示例#1
0
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
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
示例#3
0
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
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
示例#5
0
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:
        gpK = nf.gpK()
        rootof1coeff = gpK.nfrootsof1()
        rootofunityorder = int(rootof1coeff[1])
        rootof1coeff = rootof1coeff[2]
        rootofunity = web_latex(
            Ra(
                str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace(
                    'x', 'a')))
        rootofunity += ' (order $%d$)' % rootofunityorder
    else:
        rootofunity = web_latex(Ra('-1')) + ' (order $2$)'

    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': rootofunity,
        'fund_units': nf.units(),
        'grh_label': grh_label,
        'loc_alg': loc_alg
    })

    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))
    info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
    if 'dirichlet_group' in info:
        info['friends'].append(('Dirichlet character group',
                                url_for("characters.dirichlet_group_table",
                                        modulus=int(conductor),
                                        char_number_list=','.join(
                                            [str(a) for a in dirichlet_chars]),
                                        poly=info['polynomial'])))
    resinfo = []
    galois_closure = nf.galois_closure()
    if galois_closure[0] > 0:
        if len(galois_closure[1]) > 0:
            resinfo.append(('gc', galois_closure[1]))
            if len(galois_closure[2]) > 0:
                info['friends'].append(('Galois closure',
                                        url_for(".by_label",
                                                label=galois_closure[2][0])))
        else:
            resinfo.append(('gc', [dnc]))

    sextic_twins = nf.sextic_twin()
    if sextic_twins[0] > 0:
        if len(sextic_twins[1]) > 0:
            resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
        else:
            resinfo.append(('sex', dnc))

    siblings = nf.siblings()
    # [degsib list, label list]
    # first is list of [deg, num expected, list of knowls]
    if len(siblings[0]) > 0:
        for sibdeg in siblings[0]:
            if len(sibdeg[2]) == 0:
                sibdeg[2] = dnc
            else:
                sibdeg[2] = ', '.join(sibdeg[2])
                if len(sibdeg[2]) < sibdeg[1]:
                    sibdeg[2] += ', some ' + dnc

        resinfo.append(('sib', siblings[0]))
        for lab in siblings[1]:
            if lab != '':
                labparts = lab.split('.')
                info['friends'].append(("Degree %s sibling" % labparts[0],
                                        url_for(".by_label", label=lab)))

    arith_equiv = nf.arith_equiv()
    if arith_equiv[0] > 0:
        if len(arith_equiv[1]) > 0:
            resinfo.append(
                ('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
            for aelab in arith_equiv[2]:
                info['friends'].append(('Arithmetically equivalent sibling',
                                        url_for(".by_label", label=aelab)))
        else:
            resinfo.append(('ae', dnc, len(arith_equiv[1])))

    info['resinfo'] = resinfo
    learnmore = learnmore_list()
    #if info['signature'] == [0,1]:
    #    info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
    # With Galois group labels, probably not needed here
    # info['learnmore'] = [('Global number field labels',
    # url_for(".render_labels_page")), ('Galois group
    # labels',url_for(".render_groups_page")),
    # (Completename,url_for(".render_discriminants_page"))]
    title = "Global Number Field %s" % info['label']

    if npr == 1:
        primes = 'prime'
    else:
        primes = 'primes'

    properties2 = [('Label', label), ('Degree', '$%s$' % data['degree']),
                   ('Signature', '$%s$' % data['signature']),
                   ('Discriminant', '$%s$' % data['disc_factor']),
                   ('Ramified ' + primes + '', '$%s$' % ram_primes),
                   ('Class number', '%s %s' % (data['class_number'], grh_lab)),
                   ('Class group',
                    '%s %s' % (data['class_group_invs'], grh_lab)),
                   ('Galois Group', group_display_short(data['degree'], t))]
    downloads = []
    for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
        downloads.append(('Download {} code'.format(lang[0]),
                          url_for(".nf_code_download",
                                  nf=label,
                                  download_type=lang[1])))
    from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
    try:
        info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
        v = nf.factor_perm_repn(info["tim_number_field"])

        def dopow(m):
            if m == 0: return ''
            if m == 1: return '*'
            return '*<sup>%d</sup>' % m

        info["mydecomp"] = [dopow(x) for x in v]
    except AttributeError:
        pass
    return render_template("number_field.html",
                           properties2=properties2,
                           credit=NF_credit,
                           title=title,
                           bread=bread,
                           code=nf.code,
                           friends=info.pop('friends'),
                           downloads=downloads,
                           learnmore=learnmore,
                           info=info)
示例#6
0
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))
    info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
    if 'dirichlet_group' in info:
        info['friends'].append(('Dirichlet group', url_for("characters.dirichlet_group_table",
                                                           modulus=int(conductor),
                                                           char_number_list=','.join(
                                                               [str(a) for a in dirichlet_chars]),
                                                           poly=info['polynomial'])))
    info['learnmore'] = [('Global number field labels', url_for(
        ".render_labels_page")), 
        (Completename, url_for(".render_discriminants_page")),
        ('How data was computed', url_for(".how_computed_page"))]
    if info['signature'] == [0,1]:
        info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
    # With Galois group labels, probably not needed here
    # info['learnmore'] = [('Global number field labels',
    # url_for(".render_labels_page")), ('Galois group
    # labels',url_for(".render_groups_page")),
    # (Completename,url_for(".render_discriminants_page"))]
    title = "Global Number Field %s" % info['label']

