.codomain().ainvs() for phi in isogenies_2(E)]
[]
sage: E = EllipticCurve(QQbar, [9,8]); E
Elliptic Curve defined by y^2 = x^3 + 9*x + 8 over Algebraic Field
sage: isogenies_2(E) # not implemented

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_3(E)

Returns a list of all 3-isogenies with domain E.

INPUT:

• E – an elliptic curve.

OUTPUT:

(list) 3-isogenies with domain E. In general these are normalised, but over $$\QQ$$ the codomain is a minimal model.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_3
sage: E = EllipticCurve(GF(17), [1,1])
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[(0, 0, 0, 9, 7), (0, 0, 0, 0, 1)]

sage: E = EllipticCurve(GF(17^2,'a'), [1,1])
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[(0, 0, 0, 9, 7), (0, 0, 0, 0, 1), (0, 0, 0, 5*a + 1, a + 13), (0, 0, 0, 12*a + 6, 16*a + 14)]

sage: E = EllipticCurve('19a1')
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[(0, 1, 1, 1, 0), (0, 1, 1, -769, -8470)]

sage: E = EllipticCurve([1,1])
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_5_0(E)

Returns a list of all the 5-isogenies with domain E when the j-invariant is 0.

OUTPUT:

(list) 5-isogenies with codomain E. In general these are normalised, but over $$\QQ$$ the codomain is a minimal model.

Note

This implementation requires that the characteristic is not 2, 3 or 5.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(5).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_5_0
sage: E = EllipticCurve([0,12])
sage: isogenies_5_0(E)
[]

sage: E = EllipticCurve(GF(13^2,'a'),[0,-3])
sage: isogenies_5_0(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (4*a+6)*x + (2*a+10) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (12*a+5)*x + (2*a+10) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (10*a+2)*x + (2*a+10) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (3*a+12)*x + (11*a+12) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (a+4)*x + (11*a+12) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (9*a+10)*x + (11*a+12) over Finite Field in a of size 13^2]

sage: K.<a> = NumberField(x**6-320*x**3-320)
sage: E = EllipticCurve(K,[0,0,1,0,0])
sage: isogenies_5_0(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 = x^3 + (a^5-400*a^2)*x + (280*a^3-3120) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320,
Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 = x^3 + (23/2*a^5-3700*a^2)*x + (-280*a^3+86480) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_5_1728(E)

Returns a list of 5-isogenies with domain E when the j-invariant is 1728.

OUTPUT:

(list) 5-isogenies with codomain E. In general these are normalised; but if $$-1$$ is a square then there are two endomorphisms of degree $$5$$, for which the codomain is the same as the domain curve; and over $$\QQ$$, the codomain is a minimal model.

Note

This implementation requires that the characteristic is not 2, 3 or 5.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(5).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_5_1728
sage: E = EllipticCurve([7,0])
sage: isogenies_5_1728(E)
[]

sage: E = EllipticCurve(GF(13),[11,0])
sage: isogenies_5_1728(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13,
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13]


An example of endomorphisms of degree 5:

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve(K,[0,0,0,1,0])
sage: isogenies_5_1728(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1,
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1]
sage: _[0].rational_maps()
(((4/25*i + 3/25)*x^5 + (4/5*i - 2/5)*x^3 - x)/(x^4 + (-4/5*i + 2/5)*x^2 + (-4/25*i - 3/25)),
((11/125*i + 2/125)*x^6*y + (-23/125*i + 64/125)*x^4*y + (141/125*i + 162/125)*x^2*y + (3/25*i - 4/25)*y)/(x^6 + (-6/5*i + 3/5)*x^4 + (-12/25*i - 9/25)*x^2 + (2/125*i - 11/125)))


An example of 5-isogenies over a number field:

sage: K.<a> = NumberField(x**4+20*x**2-80)
sage: K(5).is_square() #necessary but not sufficient!
True
sage: E = EllipticCurve(K,[0,0,0,1,0])
sage: isogenies_5_1728(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in a with defining polynomial x^4 + 20*x^2 - 80 to Elliptic Curve defined by y^2 = x^3 + (-20*a^2-39)*x + (35*a^3+112*a) over Number Field in a with defining polynomial x^4 + 20*x^2 - 80,
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in a with defining polynomial x^4 + 20*x^2 - 80 to Elliptic Curve defined by y^2 = x^3 + (-20*a^2-39)*x + (-35*a^3-112*a) over Number Field in a with defining polynomial x^4 + 20*x^2 - 80]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_7_0(E)

Returns list of all 7-isogenies from E when the j-invariant is 0.

