# 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].

sage.schemes.elliptic_curves.isogeny_small_degree.Fricke_module()#

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 $$P(t)/t$$.

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

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 $$\ZZ[t]$$.

sage.schemes.elliptic_curves.isogeny_small_degree.Psi(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.

Note

This computation takes a negligible time for $$l=2,3,5,7$$ but more than 100s for $$l=13$$. The reason for allowing dynamic computation here instead of just using precomputed values is for testing.

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

Return 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.

EXAMPLES:

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: p = Psi2(71)                        # long time
sage: (x,u,v) = p.variables()             # long time
sage: p.coefficient({x: 0, u: 210, v: 0}) # long time
-2209380711722505179506258739515288584116147237393815266468076436521/71
sage: p.coefficient({x: 0, u: 0, v: 0})   # long time
-14790739586438315394567393301990769678157425619440464678252277649/71

sage.schemes.elliptic_curves.isogeny_small_degree.is_kernel_polynomial(E, m, f)#

Test whether E has a cyclic isogeny of degree m with kernel polynomial f.

INPUT:

• E – an elliptic curve.

• m – a positive integer.

• f – a polynomial over the base field of E.

OUTPUT:

(bool) True if E has a cyclic isogeny of degree m with kernel polynomial f, else False.

ALGORITHM:

$$f$$ must have degree $$(m-1)/2$$ (if $$m$$ is odd) or degree $$m/2$$ (if $$m$$ is even), and have the property that for each root $$x$$ of $$f$$, $$\mu(x)$$ is also a root where $$\mu$$ is the multiplication-by-$$m$$ map on $$E$$ and $$m$$ runs over a set of generators of $$(\ZZ/m\ZZ)^*/\{1,-1\}$$.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import is_kernel_polynomial
sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: x = polygen(QQ)
sage: is_kernel_polynomial(E, 5, x^2 + x - 29/5)
True
sage: is_kernel_polynomial(E, 5, (x - 16) * (x - 5))
True


An example from [KT2013], where the 13-division polynomial splits into 14 factors each of degree 6, but only two of these is a kernel polynomial for a 13-isogeny:

sage: F = GF(3)                                                                 # optional - sage.rings.finite_rings
sage: E = EllipticCurve(F, [0,0,0,-1,0])                                        # optional - sage.rings.finite_rings
sage: f13 = E.division_polynomial(13)                                           # optional - sage.rings.finite_rings
sage: factors = [f for f, e in f13.factor()]                                    # optional - sage.rings.finite_rings
sage: all(f.degree() == 6 for f in factors)                                     # optional - sage.rings.finite_rings
True
sage: [is_kernel_polynomial(E, 13, f) for f in factors]                         # optional - sage.rings.finite_rings
[True,
True,
False,
False,
False,
False,
False,
False,
False,
False,
False,
False,
False,
False]

sage: K = GF(47^2)                                                              # optional - sage.rings.finite_rings
sage: E = EllipticCurve([0, K.gen()])                                           # optional - sage.rings.finite_rings
sage: psi7 = E.division_polynomial(7)                                           # optional - sage.rings.finite_rings
sage: f = psi7.factor()                                                   # optional - sage.rings.finite_rings
sage: f                                                                         # optional - sage.rings.finite_rings
x^3 + (7*z2 + 11)*x^2 + (25*z2 + 33)*x + 25*z2
sage: f.divides(psi7)                                                           # optional - sage.rings.finite_rings
True
sage: is_kernel_polynomial(E, 7, f)                                             # optional - sage.rings.finite_rings
False

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_13_0(E, minimal_models=True)#

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

INPUT:

• E – an elliptic curve with j-invariant 0.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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)                                                # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0, r]); E                                           # optional - sage.rings.number_field
Elliptic Curve defined by y^2 = x^3 + r over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I
sage: isogenies_13_0(E)                                                         # optional - sage.rings.number_field
[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 with r = 1.732050807568878?*I
to Elliptic Curve defined by y^2 = x^3 + r over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I,
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 with r = 1.732050807568878?*I
to Elliptic Curve defined by y^2 = x^3 + r over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I]
sage: isogenies_13_0(E).rational_maps()                                      # optional - sage.rings.number_field
(((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')                                                          # optional - sage.rings.finite_rings
sage: E = EllipticCurve(j=K(0)); E                                              # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + 1
over Finite Field in a of size 19^2
sage: isogenies_13_0(E)                                                         # optional - sage.rings.finite_rings
[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).rational_maps()                                      # optional - sage.rings.finite_rings
((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)                                                                # optional - sage.rings.finite_rings
sage: E = EllipticCurve(j=K(0))                                                 # optional - sage.rings.finite_rings
sage: isogenies_13_0(E)                                                         # optional - sage.rings.finite_rings
[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)                                                               # optional - sage.rings.finite_rings
sage: E = EllipticCurve(j=K(0)); E.ainvs()                                      # optional - sage.rings.finite_rings
(0, 0, 0, 0, 1)
sage: [phi.codomain().ainvs() for phi in isogenies_13_0(E)]                     # optional - sage.rings.finite_rings
[(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)                                                    # optional - sage.rings.number_field
sage: E = EllipticCurve(j=K(0)); E.ainvs()                                      # optional - sage.rings.number_field
(0, 0, 0, 0, 1)
sage: len([phi.codomain().ainvs()           # long time (4s)                    # optional - sage.rings.number_field
....:      for phi in isogenies_13_0(E)])
2

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_13_1728(E, minimal_models=True)#

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

INPUT:

• E – an elliptic curve with j-invariant 1728.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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 $$\QQ$$ or a number field, the codomain is a global minimal model where possible.

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: K.<i> = QuadraticField(-1)                                                # optional - sage.rings.number_field
sage: E = EllipticCurve([0,0,0,i,0]); E.ainvs()                                 # optional - sage.rings.number_field
(0, 0, 0, i, 0)
sage: isogenies_13_1728(E)                                                      # optional - sage.rings.number_field
[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 with i = 1*I
to Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i
with defining polynomial x^2 + 1 with i = 1*I,
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 with i = 1*I
to Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i
with defining polynomial x^2 + 1 with i = 1*I]

sage: K = GF(83)                                                                # optional - sage.rings.finite_rings
sage: E = EllipticCurve(K, [0,0,0,5,0]); E.ainvs()                              # optional - sage.rings.finite_rings
(0, 0, 0, 5, 0)
sage: isogenies_13_1728(E)                                                      # optional - sage.rings.finite_rings
[]
sage: K = GF(89)                                                                # optional - sage.rings.finite_rings
sage: E = EllipticCurve(K, [0,0,0,5,0]); E.ainvs()                              # optional - sage.rings.finite_rings
(0, 0, 0, 5, 0)
sage: isogenies_13_1728(E)                                                      # optional - sage.rings.finite_rings
[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)                                                                # optional - sage.rings.finite_rings
sage: E = EllipticCurve(K, [1,0])                                               # optional - sage.rings.finite_rings
sage: isogenies_13_1728(E)                                                      # optional - sage.rings.finite_rings
[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)                                                    # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [1,0])                                               # optional - sage.rings.number_field
sage: [phi.codomain().ainvs()           # long time (3s)                        # optional - sage.rings.number_field
....:  for phi in isogenies_13_1728(E)]
[(0,
0,
0,
-4225010072113/3063768069807341568*a^10 - 24841071989413/15957125363579904*a^8
+ 11179537789374271/21276167151439872*a^6 - 407474562289492049/47871376090739712*a^4
+ 1608052769560747/4522994717568*a^2 + 7786720245212809/36937790193472,
-363594277511/574456513088876544*a^11 - 7213386922793/2991961005671232*a^9
- 2810970361185589/1329760446964992*a^7 + 281503836888046601/8975883017013696*a^5
- 1287313166530075/848061509544*a^3 + 9768837984886039/6925835661276*a),
(0,
0,
0,
-4225010072113/3063768069807341568*a^10 - 24841071989413/15957125363579904*a^8
+ 11179537789374271/21276167151439872*a^6 - 407474562289492049/47871376090739712*a^4
+ 1608052769560747/4522994717568*a^2 + 7786720245212809/36937790193472,
363594277511/574456513088876544*a^11 + 7213386922793/2991961005671232*a^9
+ 2810970361185589/1329760446964992*a^7 - 281503836888046601/8975883017013696*a^5
+ 1287313166530075/848061509544*a^3 - 9768837984886039/6925835661276*a)]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_2(E, minimal_models=True)#

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

INPUT:

• E – an elliptic curve.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

OUTPUT:

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

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                                        # optional - sage.rings.number_field
Elliptic Curve defined by y^2 = x^3 + 9*x + 8 over Algebraic Field
sage: isogenies_2(E)  # not implemented                                         # optional - sage.rings.number_field

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_3(E, minimal_models=True)#

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

INPUT:

• E – an elliptic curve.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

OUTPUT:

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

EXAMPLES:

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

sage: E = EllipticCurve(GF(17^2,'a'), [1,1])                                    # optional - sage.rings.finite_rings
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]                        # optional - sage.rings.finite_rings
[(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, minimal_models=True)#

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

INPUT:

• E – an elliptic curve with j-invariant 0.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

OUTPUT:

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

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])                                   # optional - sage.rings.finite_rings
sage: isogenies_5_0(E)                                                          # optional - sage.rings.finite_rings
[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: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x**6 - 320*x**3 - 320)                                # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0,0,1,0,0])                                         # optional - sage.rings.number_field
sage: isogenies_5_0(E)                                                          # optional - sage.rings.number_field
[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 + y = x^3 + (241565/32*a^5-362149/48*a^4+180281/24*a^3-9693307/4*a^2+14524871/6*a-7254985/3)*x
+ (1660391123/192*a^5-829315373/96*a^4+77680504/9*a^3-66622345345/24*a^2+33276655441/12*a-24931615912/9)
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 + y = x^3 + (47519/32*a^5-72103/48*a^4+32939/24*a^3-1909753/4*a^2+2861549/6*a-1429675/3)*x
+ (-131678717/192*a^5+65520419/96*a^4-12594215/18*a^3+5280985135/24*a^2-2637787519/12*a+1976130088/9)
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, minimal_models=True)#

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

INPUT:

• E – an elliptic curve with j-invariant 1728.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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 $$\QQ$$ or a number field, the codomain is a global minimal model where possible.

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])                                         # optional - sage.rings.finite_rings
sage: isogenies_5_1728(E)                                                       # optional - sage.rings.finite_rings
[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)                                                # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0,0,0,1,0])                                         # optional - sage.rings.number_field
sage: isogenies_5_1728(E)                                                       # optional - sage.rings.number_field
[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 with i = 1*I
to Elliptic Curve defined by y^2 = x^3 + x over Number Field in i
with defining polynomial x^2 + 1 with i = 1*I,
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 with i = 1*I
to Elliptic Curve defined by y^2 = x^3 + x over Number Field in i
with defining polynomial x^2 + 1 with i = 1*I]
sage: _.rational_maps()                                                      # optional - sage.rings.number_field
(((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: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x**4 + 20*x**2 - 80)                                  # optional - sage.rings.number_field
sage: K(5).is_square()  # necessary but not sufficient!                         # optional - sage.rings.number_field
True
sage: E = EllipticCurve(K, [0,0,0,1,0])                                         # optional - sage.rings.number_field
sage: isogenies_5_1728(E)                                                       # optional - sage.rings.number_field
[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 + (-753/4*a^2-4399)*x + (2779*a^3+65072*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 + (-753/4*a^2-4399)*x + (-2779*a^3-65072*a)
over Number Field in a with defining polynomial x^4 + 20*x^2 - 80]

sage: K.<a> = NumberField(x^4 - 5*x^2 + 5)                                      # optional - sage.rings.number_field
sage: E = EllipticCurve([a^2 + a + 1, a^3 + a^2 + a + 1, a^2 + a,
....:                    17*a^3 + 34*a^2 - 16*a - 37,
....:                    54*a^3 + 105*a^2 - 66*a - 135])
sage: len(E.isogenies_prime_degree(5))                                          # optional - sage.rings.number_field
2
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_5_1728
sage: [phi.codomain().j_invariant() for phi in isogenies_5_1728(E)]             # optional - sage.rings.number_field
[19691491018752*a^2 - 27212977933632, 19691491018752*a^2 - 27212977933632]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_7_0(E, minimal_models=True)#

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

INPUT:

• E – an elliptic curve with j-invariant 0.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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 $$\QQ$$ or a number field, the codomain is a global minimal model where possible.

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: K.<r> = QuadraticField(-3)                                                # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0,1])                                               # optional - sage.rings.number_field
sage: isogenies_7_0(E)                                                          # optional - sage.rings.number_field
[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 with r = 1.732050807568878?*I
to Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I,
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 with r = 1.732050807568878?*I
to Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I]

sage: E = EllipticCurve(GF(13^2,'a'), [0,-3])                                   # optional - sage.rings.finite_rings
sage: isogenies_7_0(E)                                                          # optional - sage.rings.finite_rings
[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)                                                               # optional - sage.rings.finite_rings
sage: E = EllipticCurve(K, [0,1])                                               # optional - sage.rings.finite_rings
sage: isogenies_7_0(E)                                                          # optional - sage.rings.finite_rings
[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: E = EllipticCurve('27a1').change_ring(QuadraticField(-3,'r'))             # optional - sage.rings.number_field
sage: isogenies_7_0(E)                                                          # optional - sage.rings.number_field
[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 with r = 1.732050807568878?*I
to Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I,
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 with r = 1.732050807568878?*I
to Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r
with defining polynomial x^2 + 3 with r = 1.732050807568878?*I]

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^6 + 1512*x^3 - 21168)                               # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0,1])                                               # optional - sage.rings.number_field
sage: isogs = isogenies_7_0(E)                                                  # optional - sage.rings.number_field
sage: [phi.codomain().a_invariants() for phi in isogs]                          # optional - sage.rings.number_field
[(0,
0,
0,
-415/98*a^5 - 675/14*a^4 + 2255/7*a^3 - 74700/7*a^2 - 25110*a - 66420,
-141163/56*a^5 + 1443453/112*a^4 - 374275/2*a^3
- 3500211/2*a^2 - 17871975/4*a - 7710065),
(0,
0,
0,
-24485/392*a^5 - 1080/7*a^4 - 2255/7*a^3 - 1340865/14*a^2 - 230040*a - 553500,
1753037/56*a^5 + 8345733/112*a^4 + 374275/2*a^3
+ 95377029/2*a^2 + 458385345/4*a + 275241835)]
sage: [phi.codomain().j_invariant() for phi in isogs]                           # optional - sage.rings.number_field
[158428486656000/7*a^3 - 313976217600000,
-158428486656000/7*a^3 - 34534529335296000]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_7_1728(E, minimal_models=True)#

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

INPUT:

• E – an elliptic curve with j-invariant 1728.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

OUTPUT:

(list) 7-isogenies with codomain E. In general these are normalised; but over $$\QQ$$ or a number field, the codomain is a global minimal model where possible.

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])                                         # optional - sage.rings.finite_rings
sage: isogenies_7_1728(E)                                                       # optional - sage.rings.finite_rings
[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])                                         # optional - sage.rings.finite_rings
sage: isogenies_7_1728(E)                                                       # optional - sage.rings.finite_rings
[]
sage: E = EllipticCurve(GF(53^2,'a'), [1, 0])                                   # optional - sage.rings.finite_rings
sage: [iso.codomain().ainvs() for iso in isogenies_7_1728(E)]                   # optional - sage.rings.finite_rings
[(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: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^8 + 84*x^6 - 1890*x^4 + 644*x^2 - 567)              # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [1, 0])                                              # optional - sage.rings.number_field
sage: isogs = isogenies_7_1728(E)                                               # optional - sage.rings.number_field
sage: [phi.codomain().j_invariant() for phi in isogs]                           # optional - sage.rings.number_field
[-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.codomain()                                                  # optional - sage.rings.number_field
sage: E2 = isogs.codomain()                                                  # optional - sage.rings.number_field
sage: E1.is_isomorphic(E2)                                                      # optional - sage.rings.number_field
False
sage: E1.is_quadratic_twist(E2)                                                 # optional - sage.rings.number_field
-1

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree(E, l, minimal_models=True)#

Return all separable l-isogenies with domain E.

INPUT:

• E – an elliptic curve.

• l – a prime.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False. Ignored except over number fields other than $$QQ$$.

OUTPUT:

A list of all separable isogenies of degree $$l$$ with domain E.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree
sage: E = EllipticCurve_from_j(GF(2^6,'a')(1))                                  # optional - sage.rings.finite_rings
sage: isogenies_prime_degree(E, 7)                                              # optional - sage.rings.finite_rings
[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))                                 # optional - sage.rings.finite_rings
sage: isogenies_prime_degree(E, 17)                                             # optional - sage.rings.finite_rings
[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):                                                 # optional - sage.rings.finite_rings
....:     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])                                # optional - sage.rings.finite_rings
sage: E.isogenies_prime_degree(71) # fast                                       # optional - sage.rings.finite_rings
[]
sage: E.isogenies_prime_degree(73)  # long time (2s)                            # optional - sage.rings.finite_rings
[]


