Functions for the computation of isogenies of small primes
degree. First: = 2, 3, 5, 7, or 13, where the modular curve
has genus 0. Second:
= 11, 17, 19, 23, 29, 31, 41, 47,
59, or 71, where
has genus 0 and
is elliptic or
hyperelliptic. Also:
= 11, 17, 19, 37, 43, 67 or 163 over
(the sporadic cases with only finitely many
-invariants each). All
the above only require factorization of a polynomial of degree
.
Finally, a generic function which works for arbitrary odd primes
(including the characteristic), but requires factorization of the
-division polynomial, of degree
.
AUTHORS:
REFERENCES:
[CW2005] |
|
[KT2013] | (1, 2, 3, 4) K. Tsukazaki, Explicit Isogenies of Elliptic Curves, PhD thesis, University of Warwick, 2013. |
Fricke module for l =2,3,5,7,13.
For these primes (and these only) the modular curve has
genus zero, and its field is generated by a single modular
function called the Fricke module (or Hauptmodul),
. There is
a classical choice of such a generator
in each case, and the
-function is a rational function of
of degree
of the
form
where
is a polynomial of degree
. Up to
scaling,
is determined by the condition that the ramification
points above
are
(with ramification degree
)
and
(with degree
). The ramification above
and
may be seen in the factorizations of
and
where
.
OUTPUT:
The rational function .
TESTS:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Fricke_module
sage: Fricke_module(2)
(t^3 + 48*t^2 + 768*t + 4096)/t
sage: Fricke_module(3)
(t^4 + 36*t^3 + 270*t^2 + 756*t + 729)/t
sage: Fricke_module(5)
(t^6 + 30*t^5 + 315*t^4 + 1300*t^3 + 1575*t^2 + 750*t + 125)/t
sage: Fricke_module(7)
(t^8 + 28*t^7 + 322*t^6 + 1904*t^5 + 5915*t^4 + 8624*t^3 + 4018*t^2 + 748*t + 49)/t
sage: Fricke_module(13)
(t^14 + 26*t^13 + 325*t^12 + 2548*t^11 + 13832*t^10 + 54340*t^9 + 157118*t^8 + 333580*t^7 + 509366*t^6 + 534820*t^5 + 354536*t^4 + 124852*t^3 + 15145*t^2 + 746*t + 13)/t
Fricke polynomial for l =2,3,5,7,13.
For these primes (and these only) the modular curve has
genus zero, and its field is generated by a single modular
function called the Fricke module (or Hauptmodul),
. There is
a classical choice of such a generator
in each case, and the
-function is a rational function of
of degree
of the
form
where
is a polynomial of degree
. Up to
scaling,
is determined by the condition that the ramification
points above
are
(with ramification degree
)
and
(with degree
). The ramification above
and
may be seen in the factorizations of
and
where
.
OUTPUT:
The polynomial as an element of
.
TESTS:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Fricke_polynomial
sage: Fricke_polynomial(2)
t^3 + 48*t^2 + 768*t + 4096
sage: Fricke_polynomial(3)
t^4 + 36*t^3 + 270*t^2 + 756*t + 729
sage: Fricke_polynomial(5)
t^6 + 30*t^5 + 315*t^4 + 1300*t^3 + 1575*t^2 + 750*t + 125
sage: Fricke_polynomial(7)
t^8 + 28*t^7 + 322*t^6 + 1904*t^5 + 5915*t^4 + 8624*t^3 + 4018*t^2 + 748*t + 49
sage: Fricke_polynomial(13)
t^14 + 26*t^13 + 325*t^12 + 2548*t^11 + 13832*t^10 + 54340*t^9 + 157118*t^8 + 333580*t^7 + 509366*t^6 + 534820*t^5 + 354536*t^4 + 124852*t^3 + 15145*t^2 + 746*t + 13
Generic kernel polynomial for genus zero primes.
For each of the primes for which
has genus zero
(namely
), we may define an elliptic curve
over
, with coefficients in
, which has good
reduction except at
and
(which lie above
) and at certain other values of
above
when
(one value) or
(two values) and above
when
(one value) or
(two
values). (These exceptional values correspond to endomorphisms of
of degree
.) The
-division polynomial of
has a
unique factor of degree
(or 1 when
), with
coefficients in
, which we call the Generic Kernel
Polynomial for
. These are used, by specialising
, in the
function isogenies_prime_degree_genus_0(), which also has to
take into account the twisting factor between
for a specific
value of
and the short Weierstrass form of an elliptic curve
with
-invariant
. 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 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:
TESTS:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Fricke_module, Psi
sage: assert Psi(2, use_stored=True) == Psi(2, use_stored=False)
sage: assert Psi(3, use_stored=True) == Psi(3, use_stored=False)
sage: assert Psi(5, use_stored=True) == Psi(5, use_stored=False)
sage: assert Psi(7, use_stored=True) == Psi(7, use_stored=False)
sage: assert Psi(13, use_stored=True) == Psi(13, use_stored=False) # not tested (very long time)
Returns the generic kernel polynomial for hyperelliptic -isogenies.
INPUT:
OUTPUT:
The generic -kernel polynomial.
TESTS:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import Psi2
sage: Psi2(11)
x^5 - 55*x^4*u + 994*x^3*u^2 - 8774*x^2*u^3 + 41453*x*u^4 - 928945/11*u^5 + 33*x^4 + 276*x^3*u - 7794*x^2*u^2 + 4452*x*u^3 + 1319331/11*u^4 + 216*x^3*v - 4536*x^2*u*v + 31752*x*u^2*v - 842616/11*u^3*v + 162*x^3 + 38718*x^2*u - 610578*x*u^2 + 33434694/11*u^3 - 4536*x^2*v + 73872*x*u*v - 2745576/11*u^2*v - 16470*x^2 + 580068*x*u - 67821354/11*u^2 - 185976*x*v + 14143896/11*u*v + 7533*x - 20437029/11*u - 12389112/11*v + 19964151/11
Returns list of all 13-isogenies from E when the j-invariant is 0.
OUTPUT:
(list) 13-isogenies with codomain E. In general these are
normalised; but if is a square then there are two
endomorphisms of degree
, for which the codomain is the same
as the domain.
Note
This implementation requires that the characteristic is not 2, 3 or 13.
Note
This function would normally be invoked indirectly via E.isogenies_prime_degree(13).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_13_0
Endomorphisms of degree 13 will exist when -3 is a square:
sage: K.<r> = QuadraticField(-3)
sage: E = EllipticCurve(K, [0, r]); E
Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3
sage: isogenies_13_0(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + r over Number Field in r with defining polynomial x^2 + 3]
sage: isogenies_13_0(E)[0].rational_maps()