OUTPUT:

(list) 7-isogenies with codomain E. In general these are normalised; but if $$-3$$ is a square then there are two endomorphisms of degree $$7$$, for which the codomain is the same as the domain; and over $$\QQ$$, the codomain is a minimal model.

Note

This implementation requires that the characteristic is not 2, 3 or 7.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(7).

EXAMPLES:

First some examples of endomorphisms:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_0
sage: E = EllipticCurve(K, [0,1])
sage: isogenies_7_0(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3,
Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3]

sage: E = EllipticCurve(GF(13^2,'a'),[0,-3])
sage: isogenies_7_0(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2]


Now some examples of 7-isogenies which are not endomorphisms:

sage: K = GF(101)
sage: E = EllipticCurve(K, [0,1])
sage: isogenies_7_0(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 101 to Elliptic Curve defined by y^2 = x^3 + 55*x + 100 over Finite Field of size 101, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 101 to Elliptic Curve defined by y^2 = x^3 + 83*x + 26 over Finite Field of size 101]


Examples over a number field:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_0
sage: isogenies_7_0(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3,
Isogeny of degree 7 from Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3]

sage: K.<a> = NumberField(x^6 + 1512*x^3 - 21168)
sage: E = EllipticCurve(K, [0,1])
sage: isogs = isogenies_7_0(E)
sage: [phi.codomain().a_invariants() for phi in isogs]
[(0, 0, 0, -5/294*a^5 - 300/7*a^2, -55/2*a^3 - 1133),
(0, 0, 0, -295/1176*a^5 - 5385/14*a^2, 55/2*a^3 + 40447)]
sage: [phi.codomain().j_invariant() for phi in isogs]
[158428486656000/7*a^3 - 313976217600000,
-158428486656000/7*a^3 - 34534529335296000]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_7_1728(E)

Returns list of all 7-isogenies from E when the j-invariant is 1728.

OUTPUT:

(list) 7-isogenies with codomain E. In general these are normalised; but over $$\QQ$$ the codomain is a minimal model.

Note

This implementation requires that the characteristic is not 2, 3, or 7.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(7).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_1728
sage: E = EllipticCurve(GF(47), [1, 0])
sage: isogenies_7_1728(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 47 to Elliptic Curve defined by y^2 = x^3 + 26 over Finite Field of size 47,
Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 47 to Elliptic Curve defined by y^2 = x^3 + 21 over Finite Field of size 47]


An example in characteristic 53 (for which an earlier implementation did not work):

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_1728
sage: E = EllipticCurve(GF(53), [1, 0])
sage: isogenies_7_1728(E)
[]
sage: E = EllipticCurve(GF(53^2,'a'), [1, 0])
sage: [iso.codomain().ainvs() for iso in isogenies_7_1728(E)]
[(0, 0, 0, 36, 19*a + 15), (0, 0, 0, 36, 34*a + 38), (0, 0, 0, 33, 39*a + 28), (0, 0, 0, 33, 14*a + 25), (0, 0, 0, 19, 45*a + 16), (0, 0, 0, 19, 8*a + 37), (0, 0, 0, 3, 45*a + 16), (0, 0, 0, 3, 8*a + 37)]