    if npr == 1:
        primes = 'prime'
    else:
        primes = 'primes'

    properties2 = [('Label', label),
                   ('Degree', '%s' % data['degree']),
                   ('Signature', '$%s$' % data['signature']),
                   ('Discriminant', '$%s$' % data['disc_factor']),
                   ('Ramified ' + primes + '', '$%s$' % ram_primes),
                   ('Class number', '%s %s' % (data['class_number'], grh_lab)),
                   ('Class group', '%s %s' % (data['class_group_invs'], grh_lab)),
                   ('Galois Group', group_display_short(data['degree'], t, C))
                   ]
    from lmfdb.math_classes import NumberFieldGaloisGroup
    try:
        info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
        v = nf.factor_perm_repn(info["tim_number_field"])
        def dopow(m):
            if m==0: return ''
            if m==1: return '*'
            return '*<sup>%d</sup>'% m

        info["mydecomp"] = [dopow(x) for x in v]
    except AttributeError:
        pass
#    del info['_id']
    return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
示例#7
0
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:
        gpK = nf.gpK()
        rootof1coeff = gpK.nfrootsof1()
        rootofunityorder = int(rootof1coeff[1])
        rootof1coeff = rootof1coeff[2]
        rootofunity = web_latex(Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a'))) 
        rootofunity += ' (order $%d$)' % rootofunityorder
    else:
        rootofunity = web_latex(Ra('-1'))+ ' (order $2$)'

    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': rootofunity,
        'fund_units': nf.units(),
        'grh_label': grh_label,
        'loc_alg': loc_alg
    })

    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))
    info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
    if 'dirichlet_group' in info:
        info['friends'].append(('Dirichlet character group', url_for("characters.dirichlet_group_table",
                                                           modulus=int(conductor),
                                                           char_number_list=','.join(
                                                               [str(a) for a in dirichlet_chars]),
                                                           poly=info['polynomial'])))
    resinfo=[]
    galois_closure = nf.galois_closure()
    if galois_closure[0]>0:
        if len(galois_closure[1])>0:
            resinfo.append(('gc', galois_closure[1]))
            if len(galois_closure[2]) > 0:
                info['friends'].append(('Galois closure',url_for(".by_label", label=galois_closure[2][0])))
        else:
            resinfo.append(('gc', [dnc]))

    sextic_twins = nf.sextic_twin()
    if sextic_twins[0]>0:
        if len(sextic_twins[1])>0:
            resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
        else:
            resinfo.append(('sex', dnc))

    siblings = nf.siblings()
    # [degsib list, label list]
    # first is list of [deg, num expected, list of knowls]
    if len(siblings[0])>0:
        for sibdeg in siblings[0]:
            if len(sibdeg[2]) ==0:
                sibdeg[2] = dnc
            else:
                sibdeg[2] = ', '.join(sibdeg[2])
                if len(sibdeg[2])<sibdeg[1]:
                    sibdeg[2] += ', some '+dnc
                
        resinfo.append(('sib', siblings[0]))
        for lab in siblings[1]:
            if lab != '':
                labparts = lab.split('.')
                info['friends'].append(("Degree %s sibling"%labparts[0] ,url_for(".by_label", label=lab)))

    arith_equiv = nf.arith_equiv()
    if arith_equiv[0]>0:
        if len(arith_equiv[1])>0:
            resinfo.append(('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
            for aelab in arith_equiv[2]:
                info['friends'].append(('Arithmetically equivalent sibling',url_for(".by_label", label=aelab)))
        else:
            resinfo.append(('ae', dnc, len(arith_equiv[1])))

    info['resinfo'] = resinfo
    learnmore = learnmore_list()
    #if info['signature'] == [0,1]:
    #    info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
    # With Galois group labels, probably not needed here
    # info['learnmore'] = [('Global number field labels',
    # url_for(".render_labels_page")), ('Galois group
    # labels',url_for(".render_groups_page")),
    # (Completename,url_for(".render_discriminants_page"))]
    title = "Global Number Field %s" % info['label']

    if npr == 1:
        primes = 'prime'
    else:
        primes = 'primes'

    properties2 = [('Label', label),
                   ('Degree', '$%s$' % data['degree']),
                   ('Signature', '$%s$' % data['signature']),
                   ('Discriminant', '$%s$' % data['disc_factor']),
                   ('Ramified ' + primes + '', '$%s$' % ram_primes),
                   ('Class number', '%s %s' % (data['class_number'], grh_lab)),
                   ('Class group', '%s %s' % (data['class_group_invs'], grh_lab)),
                   ('Galois Group', group_display_short(data['degree'], t))
                   ]
    downloads = []
    for lang in [["Magma","magma"], ["SageMath","sage"], ["Pari/GP", "gp"]]:
        downloads.append(('Download {} code'.format(lang[0]),
                          url_for(".nf_code_download", nf=label, download_type=lang[1])))
    from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
    try:
        info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
        v = nf.factor_perm_repn(info["tim_number_field"])
        def dopow(m):
            if m==0: return ''
            if m==1: return '*'
            return '*<sup>%d</sup>'% m

        info["mydecomp"] = [dopow(x) for x in v]
    except AttributeError:
        pass
    return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), downloads=downloads, learnmore=learnmore, info=info)