def __init__(self, K, F, minimal_ramification=1):
minimal_ramification = ZZ(minimal_ramification)
# print "entering WeakPadicGaloisExtension with"
# print "F = %s"%F
# if F is a polynomial, replace it by the list of its irreducible factors
# if isinstance(F, Polynomial):
# F = [f for f, m in F.factor()]
assert K.is_Qp(), "for the moment, K has to be Q_p"
# assert not K.p().divides(minimal_ramification), "minimal_ramification has to be prime to p"
if not isinstance(F, Polynomial):
if F == []:
F = PolynomialRing(K.number_field(), 'x')(1)
else:
F = prod(F)
self._base_field = K
L = K.weak_splitting_field(F)
e = ZZ(L.absolute_ramification_degree() /
K.absolute_ramification_degree())
if not minimal_ramification.divides(e):
# enlarge the absolute ramification index of vL
# such that minimal_ramification divides e(vL/vK):
m = ZZ(minimal_ramification / e.gcd(minimal_ramification))
# assert not self.p().divides(m), "p = %s, m = %s, e = %s,\nminimal_ramification = %s"%(self.p(), m, e, minimal_ramification)
L = L.ramified_extension(m)
# if m was not prime to p, L/K may not be weak Galois anymore
else:
m = ZZ(1)
self._extension_field = L
self._ramification_degree = e * m
self._degree = ZZ(L.absolute_degree() / K.absolute_degree())
self._inertia_degree = ZZ(L.absolute_inertia_degree() /
K.absolute_inertia_degree())
assert self._degree == self._ramification_degree * self._inertia_degree
def get_cusp_expansions_of_newform(k, N=1, fi=0, prec=10):
r"""
Get and return Fourier coefficients of all cusps where there exist Atkin-Lehner involutions for these cusps.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
- ''prec'' -- integer (the number of coefficients to get)
OUTPUT:
- ''s'' string giving the Atkin-Lehner eigenvalues corresponding to the Cusps (where possible)
"""
res = dict()
(t, f) = _get_newform(k, N, 0, fi)
if (not t):
return s
res[Infinity] = 1
for c in f.group().cusps():
if (c == Cusp(Infinity)):
continue
res[c] = list()
cusp = QQ(c)
q = cusp.denominator()
p = cusp.numerator()
d = ZZ(cusp * N)
if (d == 0):
ep = f.atkin_lehner_eigenvalue()
if (d.divides(N) and gcd(ZZ(N / d), ZZ(d)) == 1):
ep = f.atkin_lehner_eigenvalue(ZZ(d))
else:
# this case is not known...
res[c] = None
continue
res[c] = ep
s = html.table([res.keys(), res.values()])
return s
def get_cusp_expansions_of_newform(k, N=1, fi=0, prec=10):
r"""
Get and return Fourier coefficients of all cusps where there exist Atkin-Lehner involutions for these cusps.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
- ''prec'' -- integer (the number of coefficients to get)
OUTPUT:
- ''s'' string giving the Atkin-Lehner eigenvalues corresponding to the Cusps (where possible)
"""
res = dict()
(t, f) = _get_newform(k, N, 0, fi)
if(not t):
return s
res[Infinity] = 1
for c in f.group().cusps():
if(c == Cusp(Infinity)):
continue
res[c] = list()
cusp = QQ(c)
q = cusp.denominator()
p = cusp.numerator()
d = ZZ(cusp * N)
if(d == 0):
ep = f.atkin_lehner_eigenvalue()
if(d.divides(N) and gcd(ZZ(N / d), ZZ(d)) == 1):
ep = f.atkin_lehner_eigenvalue(ZZ(d))
else:
# this case is not known...
res[c] = None
continue
res[c] = ep
s = html.table([res.keys(), res.values()])
return s
#ar = ArtinRepresentation('2.2e3_3e2.6t5.1c1')
baselabel = label.split('c')
a = rep.find_one({'Baselabel': baselabel[0]})
ar = ArtinRepresentation(label)
outfile = open(label, 'w')
cf = a['CharacterField']
cfz = ZZ(cf)
nf = ar.nf()
nfcc = nf.conjugacy_classes()
nfcc = [int(z.order()) for z in nfcc]
nfcc = lcm(nfcc)
if not cfz.divides(nfcc):
print("Failure " + str(cfz) + " divides " + str(nfcc) + " from " + label)
sys.exit()
R, x = QQ['x'].objgen()
pol1 = R.cyclotomic_polynomial(nfcc)
K, z = NumberField(R.cyclotomic_polynomial(nfcc), 'z').objgen()
RR, y = K['y'].objgen()
zsmall = z**(nfcc / cfz)
allpols = [
sum(y**k * sum(pp[k][j] * zsmall**j for j in range(len(pp[k])))
for k in range(len(pp))) for pp in ar.local_factors_table()
]
allroots = [myroots(pp, nfcc, z) for pp in allpols]
outfile.write(str(nfcc) + "\n")
outfile.write(str(allroots) + "\n")
def char_order_valid(m, ell):
"""Return True iff the positive integer m has the form (ell^k)*m1
where m1 is 1 mod ell
"""
m1 = ZZ(m).prime_to_m_part(ell)
return m1.divides(ell-1)
baselabel=label.split('c')
a = rep.find_one({'Baselabel': baselabel[0]})
ar=ArtinRepresentation(label)
outfile=open(label, 'w')
cf=a['CharacterField']
cfz = ZZ(cf)
nf = ar.nf()
nfcc = nf.conjugacy_classes()
nfcc = [int(z.order()) for z in nfcc]
nfcc = lcm(nfcc)
if not cfz.divides(nfcc):
print "Failure "+str(cfz)+" divides "+str(nfcc)+" from "+label
sys.exit()
R,x = QQ['x'].objgen()
pol1 = R.cyclotomic_polynomial(nfcc)
K,z=NumberField(R.cyclotomic_polynomial(nfcc),'z').objgen()
RR,y = K['y'].objgen()
zsmall = z**(nfcc/cfz)
allpols = [sum(y**k * sum(pp[k][j] * zsmall**j for j in range(len(pp[k]))) for k in range(len(pp))) for pp in ar.local_factors_table()]
allroots = [myroots(pp, nfcc, z) for pp in allpols]
outfile.write(str(nfcc)+"\n")
outfile.write(str(allroots)+"\n")
j=0
p=1
while j<bound: