def systems_of_eigenvalues(self, bound):
"""
Return all systems of eigenvalues for self for primes
up to bound.
EXAMPLES::
sage: numerical_eigenforms(61).systems_of_eigenvalues(10)
[
[-1.48119430409..., 0.806063433525..., 3.15632517466..., 0.675130870567...],
[-1.0..., -2.0..., -3.0..., 1.0...],
[0.311107817466..., 2.90321192591..., -2.52542756084..., -3.21431974338...],
[2.17008648663..., -1.70927535944..., -1.63089761382..., -0.460811127189...],
[3.0, 4.0, 6.0, 8.0]
]
"""
P = prime_range(bound)
e = self.eigenvalues(P)
v = Sequence([], cr=True)
if len(e) == 0:
return v
for i in range(len(e[0])):
v.append([e[j][i] for j in range(len(e))])
v.sort()
v.set_immutable()
return v
def systems_of_abs(self, bound):
"""
Return the absolute values of all systems of eigenvalues for
self for primes up to bound.
EXAMPLES::
sage: numerical_eigenforms(61).systems_of_abs(10)
[
[0.311107817466, 2.90321192591, 2.52542756084, 3.21431974338],
[1.0, 2.0, 3.0, 1.0],
[1.48119430409, 0.806063433525, 3.15632517466, 0.675130870567],
[2.17008648663, 1.70927535944, 1.63089761382, 0.460811127189],
[3.0, 4.0, 6.0, 8.0]
]
"""
P = prime_range(bound)
e = self.eigenvalues(P)
v = Sequence([], cr=True)
if len(e) == 0:
return v
for i in range(len(e[0])):
v.append([abs(e[j][i]) for j in range(len(e))])
v.sort()
v.set_immutable()
return v
def systems_of_abs(self, bound):
"""
Return the absolute values of all systems of eigenvalues for
self for primes up to bound.
EXAMPLES::
sage: numerical_eigenforms(61).systems_of_abs(10) # rel tol 6e-14
[
[0.3111078174659775, 2.903211925911551, 2.525427560843529, 3.214319743377552],
[1.0, 2.0000000000000027, 3.000000000000003, 1.0000000000000044],
[1.4811943040920152, 0.8060634335253695, 3.1563251746586642, 0.6751308705666477],
[2.170086486626034, 1.7092753594369208, 1.63089761381512, 0.46081112718908984],
[3.0, 4.0, 6.0, 8.0]
]
"""
P = prime_range(bound)
e = self.eigenvalues(P)
v = Sequence([], cr=True)
if len(e) == 0:
return v
for i in range(len(e[0])):
v.append([abs(e[j][i]) for j in range(len(e))])
v.sort()
v.set_immutable()
return v
def systems_of_abs(self, bound):
"""
Return the absolute values of all systems of eigenvalues for
self for primes up to bound.
EXAMPLES::
sage: numerical_eigenforms(61).systems_of_abs(10)
[
[0.311107817466, 2.90321192591, 2.52542756084, 3.21431974338],
[1.0, 2.0, 3.0, 1.0],
[1.48119430409, 0.806063433525, 3.15632517466, 0.675130870567],
[2.17008648663, 1.70927535944, 1.63089761382, 0.460811127189],
[3.0, 4.0, 6.0, 8.0]
]
"""
P = prime_range(bound)
e = self.eigenvalues(P)
v = Sequence([], cr=True)
if len(e) == 0:
return v
for i in range(len(e[0])):
v.append([abs(e[j][i]) for j in range(len(e))])
v.sort()
v.set_immutable()
return v
def systems_of_eigenvalues(self, bound):
"""
Return all systems of eigenvalues for self for primes
up to bound.
EXAMPLES::
sage: numerical_eigenforms(61).systems_of_eigenvalues(10)
[
[-1.48119430409..., 0.806063433525..., 3.15632517466..., 0.675130870567...],
[-1.0..., -2.0..., -3.0..., 1.0...],
[0.311107817466..., 2.90321192591..., -2.52542756084..., -3.21431974338...],
[2.17008648663..., -1.70927535944..., -1.63089761382..., -0.460811127189...],
[3.0, 4.0, 6.0, 8.0]
]
"""
P = prime_range(bound)
e = self.eigenvalues(P)
v = Sequence([], cr=True)
if len(e) == 0:
return v
for i in range(len(e[0])):
v.append([e[j][i] for j in range(len(e))])
v.sort()
v.set_immutable()
return v
def systems_of_abs(self, bound):
"""
Return the absolute values of all systems of eigenvalues for
self for primes up to bound.
EXAMPLES::
sage: numerical_eigenforms(61).systems_of_abs(10) # rel tol 6e-14
[
[0.3111078174659775, 2.903211925911551, 2.525427560843529, 3.214319743377552],
[1.0, 2.0000000000000027, 3.000000000000003, 1.0000000000000044],
[1.4811943040920152, 0.8060634335253695, 3.1563251746586642, 0.6751308705666477],
[2.170086486626034, 1.7092753594369208, 1.63089761381512, 0.46081112718908984],
[3.0, 4.0, 6.0, 8.0]
]
"""
P = prime_range(bound)
e = self.eigenvalues(P)
v = Sequence([], cr=True)
if len(e) == 0:
return v
for i in range(len(e[0])):
v.append([abs(e[j][i]) for j in range(len(e))])
v.sort()
v.set_immutable()
return v
def isogenies_prime_degree(self, l=None, max_l=31):
"""
Generic code, valid for all fields, for arbitrary prime `l` not equal to the characteristic.
INPUT:
- ``l`` -- either None, a prime or a list of primes.
- ``max_l`` -- a bound on the primes to be tested (ignored unless `l` is None).
OUTPUT:
(list) All `l`-isogenies for the given `l` with domain self.
METHOD:
Calls the generic function
``isogenies_prime_degree()``. This requires that
certain operations have been implemented over the base field,
such as root-finding for univariate polynomials.
EXAMPLES::
sage: F = QQbar
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + x + 18 over Algebraic Field
sage: E.isogenies_prime_degree()
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for QQbar, but has not been yet.
sage: F = CC
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for general complex fields, but has not been yet.
Examples over finite fields::
sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 347438*x + 594729 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 674846*x + 7392 over Finite Field of size 1000003, Isogeny of degree 23 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 390065*x + 605596 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(7)
[]
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 4])
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(11)
[]
sage: E = EllipticCurve(GF(17),[2,0])
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 9*x over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 9 over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Finite Field of size 17]
sage: E = EllipticCurve(GF(13^4, 'a'),[2,8])
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4]
Example to show that separable isogenies of degree equal to the characteristic are now implemented::
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 6*x + 5 over Finite Field in a of size 13^4]
Examples over number fields (other than QQ)::
sage: QQroot2.<e> = NumberField(x^2-2)
sage: E = EllipticCurve(QQroot2, j=8000)
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-602112000)*x + 5035261952000 over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (903168000*e-1053696000)*x + (14161674240000*e-23288086528000) over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-903168000*e-1053696000)*x + (-14161674240000*e-23288086528000) over Number Field in e with defining polynomial x^2 - 2]
sage: E = EllipticCurve(QQroot2, [1,0,1,4, -6]); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2]
"""
F = self.base_ring()
if is_RealField(F):
raise NotImplementedError("This code could be implemented for general real fields, but has not been yet.")
if is_ComplexField(F):
raise NotImplementedError("This code could be implemented for general complex fields, but has not been yet.")
if F == rings.QQbar:
raise NotImplementedError("This code could be implemented for QQbar, but has not been yet.")
from isogeny_small_degree import isogenies_prime_degree
if l is None:
from sage.rings.all import prime_range
l = prime_range(max_l+1)
if not isinstance(l, list):
try:
l = rings.ZZ(l)
except TypeError:
raise ValueError("%s is not prime."%l)
if l.is_prime():
return isogenies_prime_degree(self, l)
else:
raise ValueError("%s is not prime."%l)
L = list(set(l))
try:
L = [rings.ZZ(l) for l in L]
except TypeError:
raise ValueError("%s is not a list of primes."%l)
L.sort()
return sum([isogenies_prime_degree(self,l) for l in L],[])
def multiple_of_order(self, maxp=None):
"""
Return a multiple of the order of this torsion group.
The multiple is computed using characteristic polynomials of Hecke
operators of odd index not dividing the level.
INPUT:
- ``maxp`` - (default: None) If maxp is None (the
default), return gcd of best bound computed so far with bound
obtained by computing GCD's of orders modulo p until this gcd
stabilizes for 3 successive primes. If maxp is given, just use all
primes up to and including maxp.
EXAMPLES::
sage: J = J0(11)
sage: G = J.rational_torsion_subgroup()
sage: G.multiple_of_order(11)
5
sage: J = J0(389)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(389) of dimension 32
sage: G.multiple_of_order()
97
sage: [G.multiple_of_order(p) for p in prime_range(3,11)]
[92645296242160800, 7275, 291]
sage: [G.multiple_of_order(p) for p in prime_range(3,13)]
[92645296242160800, 7275, 291, 97]
sage: [G.multiple_of_order(p) for p in prime_range(3,19)]
[92645296242160800, 7275, 291, 97, 97, 97]
::
sage: J = J0(33) * J0(11) ; J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0
The next example illustrates calling this function with a larger
input and how the result may be cached when maxp is None::
sage: T = J0(43)[1].rational_torsion_subgroup()
sage: T.multiple_of_order()
14
sage: T.multiple_of_order(50)
7
sage: T.multiple_of_order()
7
"""
if maxp is None:
try:
return self.__multiple_of_order
except AttributeError:
pass
bnd = ZZ(0)
A = self.abelian_variety()
if A.dimension() == 0:
T = ZZ(1)
self.__multiple_of_order = T
return T
N = A.level()
if not (len(A.groups()) == 1 and is_Gamma0(A.groups()[0])):
# to generalize to this case, you'll need to
# (1) define a charpoly_of_frob function:
# this is tricky because I don't know a simple
# way to do this for Gamma1 and GammaH. Will
# probably have to compute explicit matrix for
# <p> operator (add to modular symbols code),
# then compute some charpoly involving
# that directly...
# (2) use (1) -- see my MAGMA code.
raise NotImplementedError(
"torsion multiple only implemented for Gamma0")
cnt = 0
if maxp is None:
X = Primes()
else:
X = prime_range(maxp + 1)
for p in X:
if (2 * N) % p == 0:
continue
f = A.hecke_polynomial(p)
b = ZZ(f(p + 1))
if bnd == 0:
bnd = b
else:
bnd_last = bnd
bnd = ZZ(gcd(bnd, b))
if bnd == bnd_last:
cnt += 1
else:
cnt = 0
if maxp is None and cnt >= 2:
break
# The code below caches the computed bound and
# will be used if this function is called
# again with maxp equal to None (the default).
if maxp is None:
# maxp is None but self.__multiple_of_order is
# not set, since otherwise we would have immediately
# returned at the top of this function
self.__multiple_of_order = bnd
else:
# maxp is given -- record new info we get as
# a gcd...
try:
self.__multiple_of_order = gcd(self.__multiple_of_order, bnd)
except AttributeError:
# ... except in the case when self.__multiple_of_order
# was never set. In this case, we just set
# it as long as the gcd stabilized for 3 in a row.
if cnt >= 2:
self.__multiple_of_order = bnd
return bnd
def isogenies_prime_degree(self, l=None, max_l=31):
"""
Generic code, valid for all fields, for arbitrary prime `l` not equal to the characteristic.
INPUT:
- ``l`` -- either None, a prime or a list of primes.
- ``max_l`` -- a bound on the primes to be tested (ignored unless `l` is None).
OUTPUT:
(list) All `l`-isogenies for the given `l` with domain self.
METHOD:
Calls the generic function
``isogenies_prime_degree()``. This requires that
certain operations have been implemented over the base field,
such as root-finding for univariate polynomials.
EXAMPLES::
sage: F = QQbar
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + x + 18 over Algebraic Field
sage: E.isogenies_prime_degree()
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for QQbar, but has not been yet.
sage: F = CC
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for general complex fields, but has not been yet.
Examples over finite fields::
sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 347438*x + 594729 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 674846*x + 7392 over Finite Field of size 1000003, Isogeny of degree 23 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 390065*x + 605596 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(7)
[]
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 4])
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(11)
[]
sage: E = EllipticCurve(GF(17),[2,0])
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 9*x over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 9 over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Finite Field of size 17]
sage: E = EllipticCurve(GF(13^4, 'a'),[2,8])
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4]
Example to show that separable isogenies of degree equal to the characteristic are now implemented::
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 6*x + 5 over Finite Field in a of size 13^4]
Examples over number fields (other than QQ)::
sage: QQroot2.<e> = NumberField(x^2-2)
sage: E = EllipticCurve(QQroot2, j=8000)
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-36750)*x + 2401000 over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (220500*e-257250)*x + (54022500*e-88837000) over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-220500*e-257250)*x + (-54022500*e-88837000) over Number Field in e with defining polynomial x^2 - 2]
sage: E = EllipticCurve(QQroot2, [1,0,1,4, -6]); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-1)*x over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2]
"""
F = self.base_ring()
if is_RealField(F):
raise NotImplementedError(
"This code could be implemented for general real fields, but has not been yet."
)
if is_ComplexField(F):
raise NotImplementedError(
"This code could be implemented for general complex fields, but has not been yet."
)
if F == rings.QQbar:
raise NotImplementedError(
"This code could be implemented for QQbar, but has not been yet."
)
from isogeny_small_degree import isogenies_prime_degree
if l is None:
from sage.rings.all import prime_range
l = prime_range(max_l + 1)
if not isinstance(l, list):
try:
l = rings.ZZ(l)
except TypeError:
raise ValueError("%s is not prime." % l)
if l.is_prime():
return isogenies_prime_degree(self, l)
else:
raise ValueError("%s is not prime." % l)
L = list(set(l))
try:
L = [rings.ZZ(l) for l in L]
except TypeError:
raise ValueError("%s is not a list of primes." % l)
L.sort()
return sum([isogenies_prime_degree(self, l) for l in L], [])
def multiple_of_order(self, maxp=None):
"""
Return a multiple of the order of this torsion group.
The multiple is computed using characteristic polynomials of Hecke
operators of odd index not dividing the level.
INPUT:
- ``maxp`` - (default: None) If maxp is None (the
default), return gcd of best bound computed so far with bound
obtained by computing GCD's of orders modulo p until this gcd
stabilizes for 3 successive primes. If maxp is given, just use all
primes up to and including maxp.
EXAMPLES::
sage: J = J0(11)
sage: G = J.rational_torsion_subgroup()
sage: G.multiple_of_order(11)
5
sage: J = J0(389)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(389) of dimension 32
sage: G.multiple_of_order()
97
sage: [G.multiple_of_order(p) for p in prime_range(3,11)]
[92645296242160800, 7275, 291]
sage: [G.multiple_of_order(p) for p in prime_range(3,13)]
[92645296242160800, 7275, 291, 97]
sage: [G.multiple_of_order(p) for p in prime_range(3,19)]
[92645296242160800, 7275, 291, 97, 97, 97]
::
sage: J = J0(33) * J0(11) ; J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0
The next example illustrates calling this function with a larger
input and how the result may be cached when maxp is None::
sage: T = J0(43)[1].rational_torsion_subgroup()
sage: T.multiple_of_order()
14
sage: T.multiple_of_order(50)
7
sage: T.multiple_of_order()
7
"""
if maxp is None:
try:
return self.__multiple_of_order
except AttributeError:
pass
bnd = ZZ(0)
A = self.abelian_variety()
if A.dimension() == 0:
T = ZZ(1)
self.__multiple_of_order = T
return T
N = A.level()
if not (len(A.groups()) == 1 and is_Gamma0(A.groups()[0])):
# to generalize to this case, you'll need to
# (1) define a charpoly_of_frob function:
# this is tricky because I don't know a simple
# way to do this for Gamma1 and GammaH. Will
# probably have to compute explicit matrix for
# <p> operator (add to modular symbols code),
# then compute some charpoly involving
# that directly...
# (2) use (1) -- see my MAGMA code.
raise NotImplementedError("torsion multiple only implemented for Gamma0")
cnt = 0
if maxp is None:
X = Primes()
else:
X = prime_range(maxp+1)
for p in X:
if (2*N) % p == 0:
continue
f = A.hecke_polynomial(p)
b = ZZ(f(p+1))
if bnd == 0:
bnd = b
else:
bnd_last = bnd
bnd = ZZ(gcd(bnd, b))
if bnd == bnd_last:
cnt += 1
else:
cnt = 0
if maxp is None and cnt >= 2:
break
# The code below caches the computed bound and
# will be used if this function is called
# again with maxp equal to None (the default).
if maxp is None:
# maxp is None but self.__multiple_of_order is
# not set, since otherwise we would have immediately
# returned at the top of this function
self.__multiple_of_order = bnd
else:
# maxp is given -- record new info we get as
# a gcd...
try:
self.__multiple_of_order = gcd(self.__multiple_of_order, bnd)
except AttributeError:
# ... except in the case when self.__multiple_of_order
# was never set. In this case, we just set
# it as long as the gcd stabilized for 3 in a row.
if cnt >= 2:
self.__multiple_of_order = bnd
return bnd