sage: K.<a> = NumberField(x^8 + 84*x^6 - 1890*x^4 + 644*x^2 - 567)
sage: E = EllipticCurve(K, [1, 0])
sage: isogs = isogenies_7_1728(E)
sage: [phi.codomain().a_invariants() for phi in isogs]
[(0,
0,
0,
35/636*a^6 + 55/12*a^4 - 79135/636*a^2 + 1127/212,
155/636*a^7 + 245/12*a^5 - 313355/636*a^3 - 3577/636*a),
(0,
0,
0,
35/636*a^6 + 55/12*a^4 - 79135/636*a^2 + 1127/212,
-155/636*a^7 - 245/12*a^5 + 313355/636*a^3 + 3577/636*a)]
sage: [phi.codomain().j_invariant() for phi in isogs]
[-526110256146528/53*a^6 + 183649373229024*a^4 - 3333881559996576/53*a^2 + 2910267397643616/53,
-526110256146528/53*a^6 + 183649373229024*a^4 - 3333881559996576/53*a^2 + 2910267397643616/53]
sage: E1 = isogs[0].codomain()
sage: E2 = isogs[1].codomain()
sage: E1.is_isomorphic(E2)
False
-1

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree(E, l)

Returns a list of l -isogenies with domain E.

INPUT:

• E – an elliptic curve.
• l – a prime.

OUTPUT:

(list) a list of all isogenies of degree $$l$$. If the characteristic is $$l$$ then only separable isogenies are constructed.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree
sage: E = EllipticCurve_from_j(GF(2^6,'a')(1))
sage: isogenies_prime_degree(E, 7)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Finite Field in a of size 2^6 to Elliptic Curve defined by y^2 + x*y = x^3 + x over Finite Field in a of size 2^6]
sage: E = EllipticCurve_from_j(GF(3^12,'a')(2))
sage: isogenies_prime_degree(E, 17)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2 over Finite Field in a of size 3^12 to Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x over Finite Field in a of size 3^12, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2 over Finite Field in a of size 3^12 to Elliptic Curve defined by y^2 = x^3 + 2*x^2 + x + 2 over Finite Field in a of size 3^12]
sage: E = EllipticCurve('50a1')
sage: isogenies_prime_degree(E, 3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 126*x - 552 over Rational Field]
sage: isogenies_prime_degree(E, 5)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 76*x + 298 over Rational Field]
sage: E = EllipticCurve([0, 0, 1, -1862, -30956])
sage: isogenies_prime_degree(E, 19)
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 - 1862*x - 30956 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 672182*x + 212325489 over Rational Field]
sage: E = EllipticCurve([0, -1, 0, -6288, 211072])
sage: isogenies_prime_degree(E, 37)
[Isogeny of degree 37 from Elliptic Curve defined by y^2 = x^3 - x^2 - 6288*x + 211072 over Rational Field to Elliptic Curve defined by y^2 = x^3 - x^2 - 163137088*x - 801950801728 over Rational Field]


Isogenies of degree equal to the characteristic are computed (but only the separable isogeny). In the following example we consider an elliptic curve which is supersingular in characteristic 2 only:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree
sage: ainvs = (0,1,1,-1,-1)
sage: for l in prime_range(50):
....:     E = EllipticCurve(GF(l),ainvs)
....:     isogenies_prime_degree(E,l)
[]
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 2*x + 2 over Finite Field of size 3 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 3]
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4*x + 4 over Finite Field of size 5 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4*x + 4 over Finite Field of size 5]
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 6*x + 6 over Finite Field of size 7 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4 over Finite Field of size 7]
[Isogeny of degree 11 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 10*x + 10 over Finite Field of size 11 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + x + 1 over Finite Field of size 11]
[Isogeny of degree 13 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 12 over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 12 over Finite Field of size 13]
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 16*x + 16 over Finite Field of size 17 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 15 over Finite Field of size 17]
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 18*x + 18 over Finite Field of size 19 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 3*x + 12 over Finite Field of size 19]
[Isogeny of degree 23 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 22*x + 22 over Finite Field of size 23 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 22*x + 22 over Finite Field of size 23]
[Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 28*x + 28 over Finite Field of size 29 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 7*x + 27 over Finite Field of size 29]
[Isogeny of degree 31 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 30*x + 30 over Finite Field of size 31 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 15*x + 16 over Finite Field of size 31]
[Isogeny of degree 37 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 36*x + 36 over Finite Field of size 37 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 16*x + 17 over Finite Field of size 37]
[Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 40*x + 40 over Finite Field of size 41 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 33*x + 16 over Finite Field of size 41]
[Isogeny of degree 43 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 42*x + 42 over Finite Field of size 43 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 36 over Finite Field of size 43]
[Isogeny of degree 47 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 46*x + 46 over Finite Field of size 47 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 42*x + 34 over Finite Field of size 47]


