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(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 \(P(t)/t\).

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 \(\ZZ[t]\).

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.

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(l)

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: Psi2(71)  # long time (1 second)
-2209380711722505179506258739515288584116147237393815266468076436521/71*u^210 + ... - 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)
sage: E = EllipticCurve(F,[0,0,0,-1,0])
sage: f13 = E.division_polynomial(13)
sage: factors = [f for f,e in f13.factor()]
sage: all(f.degree() == 6 for f in factors)
True
sage: [is_kernel_polynomial(E,13,f) for f in factors]
[True,
True,
False,
False,
False,
False,
False,
False,
False,
False,
False,
False,
False,
False]

See trac ticket #22232:

sage: K =GF(47^2)
sage: E = EllipticCurve([0, K.gen()])
sage: psi7 = E.division_polynomial(7)
sage: f = psi7.factor()[4][0]
sage: f
x^3 + (7*z2 + 11)*x^2 + (25*z2 + 33)*x + 25*z2
sage: f.divides(psi7)
True
sage: is_kernel_polynomial(E,7, f)
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)
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 with r = 1.732050807568878?*I
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 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)[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,
  20360599/165164973653422080*a^11 - 3643073/41291243413355520*a^10 - 101/8789110986240*a^9 + 5557619461/573489491852160*a^8 - 82824971/11947697746920*a^7 - 19487/21127670640*a^6 - 475752603733/29409717530880*a^5 + 87205112531/7352429382720*a^4 + 8349/521670880*a^3 + 5858744881/12764634345*a^2 - 1858703809/2836585410*a + 58759402/48906645,
  -139861295/2650795873449984*a^11 - 3455957/5664093746688*a^10 - 345310571/50976843720192*a^9 - 500530795/118001953056*a^8 - 12860048113/265504394376*a^7 - 25007420461/44250732396*a^6 + 458134176455/1416023436672*a^5 + 16701880631/9077073312*a^4 + 155941666417/9077073312*a^3 + 3499310115/378211388*a^2 - 736774863/94552847*a - 21954102381/94552847,
  579363345221/13763747804451840*a^11 + 371192377511/860234237778240*a^10 + 8855090365657/1146978983704320*a^9 + 5367261541663/1633873196160*a^8 + 614883554332193/15930263662560*a^7 + 30485197378483/68078049840*a^6 - 131000897588387/2450809794240*a^5 - 203628705777949/306351224280*a^4 - 1587619388190379/204234149520*a^3 + 14435069706551/11346341640*a^2 + 7537273048614/472764235*a + 89198980034806/472764235),
 (0,
  0,
  20360599/165164973653422080*a^11 - 3643073/41291243413355520*a^10 - 101/8789110986240*a^9 + 5557619461/573489491852160*a^8 - 82824971/11947697746920*a^7 - 19487/21127670640*a^6 - 475752603733/29409717530880*a^5 + 87205112531/7352429382720*a^4 + 8349/521670880*a^3 + 5858744881/12764634345*a^2 - 1858703809/2836585410*a + 58759402/48906645,
  -6465569317/1325397936724992*a^11 - 112132307/1960647835392*a^10 - 17075412917/25488421860096*a^9 - 207832519229/531008788752*a^8 - 1218275067617/265504394376*a^7 - 9513766502551/177002929584*a^6 + 4297077855437/708011718336*a^5 + 354485975837/4538536656*a^4 + 4199379308059/4538536656*a^3 - 30841577919/189105694*a^2 - 181916484042/94552847*a - 2135779171614/94552847,
  -132601797212627/3440936951112960*a^11 - 6212467020502021/13763747804451840*a^10 - 1515926454902497/286744745926080*a^9 - 15154913741799637/4901619588480*a^8 - 576888119803859263/15930263662560*a^7 - 86626751639648671/204234149520*a^6 + 16436657569218427/306351224280*a^5 + 1540027900265659087/2450809794240*a^4 + 375782662805915809/51058537380*a^3 - 14831920924677883/11346341640*a^2 - 7237947774817724/472764235*a - 84773764066089509/472764235)]

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)
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 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)
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,
-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
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, 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])
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, 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])
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 + y = x^3 + (643/8*a^5-15779/48*a^4-32939/24*a^3-71989/2*a^2+214321/6*a-112115/3)*x + (2901961/96*a^5+4045805/48*a^4+12594215/18*a^3-30029635/6*a^2+15341626/3*a-38944312/9) 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 + (-1109/8*a^5-53873/48*a^4-180281/24*a^3-14491/2*a^2+35899/6*a-43745/3)*x + (-17790679/96*a^5-60439571/48*a^4-77680504/9*a^3+1286245/6*a^2-4961854/3*a-73854632/9) 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])
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 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: _[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 + (-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]

See trac ticket #19840:

sage: K.<a> = NumberField(x^4 - 5*x^2 + 5)
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))
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)]
[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)
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 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])
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: E = EllipticCurve('27a1').change_ring(QuadraticField(-3,'r'))
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 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: 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,
  -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]
[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])
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().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
sage: E1.is_quadratic_twist(E2)
-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))
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) # slower (2s)
[]

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))
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]

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

See trac ticket #18589: the following example took 20s before, now only 4s:

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve(K,[0,0,0,1,0])
sage: [phi.codomain().ainvs() for phi in E.isogenies_prime_degree(37)] # long time
[(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:

Cremona and Watkins [CW2005]. See also [KT2013], Chapter 4.

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')
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 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')
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 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')
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 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))
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: K = QuadraticField(5,'a')
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 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')
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 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))
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, 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)
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, 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 + 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: i = QuadraticField(-1,'i').gen()
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 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()
sage: [(p,len(isogenies_prime_degree_genus_plus_0_j1728(Emin,p))) for p in [17, 29, 41]]
[(17, 2), (29, 2), (41, 2)]

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

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')
sage: isogenies_sporadic_Q(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: isogenies_sporadic_Q(E, 13)
[]
sage: isogenies_sporadic_Q(E, 17)
[]
sage: isogenies_sporadic_Q(E)
[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_sporadic_Q(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: isogenies_sporadic_Q(E)
[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: isogenies_sporadic_Q(E, 11)
[]

sage: E = EllipticCurve([0, 0, 1, -1862, -30956])
sage: isogenies_sporadic_Q(E, 11)
[]
sage: isogenies_sporadic_Q(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: isogenies_sporadic_Q(E)
[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
sage: isogenies_sporadic_Q(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]

sage: E = EllipticCurve([1, 1, 0, -25178045, 48616918750])
sage: E.conductor()
148225
sage: isogenies_sporadic_Q(E,37)
[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
sage: isogenies_sporadic_Q(E,43)
[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
sage: isogenies_sporadic_Q(E,67)
[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
sage: isogenies_sporadic_Q(E,163)
[Isogeny of degree 163 from Elliptic Curve defined by y^2 = x^3 - 34790720*x - 78984748304 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 924354639680*x + 342062961763303088 over Rational Field]