def genus_symbol_string_to_dict(s):
sl = s.split('.')
d = dict()
for s in sl:
L1 = s.split('^')
if len(L1) > 2:
raise ValueError()
elif len(L1) == 1:
nn = 1
else:
nn = L1[1]
n = Integer(nn)
L1 = L1[0].split("_")
q = Integer(L1[0])
if len(L1) > 2: # more than one _ in item
raise ValueError
elif len(L1) == 2:
if Integer(L1[1]) in range(8):
t = Integer(L1[1])
else:
raise ValueError, "Type given, which ist not in 0..7: %s" % (
L1[1])
else:
t = None
if not (n != 0 and q != 1 and q.is_prime_power() and
(None == t or (is_even(q) and t % 2 == n % 2)) and
(not (None == t and is_even(q)) or 0 == n % 2)):
raise ValueError, "{0} is not a valid signature!".format(s)
p = q.prime_factors()[0]
r = q.factor()[0][1]
eps = sign(n)
n = abs(n)
if not d.has_key(p):
d[p] = list()
if p == 2:
if t == None:
print "eps = ", eps
if not is_even(n):
raise ValueError()
d[p].append([
r, n, 3 * (-1)**(Integer(n - 2) / 2) % 8 if eps == -1 else
(-1)**(Integer(n)), t, 4 if eps == -1 else 0
])
print d
else:
if t.kronecker(2) == eps:
det = t % 8
else:
if eps == -1:
det = 3
else:
det = 1
d[p].append([r, n, det, 1, t % 8])
else:
d[p].append([r, n, eps])
return d
def genus_symbol_string_to_dict(s):
sl = s.split('.')
d = dict()
for s in sl:
L1 = s.split('^')
if len(L1)>2:
raise ValueError()
elif len(L1) == 1:
nn = 1
else:
nn = L1[1]
n = Integer(nn)
L1= L1[0].split("_")
q = Integer(L1[0])
if len(L1) > 2: # more than one _ in item
raise ValueError
elif len(L1) == 2:
if Integer(L1[1]) in range(8):
t = Integer(L1[1])
else:
raise ValueError, "Type given, which ist not in 0..7: %s"%(L1[1])
else:
t = None
if not (n != 0 and q != 1 and q.is_prime_power()
and ( None == t or (is_even(q) and t%2 == n%2))
and ( not (None == t and is_even(q)) or 0 == n%2)
):
raise ValueError,"{0} is not a valid signature!".format(s)
p = q.prime_factors()[0]
r = q.factor()[0][1]
eps = sign(n)
n = abs(n)
if not d.has_key(p):
d[p]=list()
if p==2:
if t == None:
print "eps = ", eps
if not is_even(n):
raise ValueError()
d[p].append([r, n, 3*(-1)**(Integer(n-2)/2) % 8 if eps == -1 else (-1)**(Integer(n)),t,4 if eps == -1 else 0])
print d
else:
if t.kronecker(2) == eps:
det = t % 8
else:
if eps == -1:
det = 3
else:
det = 1
d[p].append([r,n,det,1,t % 8])
else:
d[p].append([r,n,eps])
return d
def check_character_values(self, rec, verbose=False):
"""
The x's listed in values and values_gens should be coprime to the modulus N in the label.
For x's that appear in both values and values_gens, the value should be the same.
"""
# TIME about 3000s for full table
N = Integer(rec['modulus'])
v2, u2 = N.val_unit(2)
if v2 == 1:
# Z/2 doesn't contribute generators, but 2 divides N
adjust2 = -1
elif v2 >= 3:
# Z/8 and above requires two generators
adjust2 = 1
else:
adjust2 = 0
if N == 1:
# The character stores a value in the case N=1
ngens = 1
else:
ngens = len(N.factor()) + adjust2
vals = rec['values']
val_gens = rec['values_gens']
val_gens_dict = dict(val_gens)
if len(vals) != min(12, euler_phi(N)) or len(val_gens) != ngens:
if verbose:
print "Length failure", len(vals), euler_phi(N), len(
val_gens), ngens
return False
if N > 2 and (vals[0][0] != N - 1 or vals[1][0] != 1 or vals[1][1] != 0
or vals[0][1] not in [0, rec['order'] // 2]):
if verbose:
print "Initial value failure", N, rec['order'], vals[:2]
return False
if any(N.gcd(g) > 1 for g, gval in val_gens + vals):
if verbose:
print "gcd failure", [
g for g, gval in val_gens + vals if N.gcd(g) > 1
]
return False
for g, val in vals:
if g in val_gens_dict and val != val_gens_dict[g]:
if verbose:
print "Mismatch failure", g, val, val_gens_dict[g]
return False
return True
def check_character_values(self, rec, verbose=False):
"""
The x's listed in values and values_gens should be coprime to the modulus N in the label.
For x's that appear in both values and values_gens, the value should be the same.
"""
# TIME about 3000s for full table
N = Integer(rec['modulus'])
v2, u2 = N.val_unit(2)
if v2 == 1:
# Z/2 doesn't contribute generators, but 2 divides N
adjust2 = -1
elif v2 >= 3:
# Z/8 and above requires two generators
adjust2 = 1
else:
adjust2 = 0
if N == 1:
# The character stores a value in the case N=1
ngens = 1
else:
ngens = len(N.factor()) + adjust2
vals = rec['values']
val_gens = rec['values_gens']
val_gens_dict = dict(val_gens)
if len(vals) != min(12, euler_phi(N)) or len(val_gens) != ngens:
if verbose:
print "Length failure", len(vals), euler_phi(N), len(val_gens), ngens
return False
if N > 2 and (vals[0][0] != N-1 or vals[1][0] != 1 or vals[1][1] != 0 or vals[0][1] not in [0, rec['order']//2]):
if verbose:
print "Initial value failure", N, rec['order'], vals[:2]
return False
if any(N.gcd(g) > 1 for g, gval in val_gens+vals):
if verbose:
print "gcd failure", [g for g, gval in val_gens+vals if N.gcd(g) > 1]
return False
for g, val in vals:
if g in val_gens_dict and val != val_gens_dict[g]:
if verbose:
print "Mismatch failure", g, val, val_gens_dict[g]
return False
return True
class ConreyCharacter:
"""
tiny implementation on Conrey index only
"""
def __init__(self, modulus, number):
assert gcd(modulus, number) == 1
self.modulus = Integer(modulus)
self.number = Integer(number)
@property
def texname(self):
from lmfdb.characters.web_character import WebDirichlet
return WebDirichlet.char2tex(self.modulus, self.number)
@cached_method
def modfactor(self):
return self.modulus.factor()
@cached_method
def conductor(self):
cond = Integer(1)
number = self.number
for p, e in self.modfactor():
mp = Mod(number, p**e)
if mp == 1:
continue
if p == 2:
cond = 4
if number % 4 == 3:
mp = -mp
else:
cond *= p
mp = mp**(p - 1)
while mp != 1:
cond *= p
mp = mp**p
return cond
def is_primitive(self):
return self.conductor() == self.modulus
@cached_method
def parity(self):
number = self.number
par = 0
for p, e in self.modfactor():
if p == 2:
if number % 4 == 3:
par = 1 - par
else:
phi2 = (p - 1) / Integer(2) * p**(e - 1)
if Mod(number, p**e)**phi2 != 1:
par = 1 - par
return par
def is_odd(self):
return self.parity() == 1
def is_even(self):
return self.parity() == 0
@cached_method
def multiplicative_order(self):
return Mod(self.number, self.modulus).multiplicative_order()
@property
def order(self):
return self.multiplicative_order()
@cached_method
def kronecker_symbol(self):
c = self.conductor()
p = self.parity()
return kronecker_symbol(symbol_numerator(c, p))
class ConreyCharacter:
"""
tiny implementation on Conrey index only
"""
def __init__(self, modulus, number):
assert gcd(modulus, number)==1
self.modulus = Integer(modulus)
self.number = Integer(number)
@property
def texname(self):
return WebDirichlet.char2tex(self.modulus, self.number)
@cached_method
def modfactor(self):
return self.modulus.factor()
@cached_method
def conductor(self):
cond = Integer(1);
number = self.number
for p,e in self.modfactor():
mp = Mod(number, p**e);
if mp == 1:
continue
if p == 2:
cond = 4
if number % 4 == 3:
mp = -mp
else:
cond *= p
mp = mp**(p-1)
while mp != 1:
cond *= p
mp = mp**p
return cond
def is_primitive(self):
return self.conductor() == self.modulus
@cached_method
def parity(self):
number = self.number
par = 0;
for p,e in self.modfactor():
if p == 2:
if number % 4 == 3:
par = 1 - par
else:
phi2 = (p-1)/Integer(2) * p **(e-1)
if Mod(number, p ** e)**phi2 != 1:
par = 1 - par
return par
def is_odd(self):
return self.parity() == 1
def is_even(self):
return self.parity() == 0
@cached_method
def multiplicative_order(self):
return Mod(self.number, self.modulus).multiplicative_order()
@property
def order(self):
return self.multiplicative_order()
@cached_method
def kronecker_symbol(self):
c = self.conductor()
p = self.parity()
return kronecker_symbol(symbol_numerator(c, p))
class ConreyCharacter(object):
"""
tiny implementation on Conrey index only
"""
def __init__(self, modulus, number):
assert gcd(modulus, number) == 1
self.modulus = Integer(modulus)
self.number = Integer(number)
self.G = Pari("znstar({},1)".format(modulus))
self.chi_pari = pari("znconreylog(%s,%d)" % (self.G, self.number))
self.chi_0 = None
self.indlabel = None
@property
def texname(self):
from lmfdb.characters.web_character import WebDirichlet
return WebDirichlet.char2tex(self.modulus, self.number)
@cached_method
def modfactor(self):
return self.modulus.factor()
@cached_method
def conductor(self):
B = pari("znconreyconductor(%s,%s,&chi0)" % (self.G, self.chi_pari))
if B.type() == 't_INT':
# means chi is primitive
self.chi_0 = self.chi_pari
self.indlabel = self.number
return int(B)
else:
self.chi_0 = pari("chi0")
G_0 = Pari("znstar({},1)".format(B))
self.indlabel = int(pari("znconreyexp(%s,%s)" % (G_0, self.chi_0)))
return int(B[0])
def is_primitive(self):
return self.conductor() == self.modulus
@cached_method
def parity(self):
number = self.number
par = 0
for p, e in self.modfactor():
if p == 2:
if number % 4 == 3:
par = 1 - par
else:
phi2 = (p - 1) / Integer(2) * p**(e - 1)
if Mod(number, p**e)**phi2 != 1:
par = 1 - par
return par
def is_odd(self):
return self.parity() == 1
def is_even(self):
return self.parity() == 0
@cached_method
def multiplicative_order(self):
return Mod(self.number, self.modulus).multiplicative_order()
@property
def order(self):
return self.multiplicative_order()
@cached_method
def kronecker_symbol(self):
c = self.conductor()
p = self.parity()
return kronecker_symbol(symbol_numerator(c, p))
def conreyangle(self, x):
return Rational(
pari("chareval(%s,znconreylog(%s,%d),%d)" %
(self.G, self.G, self.number, x)))
def gauss_sum_numerical(self, a):
# There seems to be a bug in pari when a is a multiple of the modulus,
# so we deal with that separately
if self.modulus.divides(a):
if self.conductor() == 1:
return euler_phi(self.modulus)
else:
return Integer(0)
else:
return pari("znchargauss(%s,%s,a=%d)" % (self.G, self.chi_pari, a))
def sage_zeta_order(self, order):
return 1 if self.modulus <= 2 else lcm(2, order)
def sage_character(self, order, genvalues):
H = DirichletGroup(self.modulus,
base_ring=CyclotomicField(
self.sage_zeta_order(order)))
M = H._module
order_corrected_genvalues = get_sage_genvalues(
self.modulus, order, genvalues, self.sage_zeta_order(order))
return DirichletCharacter(H, M(order_corrected_genvalues))