(((7/338*r + 23/338)*x^13 + (-164/13*r - 420/13)*x^10 + (720/13*r + 3168/13)*x^7 + (3840/13*r - 576/13)*x^4 + (4608/13*r + 2304/13)*x)/(x^12 + (4*r + 36)*x^9 + (1080/13*r + 3816/13)*x^6 + (2112/13*r - 5184/13)*x^3 + (-17280/169*r - 1152/169)), ((18/2197*r + 35/2197)*x^18*y + (23142/2197*r + 35478/2197)*x^15*y + (-1127520/2197*r - 1559664/2197)*x^12*y + (-87744/2197*r + 5992704/2197)*x^9*y + (-6625152/2197*r - 9085824/2197)*x^6*y + (-28919808/2197*r - 2239488/2197)*x^3*y + (-1990656/2197*r - 3870720/2197)*y)/(x^18 + (6*r + 54)*x^15 + (3024/13*r + 11808/13)*x^12 + (31296/13*r + 51840/13)*x^9 + (487296/169*r - 2070144/169)*x^6 + (-940032/169*r + 248832/169)*x^3 + (1990656/2197*r + 3870720/2197)))
An example of endomorphisms over a finite field:
sage: K = GF(19^2,'a')
sage: E = EllipticCurve(j=K(0)); E
Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2
sage: isogenies_13_0(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field in a of size 19^2]
sage: isogenies_13_0(E)[0].rational_maps()
((6*x^13 - 6*x^10 - 3*x^7 + 6*x^4 + x)/(x^12 - 5*x^9 - 9*x^6 - 7*x^3 + 5), (-8*x^18*y - 9*x^15*y + 9*x^12*y - 5*x^9*y + 5*x^6*y - 7*x^3*y + 7*y)/(x^18 + 2*x^15 + 3*x^12 - x^9 + 8*x^6 - 9*x^3 + 7))
A previous implementation did not work in some characteristics:
sage: K = GF(29)
sage: E = EllipticCurve(j=K(0))
sage: isogenies_13_0(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 29 to Elliptic Curve defined by y^2 = x^3 + 26*x + 12 over Finite Field of size 29, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 29 to Elliptic Curve defined by y^2 = x^3 + 16*x + 28 over Finite Field of size 29]
sage: K = GF(101)
sage: E = EllipticCurve(j=K(0)); E.ainvs()
(0, 0, 0, 0, 1)
sage: [phi.codomain().ainvs() for phi in isogenies_13_0(E)]
[(0, 0, 0, 64, 36), (0, 0, 0, 42, 66)]
sage: x = polygen(QQ)
sage: f = x^12 + 78624*x^9 - 130308048*x^6 + 2270840832*x^3 - 54500179968
sage: K.<a> = NumberField(f)
sage: E = EllipticCurve(j=K(0)); E.ainvs()
(0, 0, 0, 0, 1)
sage: [phi.codomain().ainvs() for phi in isogenies_13_0(E)]
[(0, 0, 0, -739946459/23857162861049856*a^11 - 2591641747/1062017577504*a^8 + 16583647773233/4248070310016*a^5 - 14310911337/378211388*a^2, 26146225/4248070310016*a^9 + 7327668845/14750244132*a^6 + 174618431365/756422776*a^3 - 378332499709/94552847), (0, 0, 0, 3501275/5964290715262464*a^11 + 24721025/531008788752*a^8 - 47974903745/1062017577504*a^5 - 6773483100/94552847*a^2, 6699581/4248070310016*a^9 + 1826193509/14750244132*a^6 - 182763866047/756422776*a^3 - 321460597/94552847)]
Returns list of all 13-isogenies from E when the j-invariant is 1728.
OUTPUT:
(list) 13-isogenies with codomain E. In general these are
normalised; but if is a square then there are two
endomorphisms of degree
, for which the codomain is the same
as the domain; and over
, the codomain is a minimal model.
Note
This implementation requires that the characteristic is not 2, 3 or 13.
