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 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 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
