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 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 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
