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 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 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 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 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 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 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)