Note that the computation is faster for degrees equal to one of the genus 0 primes (2, 3, 5, 7, 13) or one of the hyperelliptic primes (11, 17, 19, 23, 29, 31, 41, 47, 59, 71) than when the generic code must be used:

sage: E = EllipticCurve(GF(101), [-3440, 77658])
sage: E.isogenies_prime_degree(71) # fast
[]
sage: E.isogenies_prime_degree(73) # not tested (very long time: 32s)
[]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_general(E, l)

Returns a list of l -isogenies with domain E.

INPUT:

• E – an elliptic curve.
• l – a prime.

OUTPUT:

(list) a list of all isogenies of degree l.

ALGORITHM:

This algorithm factors the l-division polynomial, then combines its factors to otain kernels. See [KT2013], Chapter 3.

Note

This function works for any prime $$l$$. Normally one should use the function isogenies_prime_degree() which uses special functions for certain small primes.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree_general
sage: E = EllipticCurve_from_j(GF(2^6,'a')(1))
sage: isogenies_prime_degree_general(E, 7)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Finite Field in a of size 2^6 to Elliptic Curve defined by y^2 + x*y = x^3 + x over Finite Field in a of size 2^6]
sage: E = EllipticCurve_from_j(GF(3^12,'a')(2))
sage: isogenies_prime_degree_general(E, 17)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2 over Finite Field in a of size 3^12 to Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x over Finite Field in a of size 3^12, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2 over Finite Field in a of size 3^12 to Elliptic Curve defined by y^2 = x^3 + 2*x^2 + x + 2 over Finite Field in a of size 3^12]
sage: E = EllipticCurve('50a1')
sage: isogenies_prime_degree_general(E, 3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 126*x - 552 over Rational Field]
sage: isogenies_prime_degree_general(E, 5)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 76*x + 298 over Rational Field]
sage: E = EllipticCurve([0, 0, 1, -1862, -30956])
sage: isogenies_prime_degree_general(E, 19)
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 - 1862*x - 30956 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 672182*x + 212325489 over Rational Field]
sage: E = EllipticCurve([0, -1, 0, -6288, 211072])
sage: isogenies_prime_degree_general(E, 37)  # long time (10s)
[Isogeny of degree 37 from Elliptic Curve defined by y^2 = x^3 - x^2 - 6288*x + 211072 over Rational Field to Elliptic Curve defined by y^2 = x^3 - x^2 - 163137088*x - 801950801728 over Rational Field]

sage: E = EllipticCurve([-3440, 77658])
sage: isogenies_prime_degree_general(E, 43)  # long time (16s)
[Isogeny of degree 43 from Elliptic Curve defined by y^2 = x^3 - 3440*x + 77658 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 6360560*x - 6174354606 over Rational Field]


Isogenies of degree equal to the characteristic are computed (but only the separable isogeny). In the following example we consider an elliptic curve which is supersingular in characteristic 2 only:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree_general
sage: ainvs = (0,1,1,-1,-1)
sage: for l in prime_range(50):
....:     E = EllipticCurve(GF(l),ainvs)
....:     isogenies_prime_degree_general(E,l)
[]
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 2*x + 2 over Finite Field of size 3 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 3]
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4*x + 4 over Finite Field of size 5 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4*x + 4 over Finite Field of size 5]
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 6*x + 6 over Finite Field of size 7 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4 over Finite Field of size 7]
[Isogeny of degree 11 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 10*x + 10 over Finite Field of size 11 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + x + 1 over Finite Field of size 11]
[Isogeny of degree 13 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 12 over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 12 over Finite Field of size 13]
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 16*x + 16 over Finite Field of size 17 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 15 over Finite Field of size 17]
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 18*x + 18 over Finite Field of size 19 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 3*x + 12 over Finite Field of size 19]
[Isogeny of degree 23 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 22*x + 22 over Finite Field of size 23 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 22*x + 22 over Finite Field of size 23]
[Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 28*x + 28 over Finite Field of size 29 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 7*x + 27 over Finite Field of size 29]
[Isogeny of degree 31 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 30*x + 30 over Finite Field of size 31 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 15*x + 16 over Finite Field of size 31]
[Isogeny of degree 37 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 36*x + 36 over Finite Field of size 37 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 16*x + 17 over Finite Field of size 37]
[Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 40*x + 40 over Finite Field of size 41 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 33*x + 16 over Finite Field of size 41]
[Isogeny of degree 43 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 42*x + 42 over Finite Field of size 43 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 36 over Finite Field of size 43]
[Isogeny of degree 47 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 46*x + 46 over Finite Field of size 47 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 42*x + 34 over Finite Field of size 47]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_genus_0(E, l=None)

Returns list of l -isogenies with domain E.

INPUT:

• E – an elliptic curve.
• l – either None or 2, 3, 5, 7, or 13.

OUTPUT:

(list) When l is None a list of all isogenies of degree 2, 3, 5, 7 and 13, otherwise a list of isogenies of the given degree.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(l), which automatically calls the appropriate function.

ALGORITHM:

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree_genus_0
sage: E = EllipticCurve([0,12])
sage: isogenies_prime_degree_genus_0(E, 5)
[]

sage: E = EllipticCurve('1450c1')
sage: isogenies_prime_degree_genus_0(E)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 300*x - 1000 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 5950*x - 182250 over Rational Field]

sage: E = EllipticCurve('50a1')
sage: isogenies_prime_degree_genus_0(E)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 126*x - 552 over Rational Field,
Isogeny of degree 5 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 76*x + 298 over Rational Field]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_genus_plus_0(E, l=None)

Returns list of l -isogenies with domain E.

INPUT:

• E – an elliptic curve.
• l – either None or 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.

OUTPUT:

(list) When l is None a list of all isogenies of degree 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71, otherwise a list of isogenies of the given degree.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(l), which automatically calls the appropriate function.

ALGORITHM:

See [KT2013], Chapter 5.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree_genus_plus_0

sage: E = EllipticCurve('121a1')
sage: isogenies_prime_degree_genus_plus_0(E, 11)
[Isogeny of degree 11 from Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 30*x - 76 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 305*x + 7888 over Rational Field]

sage: E = EllipticCurve([1, 1, 0, -660, -7600])
sage: isogenies_prime_degree_genus_plus_0(E, 17)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 660*x - 7600 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 878710*x + 316677750 over Rational Field]

sage: E = EllipticCurve([0, 0, 1, -1862, -30956])
sage: isogenies_prime_degree_genus_plus_0(E, 19)
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 - 1862*x - 30956 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 672182*x + 212325489 over Rational Field]

sage: a = K.gen()
sage: E = EllipticCurve_from_j(-484650135/16777216*a + 4549855725/16777216)
sage: isogenies_prime_degree_genus_plus_0(E, 23)
[Isogeny of degree 23 from Elliptic Curve defined by y^2 = x^3 + (-14460494784192904095/140737488355328*a+270742665778826768325/140737488355328)*x + (37035998788154488846811217135/590295810358705651712*a-1447451882571839266752561148725/590295810358705651712) over Number Field in a with defining polynomial x^2 + 295 to Elliptic Curve defined by y^2 = x^3 + (-5130542435555445498495/140737488355328*a+173233955029127361005925/140737488355328)*x + (-1104699335561165691575396879260545/590295810358705651712*a+3169785826904210171629535101419675/590295810358705651712) over Number Field in a with defining polynomial x^2 + 295]

sage: a = K.gen()
sage: E = EllipticCurve_from_j(94743000*a + 269989875)
sage: isogenies_prime_degree_genus_plus_0(E, 29)
[Isogeny of degree 29 from Elliptic Curve defined by y^2 = x^3 + (-153477413215038000*a+5140130723072965125)*x + (297036215130547008455526000*a+2854277047164317800973582250) over Number Field in a with defining polynomial x^2 + 199 to Elliptic Curve defined by y^2 = x^3 + (251336161378040805000*a-3071093219933084341875)*x + (-8411064283162168580187643221000*a+34804337770798389546017184785250) over Number Field in a with defining polynomial x^2 + 199]

sage: a = K.gen()
sage: E = EllipticCurve_from_j(208438034112000*a - 3315409892960000)
sage: isogenies_prime_degree_genus_plus_0(E, 31)
[Isogeny of degree 31 from Elliptic Curve defined by y^2 = x^3 + (4146345122185433034677956608000*a-65951656549965037259634800640000)*x + (-18329111516954473474583425393698245080252416000*a+291542366110383928366510368064204147260129280000) over Number Field in a with defining polynomial x^2 - 253 to Elliptic Curve defined by y^2 = x^3 + (200339763852548615776123686912000*a-3186599019027216904280948275200000)*x + (7443671791411479629112717260182286294850207744000*a-118398847898864757209685951728838895495168655360000) over Number Field in a with defining polynomial x^2 - 253]

sage: E = EllipticCurve_from_j(GF(5)(1))
sage: isogenies_prime_degree_genus_plus_0(E, 41)
[Isogeny of degree 41 from Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 3 over Finite Field of size 5, Isogeny of degree 41 from Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 3 over Finite Field of size 5]

sage: a = K.gen()
sage: E = EllipticCurve_from_j(184068066743177379840*a - 411588709724712960000)
sage: isogenies_prime_degree_genus_plus_0(E, 47) # long time (4.3s)
[Isogeny of degree 47 from Elliptic Curve defined by y^2 = x^3 + (454562028554080355857852049849975895490560*a-1016431595837124114668689286176511361024000)*x + (-249456798429896080881440540950393713303830363999480904280965120*a+557802358738710443451273320227578156598454035482869042774016000) over Number Field in a with defining polynomial x^2 - 5 to Elliptic Curve defined by y^2 = x^3 + (39533118442361013730577638493616965245992960*a-88398740199669828340617478832005245173760000)*x + (214030321479466610282320528611562368963830105830555363061803253760*a-478586348074220699687616322532666163722004497458452316582576128000) over Number Field in a with defining polynomial x^2 - 5]

sage: a = K.gen()
sage: E = EllipticCurve_from_j(-98669236224000*a + 4401720074240000)
sage: isogenies_prime_degree_genus_plus_0(E, 59)   # long time (25s, 2012)
[Isogeny of degree 59 from Elliptic Curve defined by y^2 = x^3 + (2605886146782144762297974784000*a+1893681048912773634944634716160000)*x + (-116918454256410782232296183198067568744071168000*a+17012043538294664027185882358514011304812871680000) over Number Field in a with defining polynomial x^2 + 66827 to Elliptic Curve defined by y^2 = x^3 + (-19387084027159786821400775098368000*a-4882059104868154225052787156713472000)*x + (-25659862010101415428713331477227179429538847260672000*a-2596038148441293485938798119003462972840818381946880000) over Number Field in a with defining polynomial x^2 + 66827]

sage: E = EllipticCurve_from_j(GF(13)(5))
sage: isogenies_prime_degree_genus_plus_0(E, 71) # long time
[Isogeny of degree 71 from Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 10*x + 7 over Finite Field of size 13, Isogeny of degree 71 from Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 10*x + 7 over Finite Field of size 13]

sage: E = EllipticCurve(GF(13),[0,1,1,1,0])
sage: isogenies_prime_degree_genus_plus_0(E)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 10*x + 1 over Finite Field of size 13,
Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 4 over Finite Field of size 13,
Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 6 over Finite Field of size 13,
Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 5*x + 6 over Finite Field of size 13,
Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 4 over Finite Field of size 13,
Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 5*x + 6 over Finite Field of size 13]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_genus_plus_0_j0(E, l)

Returns a list of hyperelliptic l -isogenies with domain E when $$j(E)=0$$.

INPUT:

• E – an elliptic curve with j-invariant 0.
• l – 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.

OUTPUT:

(list) a list of all isogenies of degree 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.

Note

This implementation requires that the characteristic is not 2, 3 or l.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(l).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree_genus_plus_0_j0

sage: u = polygen(QQ)
sage: K.<a> = NumberField(u^4+228*u^3+486*u^2-540*u+225)
sage: E = EllipticCurve(K,[0,-121/5*a^3-20691/5*a^2-29403/5*a+3267])
sage: isogenies_prime_degree_genus_plus_0_j0(E,11)
[Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + (-121/5*a^3-20691/5*a^2-29403/5*a+3267) over Number Field in a with defining polynomial x^4 + 228*x^3 + 486*x^2 - 540*x + 225 to Elliptic Curve defined by y^2 = x^3 + (-44286*a^2+178596*a-32670)*x + (-17863351/5*a^3+125072739/5*a^2-74353653/5*a-682803) over Number Field in a with defining polynomial x^4 + 228*x^3 + 486*x^2 - 540*x + 225, Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + (-121/5*a^3-20691/5*a^2-29403/5*a+3267) over Number Field in a with defining polynomial x^4 + 228*x^3 + 486*x^2 - 540*x + 225 to Elliptic Curve defined by y^2 = x^3 + (-3267*a^3-740157*a^2+600039*a-277695)*x + (-17863351/5*a^3-4171554981/5*a^2+3769467867/5*a-272366523) over Number Field in a with defining polynomial x^4 + 228*x^3 + 486*x^2 - 540*x + 225]

sage: E = EllipticCurve(GF(5^6,'a'),[0,1])
sage: isogenies_prime_degree_genus_plus_0_j0(E,17)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 5^6 to Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field in a of size 5^6]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_genus_plus_0_j1728(E, l)

Returns a list of l -isogenies with domain E when $$j(E)=1728$$.

INPUT:

• E – an elliptic curve with j-invariant 1728.
• l – 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.

OUTPUT:

(list) a list of all isogenies of degree 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.

Note

This implementation requires that the characteristic is not 2, 3 or l.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(l).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree_genus_plus_0_j1728

