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].
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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 \(P(t)/t\).
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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 \(\ZZ[t]\).
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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_storedisFalse, 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.
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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
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sage.schemes.elliptic_curves.isogeny_small_degree.is_kernel_polynomial(E, m, f)¶ Test whether
Ehas a cyclic isogeny of degreemwith kernel polynomialf.INPUT:
E– an elliptic curve.m– a positive integer.f– a polynomial over the base field ofE.
OUTPUT:
(bool)
TrueifEhas a cyclic isogeny of degreemwith kernel polynomialf, elseFalse.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
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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, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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)]
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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, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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)]
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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, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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
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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, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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)] []
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sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_5_0(E, minimal_models=True)¶ Return a list of all the 5-isogenies with domain
Ewhen the j-invariant is 0.INPUT:
E– an elliptic curve with j-invariant 0.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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]
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sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_5_1728(E, minimal_models=True)¶ Return a list of 5-isogenies with domain
Ewhen the j-invariant is 1728.INPUT:
E– an elliptic curve with j-invariant 1728.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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]
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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, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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]
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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, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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
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sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree(E, l, minimal_models=True)¶ Return all separable
l-isogenies with domainE.INPUT:
E– an elliptic curve.l– a prime.minimal_models(bool, defaultTrue) – ifTrue, allcurves 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) []
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sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_general(E, l, minimal_models=True)¶ Return all separable
l-isogenies with domainE.INPUT:
E– an elliptic curve.l– a prime.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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)]
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sage.schemes.elliptic_curves.isogeny_small_degree.isogenies_prime_degree_genus_0(E, l=None, minimal_models=True)¶ Return list of
l-isogenies with domainE.INPUT:
E– an elliptic curve.l– either None or 2, 3, 5, 7, or 13.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
OUTPUT:
(list) When
lis 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]
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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 domainE.INPUT:
E– an elliptic curve.l– either None or 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
OUTPUT:
(list) When
lis 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]
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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 domainEwhen \(j(E)=0\).INPUT:
E– an elliptic curve with j-invariant 0.l– 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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]
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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 domainEwhen \(j(E)=1728\).INPUT:
E– an elliptic curve with j-invariant 1728.l– 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.minimal_models(bool, defaultTrue) – ifTrue, all curves computed will be minimal or semi-minimal models. Over fields of larger degree it can be expensive to compute these so set toFalse.
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)]
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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
lis None, a list of all isogenies with domainEand 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]