def add_row(self, nflabel):
nf = WebNumberField(nflabel)
#nflink = (nflabel, url_for('number_fields.by_label',label=nflabel))
nflink = (nflabel,
url_for('characters.render_Heckewebpage',
number_field=nflabel))
F = WebHeckeFamily(number_field=nflabel)
self._contents.append((nflink, nf.signature(), nf.web_poly()))
def vflabel(self):
order2 = self.order if self.order % 4 != 2 else self.order / 2
if order2 == 1:
return '1.1.1.1'
elif order2 == 4:
return '2.0.4.1'
valuewnf = WebNumberField.from_cyclo(order2)
if not valuewnf.is_null():
return valuewnf.label
else:
return ''
def label(self):
if "label" in self._data.keys():
return self._data["label"]
else:
# from number_fields.number_field import poly_to_field_label
# pol = PolynomialRing(QQ, 'x')(map(str,self.polynomial()))
# label = poly_to_field_label(pol)
label = WebNumberField.from_coeffs(self._data["Polynomial"]).get_label()
if label:
self._data["label"] = label
return label
def vflabel(self):
order2 = self.order if self.order % 4 != 2 else self.order / 2
if order2 == 1:
return '1.1.1.1'
elif order2 == 4:
return '2.0.4.1'
valuewnf = WebNumberField.from_cyclo(order2)
if not valuewnf.is_null():
return valuewnf.label
else:
return ''
def vflabel(self):
_ = self.valuefield # make sure valuefield was computed
order2 = self._order2
if order2 == 1:
return '1.1.1.1'
elif order2 == 4:
return '2.0.4.1'
valuewnf = WebNumberField.from_cyclo(order2)
if not valuewnf.is_null():
return valuewnf.label
else:
return ''
def label(self):
if "label" in self._data.keys():
return self._data["label"]
else:
#from number_fields.number_field import poly_to_field_label
#pol = PolynomialRing(QQ, 'x')(map(str,self.polynomial()))
#label = poly_to_field_label(pol)
label = WebNumberField.from_coeffs(
self._data["Polynomial"]).get_label()
if label:
self._data["label"] = label
return label
def G_name(self):
"""
More-or-less standardized name of the abstract group
"""
from WebNumberField import WebNumberField
import re
wnf = WebNumberField.from_polredabs(self.polredabs())
if not wnf.is_null():
mygalstring = wnf.galois_string()
if re.search('Trivial', mygalstring) is not None:
return '$C_1$'
# Have to remove dollar signs
return mygalstring
if self.polredabs().degree() < 12:
# Let pari compute it for us now
from sage.all import pari
galt = int(list(pari('polgalois(' + str(self.polredabs()) + ')'))[2])
from transitive_group import WebGaloisGroup
tg = WebGaloisGroup.from_nt(self.polredabs().degree(), galt)
return tg.display_short()
return self._data["G-Name"]
def G_name(self):
"""
More-or-less standardized name of the abstract group
"""
from WebNumberField import WebNumberField
import re
wnf = WebNumberField.from_polredabs(self.polredabs())
if not wnf.is_null():
mygalstring = wnf.galois_string()
if re.search('Trivial', mygalstring) is not None:
return '$C_1$'
# Have to remove dollar signs
return mygalstring
if self.polredabs().degree() < 12:
# Let pari compute it for us now
from sage.all import pari
galt = int(list(pari('polgalois(' + str(self.polredabs()) + ')'))[2])
from transitive_group import WebGaloisGroup
tg = WebGaloisGroup.from_nt(self.polredabs().degree(), galt)
return tg.display_short()
return self._data["G-Name"]
def add_row(self, nflabel):
nf = WebNumberField(nflabel)
#nflink = (nflabel, url_for('number_fields.by_label',label=nflabel))
nflink = (nflabel, url_for('characters.render_Heckewebpage',number_field=nflabel))
#F = WebHeckeFamily(number_field=nflabel)
self._contents.append( (nflink, nf.signature(), nf.web_poly() ) )
def dirichletcharacter(self):
#######################################################################################
## Conrey's naming convention for Dirichlet Characters
#######################################################################################
G = DirichletGroup_conrey(self.modulus)
G_sage = G.standard_dirichlet_group()
self.level = self.modulus
if self.modulus == 1 or self.number % self.modulus != 0:
chi = G[self.number]
chi_sage = chi.sage_character()
self.chi_sage = chi_sage
self.zetaorder = G_sage.zeta_order()
# if len(chi_sage.values_on_gens()) == 1:
### self.genvaluestex = latex(chi_sage.values_on_gens()[0])
# else:
### self.genvaluestex = latex(chi_sage.values_on_gens())
# chizero = G_sage[0]
self.char = str(chi)
if chi.is_primitive():
self.primitive = "True"
else:
self.primitive = "False"
self.conductor = chi.conductor()
self.order = chi.multiplicative_order()
self.vals = chi.values()
self.logvals = log_value(self.modulus, self.number)
# self.logvals = map(chi.logvalue, range(1,self.modulus+1))
# self.logvals = [latex_char_logvalue(k,True) for k in self.logvals]
Gunits = G_sage.unit_gens()
if self.modulus == 2:
Gunits = [1]
self.unitgens = latex_tuple(map(str, Gunits))
self.genvalues = chi_sage.values_on_gens() # what is the use ?
self.genlogvalues = [chi.logvalue(k) for k in Gunits]
self.genvaluestex = latex_tuple(map(latex_char_logvalue, self.genlogvalues))
self.bound = 5 * 1024
if chi.is_even():
self.parity = 'Even'
else:
self.parity = 'Odd'
if self.primitive == "False":
self.inducedchar = chi.primitive_character()
self.inducedchar_isprim = self.inducedchar.is_primitive()
self.inducedchar_modulus = self.inducedchar.modulus()
self.inducedchar_conductor = self.inducedchar.conductor()
F = DirichletGroup_conrey(self.inducedchar_modulus)
if self.number == 1:
self.inducedchar_number = 1
else:
for chi in F:
j = chi.number()
if chi == self.inducedchar:
self.inducedchar_number = j
break
self.inducedchar_tex = r"\(\chi_{%s}(%s,\cdot)\)" % (
self.inducedchar_modulus, self.inducedchar_number)
# if self.primitive == 'True':
# self.primtf = True
# else:
# self.primtf = False
# Set data for related number fields
order2 = int(self.order)
if order2 % 4 == 2:
order2 = order2 / 2
self.valuefield = r'\(\mathbb{Q}(\zeta_{%d})\)' % order2
if order2 == 1:
self.valuefield = r'\(\mathbb{Q}\)'
if order2 == 4:
self.valuefield = r'\(\mathbb{Q}(i)\)'
valuewnf = WebNumberField.from_cyclo(order2)
if not valuewnf.is_null():
self.valuefield_label = valuewnf.label
else:
self.valuefield_label = ''
self.texname = r'\chi_{%d}(%d, \cdot)' % (self.modulus, self.number)
self.kername = r'\(\mathbb{Q}(\zeta_{%d})^{\ker %s}\)' % (self.modulus, self.texname)
if self.order < 16:
pol = str(gp.galoissubcyclo(self.modulus, self.chi_sage.kernel()))
sagepol = PolynomialRing(QQ, 'x')(pol)
R = sagepol.parent()
nf_pol = R(pari(sagepol).polredabs())
self.nf_pol = "\( %s \)" % latex(nf_pol)
wnf = WebNumberField.from_coeffs([int(c) for c in nf_pol.coeffs()])
if wnf.is_null():
self.nf_friend = ''
else:
self.nf_friend = '/NumberField/' + str(wnf.label)
self.nf_label = wnf.label
else:
self.nf_pol = ''
self.nf_friend = ''
if self.order == 2:
if self.conductor % 2 == 1:
self.kronsymbol = r"\begin{equation} \chi_{%s}(a) = " % (self.number)
self.kronsymbol += r"\left(\frac{a}{%s}\right)" % (self.conductor)
self.kronsymbol += r"\end{equation}"
else:
if chi.is_even():
self.kronsymbol = r"\begin{equation} \chi_{%s}(a) = " % (self.number)
self.kronsymbol += r"\left(\frac{a}{%s}\right)" % (self.conductor)
self.kronsymbol += r"\end{equation}"
else:
self.kronsymbol = r"\begin{equation} \chi_{%s}(a) = " % (self.number)
self.kronsymbol += r"\left(\frac{a}{%s}\right)" % (self.conductor)
self.kronsymbol += r"\end{equation}"
self.credit = "Sage"
self.title = r"Dirichlet Character: \(\chi_{%s}(%s,\cdot)\)" % (self.modulus, self.number)
return chi
def label2nf(label):
return WebNumberField(label).K()
def dirichletcharacter(self):
#######################################################################################
## Conrey's naming convention for Dirichlet Characters
#######################################################################################
G = DirichletGroup_conrey(self.modulus)
G_sage = G.standard_dirichlet_group()
self.level = self.modulus
if self.modulus == 1 or self.number % self.modulus != 0:
chi = G[self.number]
chi_sage = chi.sage_character()
self.chi_sage = chi_sage
self.zetaorder = G_sage.zeta_order()
# if len(chi_sage.values_on_gens()) == 1:
### self.genvaluestex = latex(chi_sage.values_on_gens()[0])
# else:
### self.genvaluestex = latex(chi_sage.values_on_gens())
# chizero = G_sage[0]
self.char = str(chi)
if chi.is_primitive():
self.primitive = "True"
else:
self.primitive = "False"
self.conductor = chi.conductor()
self.order = chi.multiplicative_order()
self.galoisorbit = [
power_mod(self.number, k, self.modulus)
for k in xrange(1, self.order) if gcd(k, self.order) == 1
]
self.vals = chi.values()
self.logvals = log_value(self.modulus, self.number)
# self.logvals = map(chi.logvalue, range(1,self.modulus+1))
# self.logvals = [latex_char_logvalue(k,True) for k in self.logvals]
Gunits = G_sage.unit_gens()
if self.modulus == 2:
Gunits = [1]
self.unitgens = latex_tuple(map(str, Gunits))
self.genvalues = chi_sage.values_on_gens() # what is the use ?
self.genlogvalues = [chi.logvalue(k) for k in Gunits]
self.genvaluestex = latex_tuple(
map(latex_char_logvalue, self.genlogvalues))
self.bound = 5 * 1024
if chi.is_even():
self.parity = 'Even'
else:
self.parity = 'Odd'
if self.primitive == "False":
self.inducedchar = chi.primitive_character()
self.inducedchar_isprim = self.inducedchar.is_primitive()
self.inducedchar_modulus = self.inducedchar.modulus()
self.inducedchar_conductor = self.inducedchar.conductor()
F = DirichletGroup_conrey(self.inducedchar_modulus)
if self.number == 1:
self.inducedchar_number = 1
else:
for chi in F:
j = chi.number()
if chi == self.inducedchar:
self.inducedchar_number = j
break
self.inducedchar_tex = r"\(\chi_{%s}(%s,\cdot)\)" % (
self.inducedchar_modulus, self.inducedchar_number)
# if self.primitive == 'True':
# self.primtf = True
# else:
# self.primtf = False
# Set data for related number fields
order2 = int(self.order)
if order2 % 4 == 2:
order2 = order2 / 2
self.valuefield = r'\(\mathbb{Q}(\zeta_{%d})\)' % order2
if order2 == 1:
self.valuefield = r'\(\mathbb{Q}\)'
if order2 == 4:
self.valuefield = r'\(\mathbb{Q}(i)\)'
valuewnf = WebNumberField.from_cyclo(order2)
if not valuewnf.is_null():
self.valuefield_label = valuewnf.label
else:
self.valuefield_label = ''
self.texname = r'\chi_{%d}(%d, \cdot)' % (self.modulus,
self.number)
self.kername = r'\(\mathbb{Q}(\zeta_{%d})^{\ker %s}\)' % (
self.modulus, self.texname)
if self.order < 16:
pol = str(
gp.galoissubcyclo(self.modulus, self.chi_sage.kernel()))
sagepol = PolynomialRing(QQ, 'x')(pol)
R = sagepol.parent()
nf_pol = R(pari(sagepol).polredabs())
self.nf_pol = "\( %s \)" % latex(nf_pol)
wnf = WebNumberField.from_coeffs(
[int(c) for c in nf_pol.coeffs()])
if wnf.is_null():
self.nf_friend = ''
else:
self.nf_friend = '/NumberField/' + str(wnf.label)
self.nf_label = wnf.label
else:
self.nf_pol = ''
self.nf_friend = ''
if self.order == 2:
if self.conductor % 2 == 1:
self.kronsymbol = r"\begin{equation} \chi_{%s}(a) = " % (
self.number)
self.kronsymbol += r"\left(\frac{a}{%s}\right)" % (
self.conductor)
self.kronsymbol += r"\end{equation}"
else:
if chi.is_even():
self.kronsymbol = r"\begin{equation} \chi_{%s}(a) = " % (
self.number)
self.kronsymbol += r"\left(\frac{a}{%s}\right)" % (
self.conductor)
self.kronsymbol += r"\end{equation}"
else:
self.kronsymbol = r"\begin{equation} \chi_{%s}(a) = " % (
self.number)
self.kronsymbol += r"\left(\frac{a}{%s}\right)" % (
self.conductor)
self.kronsymbol += r"\end{equation}"
self.credit = "Sage"
self.title = r"Dirichlet Character: \(\chi_{%s}(%s,\cdot)\)" % (
self.modulus, self.number)
return chi