sage: u = polygen(QQ)
sage: K.<a> = NumberField(u^6 - 522*u^5 - 10017*u^4 + 2484*u^3 - 5265*u^2 + 12150*u - 5103)
sage: E = EllipticCurve(K,[-75295/1335852*a^5+13066735/445284*a^4+44903485/74214*a^3+17086861/24738*a^2+11373021/16492*a-1246245/2356,0])
sage: isogenies_prime_degree_genus_plus_0_j1728(E,11)
[Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + (-75295/1335852*a^5+13066735/445284*a^4+44903485/74214*a^3+17086861/24738*a^2+11373021/16492*a-1246245/2356)*x over Number Field in a with defining polynomial x^6 - 522*x^5 - 10017*x^4 + 2484*x^3 - 5265*x^2 + 12150*x - 5103 to Elliptic Curve defined by y^2 = x^3 + (9110695/1335852*a^5-1581074935/445284*a^4-5433321685/74214*a^3-3163057249/24738*a^2+1569269691/16492*a+73825125/2356)*x + (-3540460*a^3+30522492*a^2-7043652*a-5031180) over Number Field in a with defining polynomial x^6 - 522*x^5 - 10017*x^4 + 2484*x^3 - 5265*x^2 + 12150*x - 5103, Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + (-75295/1335852*a^5+13066735/445284*a^4+44903485/74214*a^3+17086861/24738*a^2+11373021/16492*a-1246245/2356)*x over Number Field in a with defining polynomial x^6 - 522*x^5 - 10017*x^4 + 2484*x^3 - 5265*x^2 + 12150*x - 5103 to Elliptic Curve defined by y^2 = x^3 + (9110695/1335852*a^5-1581074935/445284*a^4-5433321685/74214*a^3-3163057249/24738*a^2+1569269691/16492*a+73825125/2356)*x + (3540460*a^3-30522492*a^2+7043652*a+5031180) over Number Field in a with defining polynomial x^6 - 522*x^5 - 10017*x^4 + 2484*x^3 - 5265*x^2 + 12150*x - 5103]
sage: E = EllipticCurve([-1-2*i,0])
sage: isogenies_prime_degree_genus_plus_0_j1728(E,17)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + (-2*i-1)*x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-82*i-641)*x over Number Field in i with defining polynomial x^2 + 1,
Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + (-2*i-1)*x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-562*i+319)*x over Number Field in i with defining polynomial x^2 + 1]
sage: Emin = E.global_minimal_model()
sage: [(p,len(isogenies_prime_degree_genus_plus_0_j1728(Emin,p))) for p in [17, 29, 41]]
[(17, 2), (29, 2), (41, 2)]


Returns list of l -isogenies with domain E (defined over $$\QQ$$).

Returns a list of sporadic l-isogenies from E (l = 11, 17, 19, 37, 43, 67 or 163). Only for elliptic curves over $$\QQ$$.

INPUT:

• E – an elliptic curve defined over $$\QQ$$.
• l – either None or a prime number.

OUTPUT:

(list) If l is None, a list of all isogenies with domain E and of degree 11, 17, 19, 37, 43, 67 or 163; otherwise a list of isogenies of the given degree.

Note

This function would normally be invoked indirectly via E.isogenies_prime_degree(l), which automatically calls the appropriate function.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_sporadic_Q
sage: E = EllipticCurve('121a1')
[Isogeny of degree 11 from Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 30*x - 76 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 305*x + 7888 over Rational Field]
[]
[]
[Isogeny of degree 11 from Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 30*x - 76 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 305*x + 7888 over Rational Field]

sage: E = EllipticCurve([1, 1, 0, -660, -7600])
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 660*x - 7600 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 878710*x + 316677750 over Rational Field]
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 660*x - 7600 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 878710*x + 316677750 over Rational Field]
[]

sage: E = EllipticCurve([0, 0, 1, -1862, -30956])
[]
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 - 1862*x - 30956 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 672182*x + 212325489 over Rational Field]
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 - 1862*x - 30956 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 672182*x + 212325489 over Rational Field]

sage: E = EllipticCurve([0, -1, 0, -6288, 211072])
sage: E.conductor()
19600
[Isogeny of degree 37 from Elliptic Curve defined by y^2 = x^3 - x^2 - 6288*x + 211072 over Rational Field to Elliptic Curve defined by y^2 = x^3 - x^2 - 163137088*x - 801950801728 over Rational Field]

sage: E = EllipticCurve([1, 1, 0, -25178045, 48616918750])
sage: E.conductor()
148225
[Isogeny of degree 37 from Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 25178045*x + 48616918750 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 970*x - 13075 over Rational Field]

sage: E = EllipticCurve([-3440, 77658])
sage: E.conductor()
118336
[Isogeny of degree 43 from Elliptic Curve defined by y^2 = x^3 - 3440*x + 77658 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 6360560*x - 6174354606 over Rational Field]

sage: E = EllipticCurve([-29480, -1948226])
sage: E.conductor()
287296
[Isogeny of degree 67 from Elliptic Curve defined by y^2 = x^3 - 29480*x - 1948226 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 132335720*x + 585954296438 over Rational Field]

sage: E = EllipticCurve([-34790720, -78984748304])
sage: E.conductor()
425104

Weierstrass $$\wp$$ function for elliptic curves