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
Example #2
0
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
Example #3
0
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
Example #4
0
#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)
Example #6
0
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: