Python WebNumberField.dirichlet_group方法代码示例

# 需要导入模块: from lmfdb.WebNumberField import WebNumberField [as 别名]
# 或者: from lmfdb.WebNumberField.WebNumberField import dirichlet_group [as 别名]
    def __init__(self, **args):
        constructor_logger(self, args)
        self.motivic_weight = 0
        # Check for compulsory arguments
        if not 'label' in args.keys():
            raise Exception("You have to supply a label for a Dedekind zeta function")

        # Initialize default values

        # Put the arguments into the object dictionary
        self.__dict__.update(args)

        # Fetch the polynomial of the field from the database
        wnf = WebNumberField(self.label)
        # poly_coeffs = wnf.coeffs()

        # Extract the L-function information from the polynomial
        R = QQ['x']
        (x,) = R._first_ngens(1)
        # self.polynomial = sum([poly_coeffs[i]*x**i for i in range(len(poly_coeffs))])
        self.NF = wnf.K()  # NumberField(self.polynomial, 'a')
        self.signature = wnf.signature()  # self.NF.signature()
        self.sign = 1
        self.quasidegree = sum(self.signature)
        self.level = wnf.disc().abs()  # self.NF.discriminant().abs()
        self.degreeofN = self.NF.degree()

        self.Q_fe = float(
            sqrt(self.level) / (2 ** (self.signature[1]) * (math.pi) ** (float(self.degreeofN) / 2.0)))

        self.kappa_fe = self.signature[0] * [0.5] + self.signature[1] * [1]
        self.lambda_fe = self.quasidegree * [0]
        self.mu_fe = self.signature[0] * [0]  # not in use?
        self.nu_fe = self.signature[1] * [0]  # not in use?
        self.langlands = True
        # self.degree = self.signature[0] + 2 * self.signature[1] # N = r1 +2r2
        self.degree = self.degreeofN
        self.dirichlet_coefficients = [Integer(x) for x in self.NF.zeta_coefficients(5000)]
        self.h = wnf.class_number()  # self.NF.class_number()
        self.R = wnf.regulator()  # self.NF.regulator()
        self.w = len(self.NF.roots_of_unity())
        self.res = RR(2 ** self.signature[0] * self.h * self.R / self.w)  # r1 = self.signature[0]
        self.grh = wnf.used_grh()
        if self.degree > 1:
            if wnf.is_abelian():
                cond = wnf.conductor()
                dir_group = wnf.dirichlet_group()
                # Remove 1 from the list
                j = 0
                while dir_group[j] != 1:
                    j += 1
                dir_group.pop(j)
                self.factorization = r'\(\zeta_K(s) =\) <a href="/L/Riemann/">\(\zeta(s)\)</a>'
                fullchargroup = wnf.full_dirichlet_group()
                for j in dir_group:
                    chij = fullchargroup[j]
                    mycond = chij.conductor()
                    myj = j % mycond
                    self.factorization += r'\(\;\cdot\) <a href="/L/Character/Dirichlet/%d/%d/">\(L(s,\chi_{%d}(%d, \cdot))\)</a>' % (mycond, myj, mycond, myj)

        self.poles = [1, 0]  # poles of the Lambda(s) function
        self.residues = [self.res, -self.res]  # residues of the Lambda(s) function

        self.poles_L = [1]  # poles of L(s) used by createLcalcfile_ver2
        self.residues_L = [1234]
            # residues of L(s) used by createLcalcfile_ver2, XXXXXXXXXXXX needs to be set

        self.coefficient_period = 0
        self.selfdual = True
        self.primitive = True
        self.coefficient_type = 0
        self.texname = "\\zeta_K(s)"
        self.texnamecompleteds = "\\Lambda_K(s)"
        if self.selfdual:
            self.texnamecompleted1ms = "\\Lambda_K(1-s)"
        else:
            self.texnamecompleted1ms = "\\Lambda_K(1-s)"
        self.title = "Dedekind zeta-function: $\\zeta_K(s)$"
        self.title = self.title + ", where $K$ is the " + str(self.NF).replace("in a ", "")
        self.credit = 'Sage'
        self.citation = ''

        self.generateSageLfunction()

# 需要导入模块: from lmfdb.WebNumberField import WebNumberField [as 别名]
# 或者: from lmfdb.WebNumberField.WebNumberField import dirichlet_group [as 别名]
def render_field_webpage(args):
    data = None
    C = base.getDBConnection()
    info = {}
    bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]

    # This function should not be called unless label is set.
    label = clean_input(args['label'])
    nf = WebNumberField(label)
    data = {}
    if nf.is_null():
        bread.append(('Search results', ' '))
        info['err'] = 'There is no field with label %s in the database' % label
        info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
        return search_input_error(info, bread)

    info['wnf'] = nf
    data['degree'] = nf.degree()
    data['class_number'] = nf.class_number()
    t = nf.galois_t()
    n = nf.degree()
    data['is_galois'] = nf.is_galois()
    data['is_abelian'] = nf.is_abelian()
    if nf.is_abelian():
        conductor = nf.conductor()
        data['conductor'] = conductor
        dirichlet_chars = nf.dirichlet_group()
        if len(dirichlet_chars)>0:
            data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,&middot;)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
            data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
        if data['conductor'].is_prime() or data['conductor'] == 1:
            data['conductor'] = "\(%s\)" % str(data['conductor'])
        else:
            data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
    data['galois_group'] = group_display_knowl(n, t, C)
    data['cclasses'] = cclasses_display_knowl(n, t, C)
    data['character_table'] = character_table_display_knowl(n, t, C)
    data['class_group'] = nf.class_group()
    data['class_group_invs'] = nf.class_group_invariants()
    data['signature'] = nf.signature()
    data['coefficients'] = nf.coeffs()
    nf.make_code_snippets()
    D = nf.disc()
    ram_primes = D.prime_factors()
    data['disc_factor'] = nf.disc_factored_latex()
    if D.abs().is_prime() or D == 1:
        data['discriminant'] = "\(%s\)" % str(D)
    else:
        data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
    npr = len(ram_primes)
    ram_primes = str(ram_primes)[1:-1]
    if ram_primes == '':
        ram_primes = r'\textrm{None}'
    data['frob_data'], data['seeram'] = frobs(nf)
    data['phrase'] = group_phrase(n, t, C)
    zk = nf.zk()
    Ra = PolynomialRing(QQ, 'a')
    zk = [latex(Ra(x)) for x in zk]
    zk = ['$%s$' % x for x in zk]
    zk = ', '.join(zk)
    grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
    # Short version for properties
    grh_lab = nf.short_grh_string()
    if 'Not' in str(data['class_number']):
        grh_lab=''
        grh_label=''
    pretty_label = field_pretty(label)
    if label != pretty_label:
        pretty_label = "%s: %s" % (label, pretty_label)

    info.update(data)
    if nf.degree() > 1:
        gpK = nf.gpK()
        rootof1coeff = gpK.nfrootsof1()[2]
        rootofunity = Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a'))
    else:
        rootofunity = Ra('-1')

    info.update({
        'label': pretty_label,
        'label_raw': label,
        'polynomial': web_latex_split_on_pm(nf.poly()),
        'ram_primes': ram_primes,
        'integral_basis': zk,
        'regulator': web_latex(nf.regulator()),
        'unit_rank': nf.unit_rank(),
        'root_of_unity': web_latex(rootofunity),
        'fund_units': nf.units(),
        'grh_label': grh_label
    })

    bread.append(('%s' % info['label_raw'], ' '))
    info['downloads_visible'] = True
    info['downloads'] = [('worksheet', '/')]
    info['friends'] = []
    if nf.can_class_number():
        # hide ones that take a lond time to compute on the fly
        # note that the first degree 4 number field missed the zero of the zeta function
        if abs(D**n) < 50000000:
            info['friends'].append(('L-function', "/L/NumberField/%s" % label))
#.........这里部分代码省略.........

# 需要导入模块: from lmfdb.WebNumberField import WebNumberField [as 别名]
# 或者: from lmfdb.WebNumberField.WebNumberField import dirichlet_group [as 别名]
def render_field_webpage(args):
    data = None
    info = {}
    bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]

    # This function should not be called unless label is set.
    label = clean_input(args['label'])
    nf = WebNumberField(label)
    data = {}
    if nf.is_null():
        bread.append(('Search Results', ' '))
        info['err'] = 'There is no field with label %s in the database' % label
        info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
        return search_input_error(info, bread)

    info['wnf'] = nf
    data['degree'] = nf.degree()
    data['class_number'] = nf.class_number_latex()
    ram_primes = nf.ramified_primes()
    t = nf.galois_t()
    n = nf.degree()
    data['is_galois'] = nf.is_galois()
    data['is_abelian'] = nf.is_abelian()
    if nf.is_abelian():
        conductor = nf.conductor()
        data['conductor'] = conductor
        dirichlet_chars = nf.dirichlet_group()
        if len(dirichlet_chars)>0:
            data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,&middot;)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
            data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
        if data['conductor'].is_prime() or data['conductor'] == 1:
            data['conductor'] = "\(%s\)" % str(data['conductor'])
        else:
            factored_conductor = factor_base_factor(data['conductor'], ram_primes)
            factored_conductor = factor_base_factorization_latex(factored_conductor)
            data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), factored_conductor)
    data['galois_group'] = group_display_knowl(n, t)
    data['cclasses'] = cclasses_display_knowl(n, t)
    data['character_table'] = character_table_display_knowl(n, t)
    data['class_group'] = nf.class_group()
    data['class_group_invs'] = nf.class_group_invariants()
    data['signature'] = nf.signature()
    data['coefficients'] = nf.coeffs()
    nf.make_code_snippets()
    D = nf.disc()
    data['disc_factor'] = nf.disc_factored_latex()
    if D.abs().is_prime() or D == 1:
        data['discriminant'] = "\(%s\)" % str(D)
    else:
        data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
    data['frob_data'], data['seeram'] = frobs(nf)
    # Bad prime information
    npr = len(ram_primes)
    ramified_algebras_data = nf.ramified_algebras_data()
    if isinstance(ramified_algebras_data,str):
        loc_alg = ''
    else:
        # [label, latex, e, f, c, gal]
        loc_alg = ''
        for j in range(npr):
            if ramified_algebras_data[j] is None:
                loc_alg += '<tr><td>%s<td colspan="7">Data not computed'%str(ram_primes[j])
            else:
                mydat = ramified_algebras_data[j]
                p = ram_primes[j]
                loc_alg += '<tr><td rowspan="%d">$%s$</td>'%(len(mydat),str(p))
                mm = mydat[0]
                myurl = url_for('local_fields.by_label', label=mm[0])
                lab = mm[0]
                if mm[3]*mm[2]==1:
                    lab = r'$\Q_{%s}$'%str(p)
                loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$'%(myurl,lab,mm[1],mm[2],mm[3],mm[4],mm[5],show_slope_content(mm[8],mm[6],mm[7]))
                for mm in mydat[1:]:
                    lab = mm[0]
                    if mm[3]*mm[2]==1:
                        lab = r'$\Q_{%s}$'%str(p)
                    loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$'%(myurl,lab,mm[1],mm[2],mm[3],mm[4],mm[5],show_slope_content(mm[8],mm[6],mm[7]))
        loc_alg += '</tbody></table>'

    ram_primes = str(ram_primes)[1:-1]
    if ram_primes == '':
        ram_primes = r'\textrm{None}'
    data['phrase'] = group_phrase(n, t)
    zk = nf.zk()
    Ra = PolynomialRing(QQ, 'a')
    zk = [latex(Ra(x)) for x in zk]
    zk = ['$%s$' % x for x in zk]
    zk = ', '.join(zk)
    grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
    # Short version for properties
    grh_lab = nf.short_grh_string()
    if 'Not' in str(data['class_number']):
        grh_lab=''
        grh_label=''
    pretty_label = field_pretty(label)
    if label != pretty_label:
        pretty_label = "%s: %s" % (label, pretty_label)

    info.update(data)
    if nf.degree() > 1:
#.........这里部分代码省略.........