def upper_bound_on_elliptic_factors(self, p=None, ellmax=2):
r"""
Return an upper bound (provably correct) on the number of
elliptic curves of conductor equal to the level of this
supersingular module.
INPUT:
- ``p`` - (default: 997) prime to work modulo
ALGORITHM: Currently we only use `T_2`. Function will be
extended to use more Hecke operators later.
The prime p is replaced by the smallest prime that doesn't
divide the level.
EXAMPLE::
sage: SupersingularModule(37).upper_bound_on_elliptic_factors()
2
(There are 4 elliptic curves of conductor 37, but only 2 isogeny
classes.)
"""
# NOTE: The heuristic runtime is *very* roughly `p^2/(2\cdot 10^6)`.
#ellmax -- (default: 2) use Hecke operators T_ell with ell <= ellmax
if p is None:
p = 997
while self.level() % p == 0:
p = rings.next_prime(p)
ell = 2
t = self.hecke_matrix(ell).change_ring(rings.GF(p))
# TODO: temporarily try using sparse=False
# turn this off when sparse rank is optimized.
t = t.dense_matrix()
B = 2 * math.sqrt(ell)
bnd = 0
lower = -int(math.floor(B))
upper = int(math.floor(B)) + 1
for a in range(lower, upper):
tm = verbose("computing T_%s - %s" % (ell, a))
if a == lower:
c = a
else:
c = 1
for i in range(t.nrows()):
t[i, i] += c
tm = verbose("computing kernel", tm)
#dim = t.kernel().dimension()
dim = t.nrows() - t.rank()
bnd += dim
verbose('got dimension = %s; new bound = %s' % (dim, bnd), tm)
return bnd
def upper_bound_on_elliptic_factors(self, p=None, ellmax=2):
r"""
Return an upper bound (provably correct) on the number of
elliptic curves of conductor equal to the level of this
supersingular module.
INPUT:
- ``p`` - (default: 997) prime to work modulo
ALGORITHM: Currently we only use `T_2`. Function will be
extended to use more Hecke operators later.
The prime p is replaced by the smallest prime that doesn't
divide the level.
EXAMPLE::
sage: SupersingularModule(37).upper_bound_on_elliptic_factors()
2
(There are 4 elliptic curves of conductor 37, but only 2 isogeny
classes.)
"""
# NOTE: The heuristic runtime is *very* roughly `p^2/(2\cdot 10^6)`.
# ellmax -- (default: 2) use Hecke operators T_ell with ell <= ellmax
if p is None:
p = 997
while self.level() % p == 0:
p = rings.next_prime(p)
ell = 2
t = self.hecke_matrix(ell).change_ring(rings.GF(p))
# TODO: temporarily try using sparse=False
# turn this off when sparse rank is optimized.
t = t.dense_matrix()
B = 2 * math.sqrt(ell)
bnd = 0
lower = -int(math.floor(B))
upper = int(math.floor(B)) + 1
for a in range(lower, upper):
tm = verbose("computing T_%s - %s" % (ell, a))
if a == lower:
c = a
else:
c = 1
for i in range(t.nrows()):
t[i, i] += c
tm = verbose("computing kernel", tm)
# dim = t.kernel().dimension()
dim = t.nrows() - t.rank()
bnd += dim
verbose("got dimension = %s; new bound = %s" % (dim, bnd), tm)
return bnd
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def max_det_prime(n):
"""
Return the largest prime so that it is reasonably efficiency to
compute modulo that prime with n x n matrices in LinBox.
INPUT:
n -- a positive integer
OUTPUT:
a prime number
EXAMPLES:
sage: from sage.matrix.matrix_integer_dense_hnf import max_det_prime
sage: max_det_prime(10000)
524309
sage: max_det_prime(1000)
2097169
sage: max_det_prime(10)
16777259
"""
k = int(26 - math.ceil(math.log(n)*0.7213475205))
return next_prime(2**k)
def modular_symbols_from_curve(C, N, num_factors=3):
"""
Find the modular symbols spaces that shoudl correspond to the
Jacobian of the given hyperelliptic curve, up to the number of
factors we consider.
INPUT:
- C -- a hyperelliptic curve over QQ
- N -- a positive integer
- num_factors -- number of Euler factors to verify match up; this is
important, because if, e.g., there is only one factor of degree g(C), we
don't want to just immediately conclude that Jac(C) = A_f.
OUTPUT:
- list of all sign 1 simple modular symbols factor of level N
that correspond to a simple modular abelian A_f
that is isogenous to Jac(C).