Test that github issue #32269 is fixed:

sage: K = QuadraticField(-11)                                                   # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0,1,0,-117,-541])                                   # optional - sage.rings.number_field
sage: E.isogenies_prime_degree(37)  # long time (9s)                            # optional - sage.rings.number_field
[Isogeny of degree 37
from Elliptic Curve defined by y^2 = x^3 + x^2 + (-117)*x + (-541)
over Number Field in a with defining polynomial x^2 + 11
with a = 3.316624790355400?*I
to Elliptic Curve defined by
y^2 = x^3 + x^2 + (30800*a+123963)*x + (3931312*a-21805005)
over Number Field in a with defining polynomial x^2 + 11
with a = 3.316624790355400?*I,
Isogeny of degree 37
from Elliptic Curve defined by y^2 = x^3 + x^2 + (-117)*x + (-541)
over Number Field in a with defining polynomial x^2 + 11
with a = 3.316624790355400?*I
to Elliptic Curve defined by
y^2 = x^3 + x^2 + (-30800*a+123963)*x + (-3931312*a-21805005)
over Number Field in a with defining polynomial x^2 + 11
with a = 3.316624790355400?*I]

sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_general(E, l, minimal_models=True)#

Return all separable l-isogenies with domain E.

INPUT:

• E – an elliptic curve.

• l – a prime.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

OUTPUT:

A list of all separable isogenies of degree $$l$$ with domain E.

ALGORITHM:

This algorithm factors the l-division polynomial, then combines its factors to obtain 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))                                  # optional - sage.rings.finite_rings
sage: isogenies_prime_degree_general(E, 7)                                      # optional - sage.rings.finite_rings
[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))                                 # optional - sage.rings.finite_rings
sage: isogenies_prime_degree_general(E, 17)                                     # optional - sage.rings.finite_rings
[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 (2s)
[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 (2s)
[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):                                                 # optional - sage.rings.finite_rings
....:     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]


Note that not all factors of degree $$(l-1)/2$$ of the $$l$$-division polynomial are kernel polynomials. In this example, the 13-division polynomial factors as a product of 14 irreducible factors of degree 6 each, but only two those are kernel polynomials:

sage: F3 = GF(3)                                                                # optional - sage.rings.finite_rings
sage: E = EllipticCurve(F3, [0,0,0,-1,0])                                       # optional - sage.rings.finite_rings
sage: Psi13 = E.division_polynomial(13)                                         # optional - sage.rings.finite_rings
sage: len([f for f, e in Psi13.factor() if f.degree() == 6])                    # optional - sage.rings.finite_rings
14
sage: len(E.isogenies_prime_degree(13))                                         # optional - sage.rings.finite_rings
2


Over GF(9) the other factors of degree 6 split into pairs of cubics which can be rearranged to give the remaining 12 kernel polynomials:

sage: len(E.change_ring(GF(3^2,'a')).isogenies_prime_degree(13))                # optional - sage.rings.finite_rings
14


See github issue #18589: the following example took 20s before, now only 4s:

sage: K.<i> = QuadraticField(-1)                                                # optional - sage.rings.number_field
sage: E = EllipticCurve(K,[0,0,0,1,0])                                          # optional - sage.rings.number_field
sage: [phi.codomain().ainvs()                   # long time (6s)                # optional - sage.rings.number_field
....:  for phi in E.isogenies_prime_degree(37)]
[(0, 0, 0, -840*i + 1081, 0),
(0, 0, 0, 840*i + 1081, 0)]

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

Return list of l -isogenies with domain E.

INPUT:

• E – an elliptic curve.

• l – either None or 2, 3, 5, 7, or 13.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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, minimal_models=True)#

Return 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.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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: K = QuadraticField(-295,'a')                                              # optional - sage.rings.number_field
sage: a = K.gen()                                                               # optional - sage.rings.number_field
sage: E = EllipticCurve_from_j(-484650135/16777216*a + 4549855725/16777216)     # optional - sage.rings.number_field
sage: isogenies_prime_degree_genus_plus_0(E, 23)                                # optional - sage.rings.number_field
[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
with a = 17.17556403731767?*I
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
with a = 17.17556403731767?*I]

sage: K = QuadraticField(-199,'a')                                              # optional - sage.rings.number_field
sage: a = K.gen()                                                               # optional - sage.rings.number_field
sage: E = EllipticCurve_from_j(94743000*a + 269989875)                          # optional - sage.rings.number_field
sage: isogenies_prime_degree_genus_plus_0(E, 29)                                # optional - sage.rings.number_field
[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
with a = 14.106735979665884?*I
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
with a = 14.106735979665884?*I]

sage: K = QuadraticField(253,'a')                                               # optional - sage.rings.number_field
sage: a = K.gen()                                                               # optional - sage.rings.number_field
sage: E = EllipticCurve_from_j(208438034112000*a - 3315409892960000)            # optional - sage.rings.number_field
sage: isogenies_prime_degree_genus_plus_0(E, 31)                                # optional - sage.rings.number_field
[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
with a = 15.905973720586867?
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
with a = 15.905973720586867?]

sage: E = EllipticCurve_from_j(GF(5)(1))                                        # optional - sage.rings.finite_rings
sage: isogenies_prime_degree_genus_plus_0(E, 41)                                # optional - sage.rings.finite_rings
[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: K = QuadraticField(5,'a')                                                 # optional - sage.rings.number_field
sage: a = K.gen()                                                               # optional - sage.rings.number_field
sage: E = EllipticCurve_from_j(184068066743177379840*a                          # optional - sage.rings.number_field
....:                          - 411588709724712960000)
sage: isogenies_prime_degree_genus_plus_0(E, 47)  # long time (2s)              # optional - sage.rings.number_field
[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
with a = 2.236067977499790?
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
with a = 2.236067977499790?]

sage: K = QuadraticField(-66827,'a')                                            # optional - sage.rings.number_field
sage: a = K.gen()                                                               # optional - sage.rings.number_field
sage: E = EllipticCurve_from_j(-98669236224000*a + 4401720074240000)            # optional - sage.rings.number_field
sage: isogenies_prime_degree_genus_plus_0(E, 59)   # long time (5s)
[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
with a = 258.5091874576221?*I
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
with a = 258.5091874576221?*I]

sage: E = EllipticCurve_from_j(GF(13)(5))                                       # optional - sage.rings.finite_rings
sage: isogenies_prime_degree_genus_plus_0(E, 71)                                # optional - sage.rings.finite_rings
[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])                                    # optional - sage.rings.finite_rings
sage: isogenies_prime_degree_genus_plus_0(E)                                    # optional - sage.rings.finite_rings
[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, minimal_models=True)#

Return 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.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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)                # optional - sage.rings.number_field
sage: E = EllipticCurve(K, [0, -121/5*a^3 - 20691/5*a^2 - 29403/5*a + 3267])    # optional - sage.rings.number_field
sage: isogenies_prime_degree_genus_plus_0_j0(E, 11)                             # optional - sage.rings.number_field
[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, minimal_models=True)#

Return 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.

• minimal_models (bool, default True) – if True, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set to False.

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                             # optional - sage.rings.number_field
....:                     + 2484*u^3 - 5265*u^2 + 12150*u - 5103)
sage: E = EllipticCurve(K, [-75295/1335852*a^5 + 13066735/445284*a^4            # optional - sage.rings.number_field
....:                       + 44903485/74214*a^3 + 17086861/24738*a^2
....:                       + 11373021/16492*a - 1246245/2356, 0])
sage: isogenies_prime_degree_genus_plus_0_j1728(E, 11)                          # optional - sage.rings.number_field
[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: i = QuadraticField(-1,'i').gen()                                          # optional - sage.rings.number_field
sage: E = EllipticCurve([-1 - 2*i, 0])                                          # optional - sage.rings.number_field
sage: isogenies_prime_degree_genus_plus_0_j1728(E, 17)                          # optional - sage.rings.number_field
[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 with i = 1*I
to Elliptic Curve defined by y^2 = x^3 + (-82*i-641)*x
over Number Field in i with defining polynomial x^2 + 1 with i = 1*I,
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 with i = 1*I
to Elliptic Curve defined by y^2 = x^3 + (-562*i+319)*x
over Number Field in i with defining polynomial x^2 + 1 with i = 1*I]
sage: Emin = E.global_minimal_model()                                           # optional - sage.rings.number_field
sage: [(p, len(isogenies_prime_degree_genus_plus_0_j1728(Emin, p)))             # optional - sage.rings.number_field
....:  for p in [17, 29, 41]]
[(17, 2), (29, 2), (41, 2)]


Return 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