def class_nr_pos_def_qf(D):
r"""
Compute the class number of positive definite quadratic forms.
For fundamental discriminants this is the class number of Q(sqrt(D)),
otherwise it is computed using: Cohen 'A course in Computational Algebraic Number Theory', p. 233
"""
if D>0:
return 0
D4 = D % 4
if D4 == 3 or D4==2:
return 0
K = QuadraticField(D)
if is_fundamental_discriminant(D):
return K.class_number()
else:
D0 = K.discriminant()
Df = ZZ(D).divide_knowing_divisible_by(D0)
if not is_square(Df):
raise ArithmeticError("Did not get a discrimimant * square! D={0} disc(D)={1}".format(D,D0))
D2 = sqrt(Df)
h0 = QuadraticField(D0).class_number()
w0 = _get_w(D0)
w = _get_w(D)
#print "w,w0=",w,w0
#print "h0=",h0
h = 1
for p in prime_divisors(D2):
h = QQ(h)*(1-kronecker(D0,p)/QQ(p))
#print "h=",h
#print "fak=",
h=QQ(h*h0*D2*w)/QQ(w0)
return h
def lpp_table(self):
"""generate the large putative primes table"""
output_str = r"${Delta_K}$ & $\Q(\sqrt{{{D}}})$ & {large_putative_primes}\\"
latex_output = []
for D in range(-self.range, self.range + 1):
if Integer(D).is_squarefree():
if not D in CLASS_NUMBER_ONE_DISCS:
if D != 1:
K = QuadraticField(D)
Delta_K = K.discriminant()
candidates = get_isogeny_primes(K,
aux_prime_count=25,
bound=2000,
loop_curves=False)
candidates = [
c for c in candidates
if c not in EC_Q_ISOGENY_PRIMES
]
candidates = [c for c in candidates if c > 71]
candidates.sort()
large_putative_primes = ", ".join(map(str, candidates))
output_here = output_str.format(
Delta_K=Delta_K,
D=D,
large_putative_primes=large_putative_primes,
)
latex_output.append(output_here)
for one_line in latex_output:
print(one_line)
def class_nr_pos_def_qf(D):
r"""
Compute the class number of positive definite quadratic forms.
For fundamental discriminants this is the class number of Q(sqrt(D)),
otherwise it is computed using: Cohen 'A course in Computational Algebraic Number Theory', p. 233
"""
if D>0:
return 0
D4 = D % 4
if D4 == 3 or D4==2:
return 0
K = QuadraticField(D)
if is_fundamental_discriminant(D):
return K.class_number()
else:
D0 = K.discriminant()
Df = ZZ(D).divide_knowing_divisible_by(D0)
if not is_square(Df):
raise ArithmeticError,"DId not get a discrinimant * square! D={0} disc(D)={1}".format(D,D0)
D2 = sqrt(Df)
h0 = QuadraticField(D0).class_number()
w0 = _get_w(D0)
w = _get_w(D)
#print "w,w0=",w,w0
#print "h0=",h0
h = 1
for p in prime_divisors(D2):
h = QQ(h)*(1-kronecker(D0,p)/QQ(p))
#print "h=",h
#print "fak=",
h=QQ(h*h0*D2*w)/QQ(w0)
return h
def test_interval(D):
K = QuadraticField(D)
if not K.discriminant() in CLASS_NUMBER_ONE_DISCS:
superset, _ = get_isogeny_primes(K, **TEST_SETTINGS)
# test that there are not too many primes left over
todo = set(superset).difference(EC_Q_ISOGENY_PRIMES)
assert len(todo) == 0 or max(todo) <= 109
assert len(todo) <= 2 or max(todo) <= 31
def oezman_sieve(p, N):
"""If p is unramified in Q(sqrt(-N)) this always returns True.
Otherwise returns True iff p is in S_N or . Only makes sense if p ramifies in K"""
M = QuadraticField(-N)
if p.divides(M.discriminant()):
return True
pp = (M * p).factor()[0][0]
C_M = M.class_group()
if C_M(pp).order() == 1:
return True
return False
def dimension_cuspforms_sqrt5(k1, k2):
'''
Return the dimension of hilbert cusp forms of
weight (k1, k2) where k1 > 2 and k2 > 2.
cf. K. Takase, on the trace formula of the Hecke operators and the special
values of the second L-functions attached to the Hilbert modular forms.
manuscripta math. 55, 137 -- 170 (1986).
'''
k = (k1, k2)
F = QuadraticField(5)
rho = (1 + F.gen()) / ZZ(2)
a = c_km2_1_rho(k, rho)
return (ZZ((k1 - 1) * (k2 - 1)) / ZZ(60) + ZZ(c_km2_1_01(k)) / ZZ(4) +
ZZ(c_km2_1_11(k)) / ZZ(3) +
ZZ(a + F(a).galois_conjugate()) / ZZ(5))
def test_get_dirichlet_character():
K = QuadraticField(-31)
chi = get_dirichlet_character(K)
assert (3 * K).is_prime()
assert chi(3) == -1
assert not (5 * K).is_prime()
assert chi(5) == 1
assert (73 * K).is_prime()
assert chi(73) == -1
def main():
for D in square_free_D:
K = QuadraticField(D)
if not K.discriminant() in CLASS_NUMBER_ONE_DISCS:
superset, _ = get_isogeny_primes(
K,
bound=1000,
ice_filter=True,
appendix_bound=1000,
norm_bound=50,
auto_stop_strategy=True,
repeat_bound=4,
)
possible_new_isog_primes = superset - EC_Q_ISOGENY_PRIMES
possible_new_isog_primes_list = list(possible_new_isog_primes)
possible_new_isog_primes_list.sort()
if possible_new_isog_primes_list:
print(f"D = {D} possible isogenies = {possible_new_isog_primes_list}")
def test_interval(self):
R = 100
for D in range(-R, R + 1):
if Integer(D).is_squarefree():
if not D in CLASS_NUMBER_ONE_DISCS:
if D != 1:
K = QuadraticField(D)
superset = get_isogeny_primes(K, AUX_PRIME_COUNT)
self.assertTrue(
set(superset).issuperset(EC_Q_ISOGENY_PRIMES))
def c_km2_1_sqrt2(k):
K = QuadraticField(2)
a = K.gen()
k1, k2 = k
k1 = k1 % 8
k2 = k2 % 8
k = (k1, k2)
if k in [(0, 0), (2, 2), (4, 4), (6, 6), (0, 6), (6, 0), (2, 4), (4, 2)]:
return 1
if k in [(3, 7), (7, 3)]:
return 2
if k in [(0, 7), (2, 3), (3, 0), (3, 6), (4, 3), (6, 7), (7, 2), (7, 4)]:
return a
if k1 in [1, 5] or k2 in [1, 5]:
return 0
if k in [(0, 3), (2, 7), (3, 2), (3, 4), (4, 7), (6, 3), (7, 0), (7, 6)]:
return -a
if k in [(3, 3), (7, 7)]:
return -2
else:
return -1
def oezman_sieve(p, N):
"""Returns True iff p is in S_N. Only makes sense if p ramifies in K"""
M = QuadraticField(-N)
h_M = M.class_number()
H = M.hilbert_class_field("b")
primes_above_p = M.primes_above(p)
primes_tot_split_in_hcf = []
for P in primes_above_p:
if len(H.primes_above(P)) == h_M:
primes_tot_split_in_hcf.append(P)
if not primes_tot_split_in_hcf:
return False
f = R(hilbert_class_polynomial(M.discriminant()))
B = NumberField(f, name="t")
assert B.degree() == h_M
possible_nus = B.primes_above(p)
for nu in possible_nus:
if nu.residue_class_degree() == 1:
return True
return False
def quadratic_number_field():
"""
Return a quadratic extension of QQ.
EXAMPLES:
sage: sage.rings.tests.quadratic_number_field()
Number Field in a with defining polynomial x^2 - 61099
"""
from sage.all import ZZ, QuadraticField
while True:
d = ZZ.random_element(x=-10**5, y=10**5)
if not d.is_square():
return QuadraticField(d, 'a')
def test_from_literature(D, extra_isogeny, appendix_bound, potenial_isogenies):
K = QuadraticField(D)
upperbound = potenial_isogenies.union(EC_Q_ISOGENY_PRIMES).union(
{extra_isogeny})
superset, _ = get_isogeny_primes(K,
appendix_bound=appendix_bound,
**TEST_SETTINGS)
assert set(EC_Q_ISOGENY_PRIMES).difference(superset) == set()
assert extra_isogeny in superset
assert set(
superset.difference(upperbound)) == set(), "We got worse at filtering"
assert set(
upperbound.difference(superset)) == set(), "We got better at filtering"
def dlmv_table(self):
"""generate the dlmv table"""
output_str = (
r"${Delta_K}$ & $\Q(\sqrt{{{D}}})$ & ${rem} \times 10^{{{exp_at_10}}}$\\"
)
for D in range(-self.range, self.range + 1):
if Integer(D).is_squarefree():
if not D in CLASS_NUMBER_ONE_DISCS:
if D != 1:
K = QuadraticField(D)
Delta_K = K.discriminant()
dlmv_bound = RR(DLMV(K))
log_dlmv_bound = dlmv_bound.log10()
exp_at_10 = int(log_dlmv_bound)
rem = log_dlmv_bound - exp_at_10
rem = 10**rem
rem = rem.numerical_approx(digits=3)
output_here = output_str.format(Delta_K=Delta_K,
D=D,
rem=rem,
exp_at_10=exp_at_10)
print(output_here)
def compare_formulas_1(D, k):
DG = DirichletGroup(abs(D))
chi = DG(kronecker_character(D))
d1 = dimension_new_cusp_forms(chi, k)
#if D>0:
# lvals=sage.lfunctions.all.lcalc.twist_values(1,2,D)
#else:
# lvals=sage.lfunctions.all.lcalc.twist_values(1,D,0)
#s1=RR(sum([sqrt(abs(lv[0]))*lv[1]*2**len(prime_factors(D/lv[0])) for lv in lvals if lv[0].divides(D) and Zmod(lv[0])(abs(D/lv[0])).is_square()]))
#d2=RR(1/pi*s1)
d2 = 0
for d in divisors(D):
if is_fundamental_discriminant(-d):
K = QuadraticField(-d)
DD = old_div(ZZ(D), ZZ(d))
ep = euler_phi((chi * DG(kronecker_character(-d))).conductor())
#ep=euler_phi(squarefree_part(abs(D*d)))
print("ep=", ep, D, d)
ids = [a for a in K.ideals_of_bdd_norm(-DD)[-DD]]
eulers1 = []
for a in ids:
e = a.euler_phi()
if e != 1 and ep == 1:
if K(-1).mod(a) != K(1).mod(a):
e = old_div(e, (2 * ep))
else:
e = old_div(e, ep)
eulers1.append(e)
print(eulers1, ep)
s = sum(eulers1)
if ep == 1 and not (d.divides(DD) or abs(DD) == 1):
continue
print(d, s)
if len(eulers1) > 0:
d2 += s * K.class_number()
return d1 - d2
def cli_handler(args):
if not Integer(args.D).is_squarefree():
msg = "Your D is not squarefree. Please choose a " "squarefree D. Exiting."
print(msg)
return
if args.D in CLASS_NUMBER_ONE_DISCS:
msg = (
"Your D yields an imaginary quadratic field of class "
"number one. These fields have infinitely many isogeny primes. "
"Exiting."
)
print(msg)
return
K = QuadraticField(args.D)
if args.dlmv:
dlmv_bound = DLMV(K)
print(
"DLMV bound for {} is:\n\n{}\n\nwhich is approximately {}".format(
K, dlmv_bound, RR(dlmv_bound)
)
)
else:
if args.rigorous:
bound = None
print("Checking all Type 2 primes up to conjectural bound")
else:
bound = args.bound
print("WARNING: Only checking Type 2 primes up to {}.\n".format(bound))
print(
(
"To check all, run with '--rigorous', but be advised that "
"this will take ages and require loads of memory"
)
)
superset = get_isogeny_primes(K, args.aux_prime_count, bound, args.loop_curves)
print("superset = {}".format(superset))
def works_method_of_appendix(p, K):
"""This implements the method of the appendix, returns True if that
method is able to remove p as an isogeny prime for K."""
if QuadraticField(-p).class_number() <= K.degree():
return False
if p in SMALL_GONALITIES:
return False
chi = get_dirichlet_character(K)
if K.degree() == 2 and chi(p) == -1:
return False
logger.debug("Testing whether torsion is same")
if is_torsion_same(p, K, chi):
logger.debug("Torsion is same test passed")
logger.debug("Testing whether rank is same")
if is_rank_of_twist_zero(p, chi):
return True
return False
def test_73_Box(self):
K = QuadraticField(-31)
superset = get_isogeny_primes(K, AUX_PRIME_COUNT)
self.assertTrue(set(superset).issuperset(EC_Q_ISOGENY_PRIMES))
self.assertIn(73, superset)
def test_103(self):
K = QuadraticField(5 * 577)
superset = get_isogeny_primes(K, AUX_PRIME_COUNT)
self.assertTrue(set(superset).issuperset(EC_Q_ISOGENY_PRIMES))
self.assertIn(103, superset)
def test_73(self):
K = QuadraticField(-127)
superset = get_isogeny_primes(K, AUX_PRIME_COUNT, loop_curves=True)
self.assertTrue(set(superset).issuperset(EC_Q_ISOGENY_PRIMES))
self.assertIn(73, superset)
def test_191(self):
K = QuadraticField(61 * 229 * 145757)
superset = get_isogeny_primes(K, AUX_PRIME_COUNT)
self.assertTrue(set(superset).issuperset(EC_Q_ISOGENY_PRIMES))
self.assertIn(191, superset)
def test_is_rank_of_twist_zero():
p = 73
K = QuadraticField(-31)
chi = get_dirichlet_character(K)
assert not is_rank_of_twist_zero(p, chi)
def test_311(self):
K = QuadraticField(11 * 17 * 9011 * 23629)
superset = get_isogeny_primes(K, AUX_PRIME_COUNT)
self.assertTrue(set(superset).issuperset(EC_Q_ISOGENY_PRIMES))
self.assertIn(311, superset)
def _test(self):
from sage.all import QuadraticField, loads, dumps
K = QuadraticField(-1, 'i')
if loads(dumps(K.maximal_order())) is not K.maximal_order():
raise Exception("#24934 has not been fixed")
# -*- coding: utf-8; mode: sage -*-
from itertools import takewhile
from pickle import Pickler
from os.path import join
from sage.all import (ZZ, FreeModule, PolynomialRing, QuadraticField,
TermOrder, cached_method, flatten, gcd, load, QQ, cached_function,
PowerSeriesRing, O)
from sage.libs.singular.function import singular_function
K = QuadraticField(5)
Monomial_Wts = (6, 5, 2)
R = PolynomialRing(K, names='g6, g5, g2', order=TermOrder('wdegrevlex', Monomial_Wts))
g6, g5, g2 = R.gens()
DATA_DIR = "/home/sho/work/rust/hilbert_sqrt5/data/brackets"
smodule = singular_function("module")
sideal = singular_function("ideal")
squotient = singular_function("quotient")
smres = singular_function("mres")
slist = singular_function("list")
sintersect = singular_function("intersect")
ssyz = singular_function("syz")
def diag_res(f):
R_el = PolynomialRing(QQ, "E4, E6")
E4, E6 = R_el.gens()
Delta = (E4**3 - E6**2) / 1728
d = {g2: E4, g5: 0, g6: 2 * Delta}
return f.subs(d)
[email protected]. ==================================================================== """ import pytest from sage.all import EllipticCurve_from_j, GF, QQ, QuadraticField from sage.rings.polynomial.polynomial_ring import polygen from sage.schemes.elliptic_curves.constructor import EllipticCurve from sage_code.frobenius_polynomials import ( semi_stable_frobenius_polynomial, isogeny_character_values_12, ) K = QuadraticField(-127, "D") D = K.gen(0) j = 20 * (3 * (-26670989 - 15471309 * D) / 2**26)**3 # this is one of the curves from the Gonzaléz, Lario, and Quer article E = EllipticCurve_from_j(j) def test_semi_stable_frobenius_polynomial(): # test that we indeed have a 73 isogeny mod p for p in 2, 3, 5, 7, 11, 19: for pp, e in (p * K).factor(): f = semi_stable_frobenius_polynomial(E, pp) assert not f.change_ring(GF(73)).is_irreducible()
def test_works_method_of_appendix(D, p, works):
K = QuadraticField(D)
result = works_method_of_appendix(p, K)
assert result == works