EXAMPLES::
sage: from psage.modform.rational.unfiled import modular_symbols_from_curve
sage: R.<x> = ZZ[]
sage: f = x^7+4*x^6+5*x^5+x^4-3*x^3-2*x^2+1
sage: C1 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C1, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
sage: f = x^7-7*x^5-11*x^4+5*x^3+18*x^2+4*x-11
sage: C2 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C2, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
"""
# We will use the Eichler-Shimura relation and David Harvey's
# p-adic point counting hypellfrob. Harvey's code has the
# constraint: p > (2*g + 1)*(2*prec - 1).
# So, find the smallest p not dividing N that satisfies the
# above constraint, for our given choice of prec.
f, f2 = C.hyperelliptic_polynomials()
if f2 != 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x)"
if f.degree() % 2 == 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x) with f(x) odd"
prec = 1
g = C.genus()
B = (2 * g + 1) * (2 * prec - 1)
from sage.rings.all import next_prime
p = B
# We use that if F(X) is the characteristic polynomial of the
# Hecke operator T_p, then X^g*F(X+p/X) is the characteristic
# polynomial of Frob_p, because T_p = Frob_p + p/Frob_p, according
# to Eichler-Shimura. Use this to narrow down the factors.
from sage.all import ModularSymbols, Integers, get_verbose
D = ModularSymbols(
N, sign=1).cuspidal_subspace().new_subspace().decomposition()
D = [A for A in D if A.dimension() == g]
from sage.schemes.hyperelliptic_curves.hypellfrob import hypellfrob
while num_factors > 0:
p = next_prime(p)
while N % p == 0:
p = next_prime(p)
R = Integers(p**prec)['X']
X = R.gen()
D2 = []
# Compute the charpoly of Frobenius using hypellfrob
M = hypellfrob(p, 1, f)
H = R(M.charpoly())
for A in D:
F = R(A.hecke_polynomial(p))
# Compute charpoly of Frobenius from F(X)
G = R(F.parent()(X**g * F(X + p / X)))
if get_verbose(): print(p, G, H)
if G == H:
D2.append(A)
D = D2
num_factors -= 1
return D
def next_prime_of_characteristic_coprime_to(P, I):
p = next_prime(P.smallest_integer())
N = ZZ(I.norm())
while N%p == 0:
p = next_prime(p)
return F.primes_above(p)[0]
def next_prime_of_characteristic_coprime_to(P, I):
p = next_prime(P.smallest_integer())
N = ZZ(I.norm())
while N%p == 0:
p = next_prime(p)
return F.primes_above(p)[0]
def modular_symbols_from_curve(C, N, num_factors=3):
"""
Find the modular symbols spaces that shoudl correspond to the
Jacobian of the given hyperelliptic curve, up to the number of
factors we consider.
INPUT:
- C -- a hyperelliptic curve over QQ
- N -- a positive integer
- num_factors -- number of Euler factors to verify match up; this is
important, because if, e.g., there is only one factor of degree g(C), we
don't want to just immediately conclude that Jac(C) = A_f.
OUTPUT:
- list of all sign 1 simple modular symbols factor of level N
that correspond to a simple modular abelian A_f
that is isogenous to Jac(C).
EXAMPLES::
sage: from psage.modform.rational.unfiled import modular_symbols_from_curve
sage: R.<x> = ZZ[]
sage: f = x^7+4*x^6+5*x^5+x^4-3*x^3-2*x^2+1
sage: C1 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C1, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
sage: f = x^7-7*x^5-11*x^4+5*x^3+18*x^2+4*x-11
sage: C2 = HyperellipticCurve(f)
sage: modular_symbols_from_curve(C2, 284)
[Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
"""
# We will use the Eichler-Shimura relation and David Harvey's
# p-adic point counting hypellfrob. Harvey's code has the
# constraint: p > (2*g + 1)*(2*prec - 1).
# So, find the smallest p not dividing N that satisfies the
# above constraint, for our given choice of prec.
f, f2 = C.hyperelliptic_polynomials()
if f2 != 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x)"
if f.degree() % 2 == 0:
raise NotImplementedError, "curve must be of the form y^2 = f(x) with f(x) odd"
prec = 1
g = C.genus()
B = (2*g + 1)*(2*prec - 1)
from sage.rings.all import next_prime
p = B
# We use that if F(X) is the characteristic polynomial of the
# Hecke operator T_p, then X^g*F(X+p/X) is the characteristic
# polynomial of Frob_p, because T_p = Frob_p + p/Frob_p, according
# to Eichler-Shimura. Use this to narrow down the factors.
from sage.all import ModularSymbols, Integers, get_verbose
D = ModularSymbols(N,sign=1).cuspidal_subspace().new_subspace().decomposition()
D = [A for A in D if A.dimension() == g]
from sage.schemes.hyperelliptic_curves.hypellfrob import hypellfrob
while num_factors > 0:
p = next_prime(p)
while N % p == 0: p = next_prime(p)
R = Integers(p**prec)['X']
X = R.gen()
D2 = []
# Compute the charpoly of Frobenius using hypellfrob
M = hypellfrob(p, 1, f)
H = R(M.charpoly())
for A in D:
F = R(A.hecke_polynomial(p))
# Compute charpoly of Frobenius from F(X)
G = R(F.parent()(X**g * F(X + p/X)))
if get_verbose(): print (p, G, H)
if G == H:
D2.append(A)
D = D2
num_factors -= 1
return D
