Python WebNumberField.coeffs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 coeffs [as 别名]
def download_hmf_magma(**args):
    label = str(args['label'])
    f = get_hmf(label)
    if f is None:
        return "No such form"

    F = WebNumberField(f['field_label'])
    F_hmf = get_hmf_field(f['field_label'])

    hecke_pol  = f['hecke_polynomial']
    hecke_eigs = f['hecke_eigenvalues']
    AL_eigs    = f['AL_eigenvalues']

    outstr = 'P<x> := PolynomialRing(Rationals());\n'
    outstr += 'g := P!' + str(F.coeffs()) + ';\n'
    outstr += 'F<w> := NumberField(g);\n'
    outstr += 'ZF := Integers(F);\n\n'
#    outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
#    outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'

    outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'

    outstr += 'primesArray := [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
    outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'

    if hecke_pol != 'x':
        outstr += 'heckePol := ' + hecke_pol + ';\n'
        outstr += 'K<e> := NumberField(heckePol);\n'
    else:
        outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'

    outstr += '\nheckeEigenvaluesArray := [' + ', '.join([st for st in hecke_eigs]) + '];'
    outstr += '\nheckeEigenvalues := AssociativeArray();\n'
    outstr += 'for i := 1 to #heckeEigenvaluesArray do\n  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'

    outstr += 'ALEigenvalues := AssociativeArray();\n'
    for s in AL_eigs:
        outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + s[1] + ';\n'

    outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'

    outstr += '/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):\n'
    outstr += 'M := HilbertCuspForms(F, NN);\n'
    outstr += 'S := NewSubspace(M);\n'
    outstr += '// SetVerbose("ModFrmHil", 1);\n'
    outstr += 'newspaces := NewformDecomposition(S);\n'
    outstr += 'newforms := [Eigenform(U) : U in newspaces];\n'
    outstr += 'ppind := 0;\n'
    outstr += 'while #newforms gt 1 do\n'
    outstr += '  pp := primes[ppind];\n'
    outstr += '  newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];\n'
    outstr += 'end while;\n'
    outstr += 'f := newforms[1];\n'
    outstr += '// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;\n'
    outstr += '*/\n'

    return outstr

# 需要导入模块: from lmfdb.WebNumberField import WebNumberField [as 别名]
# 或者: from lmfdb.WebNumberField.WebNumberField import coeffs [as 别名]
def download_hmf_sage(**args):
    label = str(args['label'])
    f = get_hmf(label)
    if f is None:
        return "No such form"

    hecke_pol  = f['hecke_polynomial']
    hecke_eigs = f['hecke_eigenvalues']
    AL_eigs    = f['AL_eigenvalues']

    F = WebNumberField(f['field_label'])
    F_hmf = get_hmf_field(f['field_label'])

    outstr = 'P.<x> = PolynomialRing(QQ)\n'
    outstr += 'g = P(' + str(F.coeffs()) + ')\n'
    outstr += 'F.<w> = NumberField(g)\n'
    outstr += 'ZF = F.ring_of_integers()\n\n'

    outstr += 'NN = ZF.ideal(' + f["level_ideal"] + ')\n\n'

    outstr += 'primes_array = [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[',
                                                                                      '],\\\n[') + ']\n'
    outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'

    if hecke_pol != 'x':
        outstr += 'heckePol = ' + hecke_pol + '\n'
        outstr += 'K.<e> = NumberField(heckePol)\n'
    else:
        outstr += 'heckePol = x\nK = QQ\ne = 1\n'

    outstr += '\nhecke_eigenvalues_array = [' + ', '.join([st for st in hecke_eigs]) + ']'
    outstr += '\nhecke_eigenvalues = {}\n'
    outstr += 'for i in range(len(hecke_eigenvalues_array)):\n    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'

    outstr += 'AL_eigenvalues = {}\n'
    for s in AL_eigs:
        outstr += 'AL_eigenvalues[ZF.ideal(%s)] = %s\n' % (s[0],s[1])

    outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'

    return outstr

# 需要导入模块: from lmfdb.WebNumberField import WebNumberField [as 别名]
# 或者: from lmfdb.WebNumberField.WebNumberField import coeffs [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 coeffs [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.........