Note
This function would normally be invoked indirectly via E.isogenies_prime_degree(13).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_13_1728
sage: 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 to Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + i*x over Number Field in i with defining polynomial x^2 + 1]
sage: K = GF(83)
sage: E = EllipticCurve(K, [0,0,0,5,0]); E.ainvs()
(0, 0, 0, 5, 0)
sage: isogenies_13_1728(E)
[]
sage: K = GF(89)
sage: E = EllipticCurve(K, [0,0,0,5,0]); E.ainvs()
(0, 0, 0, 5, 0)
sage: isogenies_13_1728(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89 to Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89 to Elliptic Curve defined by y^2 = x^3 + 5*x over Finite Field of size 89]
sage: K = GF(23)
sage: E = EllipticCurve(K, [1,0])
sage: isogenies_13_1728(E)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 23 to Elliptic Curve defined by y^2 = x^3 + 16 over Finite Field of size 23, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 23 to Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 23]
sage: x = polygen(QQ)
sage: f = x^12 + 1092*x^10 - 432432*x^8 + 6641024*x^6 - 282896640*x^4 - 149879808*x^2 - 349360128
sage: K.<a> = NumberField(f)
sage: E = EllipticCurve(K, [1,0])
sage: [phi.codomain().ainvs() for phi in isogenies_13_1728(E)]
[(0,
0,
0,
11090413835/20943727039698624*a^10 + 32280103535965/55849938772529664*a^8 - 355655987835845/1551387188125824*a^6 + 19216954517530195/5235931759924656*a^4 - 1079766118721735/5936430566808*a^2 + 156413528482727/8080141604822,
214217013065/82065216155553792*a^11 + 1217882637605/427423000810176*a^9 - 214645003230565/189965778137856*a^7 + 22973355421236025/1282269002430528*a^5 - 2059145797340695/2544184528632*a^3 - 23198483147321/989405094468*a),
(0,
0,
0,
11090413835/20943727039698624*a^10 + 32280103535965/55849938772529664*a^8 - 355655987835845/1551387188125824*a^6 + 19216954517530195/5235931759924656*a^4 - 1079766118721735/5936430566808*a^2 + 156413528482727/8080141604822,
-214217013065/82065216155553792*a^11 - 1217882637605/427423000810176*a^9 + 214645003230565/189965778137856*a^7 - 22973355421236025/1282269002430528*a^5 + 2059145797340695/2544184528632*a^3 + 23198483147321/989405094468*a)]
Returns a list of all 2-isogenies with domain E.
INPUT:
OUTPUT:
(list) 2-isogenies with domain E. In general these are
normalised, but over the codomain is a minimal model.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_2
sage: E = EllipticCurve('14a1'); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field
sage: [phi.codomain().ainvs() for phi in isogenies_2(E)]
[(1, 0, 1, -36, -70)]
sage: E = EllipticCurve([1,2,3,4,5]); E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
sage: [phi.codomain().ainvs() for phi in isogenies_2(E)]
[]
sage: E = EllipticCurve(QQbar, [9,8]); E
Elliptic Curve defined by y^2 = x^3 + 9*x + 8 over Algebraic Field
sage: isogenies_2(E) # not implemented
Returns a list of all 3-isogenies with domain E.
INPUT:
OUTPUT:
(list) 3-isogenies with domain E. In general these are
normalised, but over the codomain is a minimal model.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_3
sage: E = EllipticCurve(GF(17), [1,1])
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[(0, 0, 0, 9, 7), (0, 0, 0, 0, 1)]
sage: E = EllipticCurve(GF(17^2,'a'), [1,1])
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[(0, 0, 0, 9, 7), (0, 0, 0, 0, 1), (0, 0, 0, 5*a + 1, a + 13), (0, 0, 0, 12*a + 6, 16*a + 14)]
sage: E = EllipticCurve('19a1')
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[(0, 1, 1, 1, 0), (0, 1, 1, -769, -8470)]
sage: E = EllipticCurve([1,1])
sage: [phi.codomain().ainvs() for phi in isogenies_3(E)]
[]
Returns a list of all the 5-isogenies with domain E when the j-invariant is 0.
OUTPUT:
(list) 5-isogenies with codomain E. In general these are
normalised, but over the codomain is a minimal model.
Note
This implementation requires that the characteristic is not 2, 3 or 5.
Note
This function would normally be invoked indirectly via E.isogenies_prime_degree(5).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_5_0
sage: E = EllipticCurve([0,12])
sage: isogenies_5_0(E)
[]
sage: E = EllipticCurve(GF(13^2,'a'),[0,-3])
sage: isogenies_5_0(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (4*a+6)*x + (2*a+10) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (12*a+5)*x + (2*a+10) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (10*a+2)*x + (2*a+10) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (3*a+12)*x + (11*a+12) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (a+4)*x + (11*a+12) over Finite Field in a of size 13^2, Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + (9*a+10)*x + (11*a+12) over Finite Field in a of size 13^2]
sage: K.<a> = NumberField(x**6-320*x**3-320)
sage: E = EllipticCurve(K,[0,0,1,0,0])
sage: isogenies_5_0(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 = x^3 + (a^5-400*a^2)*x + (280*a^3-3120) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320,
Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^6 - 320*x^3 - 320 to Elliptic Curve defined by y^2 = x^3 + (23/2*a^5-3700*a^2)*x + (-280*a^3+86480) over Number Field in a with defining polynomial x^6 - 320*x^3 - 320]
Returns a list of 5-isogenies with domain E when the j-invariant is 1728.
OUTPUT:
(list) 5-isogenies with codomain E. In general these are
normalised; but if is a square then there are two
endomorphisms of degree
, for which the codomain is the same as
the domain curve; and over
, the codomain is a minimal model.
Note
This implementation requires that the characteristic is not 2, 3 or 5.
Note
This function would normally be invoked indirectly via E.isogenies_prime_degree(5).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_5_1728
sage: E = EllipticCurve([7,0])
sage: isogenies_5_1728(E)
[]
sage: E = EllipticCurve(GF(13),[11,0])
sage: isogenies_5_1728(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13,
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 11*x over Finite Field of size 13]
An example of endomorphisms of degree 5:
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve(K,[0,0,0,1,0])
sage: isogenies_5_1728(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1,
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1]
sage: _[0].rational_maps()
(((4/25*i + 3/25)*x^5 + (4/5*i - 2/5)*x^3 - x)/(x^4 + (-4/5*i + 2/5)*x^2 + (-4/25*i - 3/25)),
((11/125*i + 2/125)*x^6*y + (-23/125*i + 64/125)*x^4*y + (141/125*i + 162/125)*x^2*y + (3/25*i - 4/25)*y)/(x^6 + (-6/5*i + 3/5)*x^4 + (-12/25*i - 9/25)*x^2 + (2/125*i - 11/125)))
An example of 5-isogenies over a number field:
sage: K.<a> = NumberField(x**4+20*x**2-80)
sage: K(5).is_square() #necessary but not sufficient!
True
sage: E = EllipticCurve(K,[0,0,0,1,0])
sage: isogenies_5_1728(E)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in a with defining polynomial x^4 + 20*x^2 - 80 to Elliptic Curve defined by y^2 = x^3 + (-20*a^2-39)*x + (35*a^3+112*a) over Number Field in a with defining polynomial x^4 + 20*x^2 - 80,
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in a with defining polynomial x^4 + 20*x^2 - 80 to Elliptic Curve defined by y^2 = x^3 + (-20*a^2-39)*x + (-35*a^3-112*a) over Number Field in a with defining polynomial x^4 + 20*x^2 - 80]
Returns list of all 7-isogenies from E when the j-invariant is 0.
OUTPUT:
(list) 7-isogenies with codomain E. In general these are
normalised; but if is a square then there are two
endomorphisms of degree
, for which the codomain is the same as
the domain; and over
, the codomain is a minimal model.
Note
This implementation requires that the characteristic is not 2, 3 or 7.
Note
This function would normally be invoked indirectly via E.isogenies_prime_degree(7).
EXAMPLES:
First some examples of endomorphisms:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_0
sage: 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 to Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3,
Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in r with defining polynomial x^2 + 3]
sage: E = EllipticCurve(GF(13^2,'a'),[0,-3])
sage: isogenies_7_0(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2 to Elliptic Curve defined by y^2 = x^3 + 10 over Finite Field in a of size 13^2]
Now some examples of 7-isogenies which are not endomorphisms:
sage: K = GF(101)
sage: E = EllipticCurve(K, [0,1])
sage: isogenies_7_0(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 101 to Elliptic Curve defined by y^2 = x^3 + 55*x + 100 over Finite Field of size 101, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 101 to Elliptic Curve defined by y^2 = x^3 + 83*x + 26 over Finite Field of size 101]
Examples over a number field:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_0
sage: 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 to Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3,
Isogeny of degree 7 from Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3 to Elliptic Curve defined by y^2 + y = x^3 + (-7) over Number Field in r with defining polynomial x^2 + 3]
sage: K.<a> = NumberField(x^6 + 1512*x^3 - 21168)
sage: E = EllipticCurve(K, [0,1])
sage: isogs = isogenies_7_0(E)
sage: [phi.codomain().a_invariants() for phi in isogs]
[(0, 0, 0, -5/294*a^5 - 300/7*a^2, -55/2*a^3 - 1133),
(0, 0, 0, -295/1176*a^5 - 5385/14*a^2, 55/2*a^3 + 40447)]
sage: [phi.codomain().j_invariant() for phi in isogs]
[158428486656000/7*a^3 - 313976217600000,
-158428486656000/7*a^3 - 34534529335296000]
Returns list of all 7-isogenies from E when the j-invariant is 1728.
OUTPUT:
(list) 7-isogenies with codomain E. In general these are
normalised; but over the codomain is a minimal model.
Note
This implementation requires that the characteristic is not 2, 3, or 7.
Note
This function would normally be invoked indirectly via E.isogenies_prime_degree(7).
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_1728
sage: E = EllipticCurve(GF(47), [1, 0])
sage: isogenies_7_1728(E)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 47 to Elliptic Curve defined by y^2 = x^3 + 26 over Finite Field of size 47,
Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 47 to Elliptic Curve defined by y^2 = x^3 + 21 over Finite Field of size 47]
An example in characteristic 53 (for which an earlier implementation did not work):
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_7_1728
sage: E = EllipticCurve(GF(53), [1, 0])
sage: isogenies_7_1728(E)
[]
sage: E = EllipticCurve(GF(53^2,'a'), [1, 0])
sage: [iso.codomain().ainvs() for iso in isogenies_7_1728(E)]
[(0, 0, 0, 36, 19*a + 15), (0, 0, 0, 36, 34*a + 38), (0, 0, 0, 33, 39*a + 28), (0, 0, 0, 33, 14*a + 25), (0, 0, 0, 19, 45*a + 16), (0, 0, 0, 19, 8*a + 37), (0, 0, 0, 3, 45*a + 16), (0, 0, 0, 3, 8*a + 37)]
sage: K.<a> = NumberField(x^8 + 84*x^6 - 1890*x^4 + 644*x^2 - 567)
sage: E = EllipticCurve(K, [1, 0])
sage: isogs = isogenies_7_1728(E)
sage: [phi.codomain().a_invariants() for phi in isogs]
[(0,
0,
0,
35/636*a^6 + 55/12*a^4 - 79135/636*a^2 + 1127/212,
155/636*a^7 + 245/12*a^5 - 313355/636*a^3 - 3577/636*a),
(0,
0,
0,
35/636*a^6 + 55/12*a^4 - 79135/636*a^2 + 1127/212,
-155/636*a^7 - 245/12*a^5 + 313355/636*a^3 + 3577/636*a)]
sage: [phi.codomain().j_invariant() for phi in isogs]
[-526110256146528/53*a^6 + 183649373229024*a^4 - 3333881559996576/53*a^2 + 2910267397643616/53,
-526110256146528/53*a^6 + 183649373229024*a^4 - 3333881559996576/53*a^2 + 2910267397643616/53]
sage: E1 = isogs[0].codomain()
sage: E2 = isogs[1].codomain()
sage: E1.is_isomorphic(E2)
False
sage: E1.is_quadratic_twist(E2)
-1
Returns a list of l -isogenies with domain E.
INPUT:
OUTPUT:
(list) a list of all isogenies of degree . If the
characteristic is
then only separable isogenies are
constructed.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree
sage: E = EllipticCurve_from_j(GF(2^6,'a')(1))
sage: isogenies_prime_degree(E, 7)
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Finite Field in a of size 2^6 to Elliptic Curve defined by y^2 + x*y = x^3 + x over Finite Field in a of size 2^6]
sage: E = EllipticCurve_from_j(GF(3^12,'a')(2))
sage: isogenies_prime_degree(E, 17)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2 over Finite Field in a of size 3^12 to Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x over Finite Field in a of size 3^12, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2 over Finite Field in a of size 3^12 to Elliptic Curve defined by y^2 = x^3 + 2*x^2 + x + 2 over Finite Field in a of size 3^12]
sage: E = EllipticCurve('50a1')
sage: isogenies_prime_degree(E, 3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 126*x - 552 over Rational Field]
sage: isogenies_prime_degree(E, 5)
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 76*x + 298 over Rational Field]
sage: E = EllipticCurve([0, 0, 1, -1862, -30956])
sage: isogenies_prime_degree(E, 19)
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 - 1862*x - 30956 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 672182*x + 212325489 over Rational Field]
sage: E = EllipticCurve([0, -1, 0, -6288, 211072])
sage: isogenies_prime_degree(E, 37)
[Isogeny of degree 37 from Elliptic Curve defined by y^2 = x^3 - x^2 - 6288*x + 211072 over Rational Field to Elliptic Curve defined by y^2 = x^3 - x^2 - 163137088*x - 801950801728 over Rational Field]
Isogenies of degree equal to the characteristic are computed (but only the separable isogeny). In the following example we consider an elliptic curve which is supersingular in characteristic 2 only:
sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_prime_degree
sage: ainvs = (0,1,1,-1,-1)
sage: for l in prime_range(50):
....: E = EllipticCurve(GF(l),ainvs)
....: isogenies_prime_degree(E,l)
[]
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 2*x + 2 over Finite Field of size 3 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 3]
[Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4*x + 4 over Finite Field of size 5 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4*x + 4 over Finite Field of size 5]
[Isogeny of degree 7 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 6*x + 6 over Finite Field of size 7 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 4 over Finite Field of size 7]
[Isogeny of degree 11 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 10*x + 10 over Finite Field of size 11 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + x + 1 over Finite Field of size 11]
[Isogeny of degree 13 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 12 over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 12 over Finite Field of size 13]
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 16*x + 16 over Finite Field of size 17 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 15 over Finite Field of size 17]
[Isogeny of degree 19 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 18*x + 18 over Finite Field of size 19 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 3*x + 12 over Finite Field of size 19]
[Isogeny of degree 23 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 22*x + 22 over Finite Field of size 23 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 22*x + 22 over Finite Field of size 23]
[Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 28*x + 28 over Finite Field of size 29 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 7*x + 27 over Finite Field of size 29]
[Isogeny of degree 31 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 30*x + 30 over Finite Field of size 31 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 15*x + 16 over Finite Field of size 31]
[Isogeny of degree 37 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 36*x + 36 over Finite Field of size 37 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 16*x + 17 over Finite Field of size 37]
[Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 40*x + 40 over Finite Field of size 41 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 33*x + 16 over Finite Field of size 41]
[Isogeny of degree 43 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 42*x + 42 over Finite Field of size 43 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 36 over Finite Field of size 43]
[Isogeny of degree 47 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + 46*x + 46 over Finite Field of size 47 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 42*x + 34 over Finite Field of size 47]
Note that the computation is faster for degrees equal to one of the genus 0 primes (2, 3, 5, 7, 13) or one of the hyperelliptic primes (11, 17, 19, 23, 29, 31, 41, 47, 59, 71) than when the generic code must be used:
sage: E = EllipticCurve(GF(101), [-3440, 77658])
sage: E.isogenies_prime_degree(71) # fast
[]
sage: E.isogenies_prime_degree(73) # not tested (very long time: 32s)
[]
Returns a list of l -isogenies with domain E.
INPUT:
OUTPUT:
(list) a list of all isogenies of degree l.
ALGORITHM:
This algorithm factors the l-division polynomial, then combines its factors to otain kernels. See [KT2013], Chapter 3.
Note
This function works for any prime . 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]
Returns list of l -isogenies with domain E.
INPUT:
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]
Returns list of l -isogenies with domain E.
INPUT:
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 to Elliptic Curve defined by y^2 = x^3 + (-5130542435555445498495/140737488355328*a+173233955029127361005925/140737488355328)*x + (-1104699335561165691575396879260545/590295810358705651712*a+3169785826904210171629535101419675/590295810358705651712) over Number Field in a with defining polynomial x^2 + 295]
sage: 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 to Elliptic Curve defined by y^2 = x^3 + (251336161378040805000*a-3071093219933084341875)*x + (-8411064283162168580187643221000*a+34804337770798389546017184785250) over Number Field in a with defining polynomial x^2 + 199]
sage: 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 to Elliptic Curve defined by y^2 = x^3 + (200339763852548615776123686912000*a-3186599019027216904280948275200000)*x + (7443671791411479629112717260182286294850207744000*a-118398847898864757209685951728838895495168655360000) over Number Field in a with defining polynomial x^2 - 253]
sage: E = EllipticCurve_from_j(GF(5)(1))
sage: isogenies_prime_degree_genus_plus_0(E, 41)
[Isogeny of degree 41 from Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 3 over Finite Field of size 5, Isogeny of degree 41 from Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 3 over Finite Field of size 5]
sage: 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 to Elliptic Curve defined by y^2 = x^3 + (39533118442361013730577638493616965245992960*a-88398740199669828340617478832005245173760000)*x + (214030321479466610282320528611562368963830105830555363061803253760*a-478586348074220699687616322532666163722004497458452316582576128000) over Number Field in a with defining polynomial x^2 - 5]
sage: 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 to Elliptic Curve defined by y^2 = x^3 + (-19387084027159786821400775098368000*a-4882059104868154225052787156713472000)*x + (-25659862010101415428713331477227179429538847260672000*a-2596038148441293485938798119003462972840818381946880000) over Number Field in a with defining polynomial x^2 + 66827]
sage: E = EllipticCurve_from_j(GF(13)(5))
sage: isogenies_prime_degree_genus_plus_0(E, 71) # long time
[Isogeny of degree 71 from Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 10*x + 7 over Finite Field of size 13, Isogeny of degree 71 from Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 13 to Elliptic Curve defined by y^2 = x^3 + 10*x + 7 over Finite Field of size 13]
sage: E = EllipticCurve(GF(13),[0,1,1,1,0])
sage: isogenies_prime_degree_genus_plus_0(E)
[Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 10*x + 1 over Finite Field of size 13,
Isogeny of degree 17 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 4 over Finite Field of size 13,
Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 6 over Finite Field of size 13,
Isogeny of degree 29 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 5*x + 6 over Finite Field of size 13,
Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 12*x + 4 over Finite Field of size 13,
Isogeny of degree 41 from Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 13 to Elliptic Curve defined by y^2 + y = x^3 + x^2 + 5*x + 6 over Finite Field of size 13]
Returns a list of hyperelliptic l -isogenies with domain E when .
INPUT:
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]
Returns a list of l -isogenies with domain E when .
INPUT:
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 to Elliptic Curve defined by y^2 = x^3 + (-82*i-641)*x over Number Field in i with defining polynomial x^2 + 1,
Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + (-2*i-1)*x over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-562*i+319)*x over Number Field in i with defining polynomial x^2 + 1]
sage: Emin = E.global_minimal_model()
sage: [(p,len(isogenies_prime_degree_genus_plus_0_j1728(Emin,p))) for p in [17, 29, 41]]
[(17, 2), (29, 2), (41, 2)]
Returns list of l -isogenies with domain E (defined over ).
Returns a list of sporadic l-isogenies from E (l = 11, 17, 19, 37,
43, 67 or 163). Only for elliptic curves over .
INPUT:
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]