Python lmfdb.WebNumberField类代码示例

def download_hmf_magma(**args):
    C = getDBConnection()
    data = None
    label = str(args['label'])
    f = C.hmfs.forms.find_one({'label': label})
    if f is None:
        return "No such form"

    F = WebNumberField(f['field_label'])
    F_hmf = C.hmfs.fields.find_one({'label': f['field_label']})

    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 f["hecke_polynomial"] != 'x':
        outstr += 'heckePol := ' + f["hecke_polynomial"] + ';\n'
        outstr += 'K<e> := NumberField(heckePol);\n'
    else:
        outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'

    outstr += '\nheckeEigenvaluesArray := [' + ', '.join([st for st in f["hecke_eigenvalues"]]) + '];'
    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 f["AL_eigenvalues"]:
        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):
    C = getDBConnection()
    data = None
    label = str(args["label"])
    f = C.hmfs.forms.find_one({"label": label})
    if f is None:
        return "No such form"

    F = WebNumberField(f["field_label"])
    F_hmf = C.hmfs.fields.find_one({"label": f["field_label"]})

    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 f["hecke_polynomial"] != "x":
        outstr += "heckePol := " + f["hecke_polynomial"] + ";\n"
        outstr += "K<e> := NumberField(heckePol);\n"
    else:
        outstr += "heckePol := x;\nK := Rationals(); e := 1;\n"

    outstr += "\nheckeEigenvaluesArray := [" + ", ".join([st for st in f["hecke_eigenvalues"]]) + "];"
    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 f["AL_eigenvalues"]:
        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_sage(**args):
    C = getDBConnection()
    data = None
    label = str(args['label'])
    f = C.hmfs.forms.find_one({'label': label})
    if f is None:
        return "No such form"

    F = WebNumberField(f['field_label'])
    F_hmf = C.hmfs.fields.find_one({'label': 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 f["hecke_polynomial"] != 'x':
        outstr += 'hecke_pol = ' + f["hecke_polynomial"] + '\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 f["hecke_eigenvalues"]]) + ']'
    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 f["AL_eigenvalues"]:
        outstr += 'ALEigenvalues[ZF.ideal(s[0])] = s[1]\n'

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

    return outstr

def download_hmf_sage(**args):
    C = getDBConnection()
    data = None
    label = str(args["label"])
    f = C.hmfs.forms.find_one({"label": label})
    if f is None:
        return "No such form"

    F = WebNumberField(f["field_label"])
    F_hmf = C.hmfs.fields.find_one({"label": 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 f["hecke_polynomial"] != "x":
        outstr += "hecke_pol = " + f["hecke_polynomial"] + "\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 f["hecke_eigenvalues"]]) + "]"
    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 f["AL_eigenvalues"]:
        outstr += "ALEigenvalues[ZF.ideal(s[0])] = s[1]\n"

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

    return outstr

def render_hmf_webpage(**args):
    C = getDBConnection()
    if 'data' in args:
        data = args['data']
        label = data['label']
    else:
        label = str(args['label'])
        data = C.hmfs.forms.find_one({'label': label})
    if data is None:
        return "No such form"
    info = {}
    try:
        info['count'] = args['count']
    except KeyError:
        info['count'] = 10

    try:
        numeigs = request.args['numeigs']
        numeigs = int(numeigs)
    except:
        numeigs = 20

    hmf_field = C.hmfs.fields.find_one({'label': data['field_label']})
    gen_name = findvar(hmf_field['ideals'])
    nf = WebNumberField(data['field_label'], gen_name=gen_name)
    info['field'] = nf
    info['base_galois_group'] = nf.galois_string()
    info['field_degree'] = nf.degree()
    info['field_disc'] = str(nf.disc())
    info['field_poly'] = teXify_pol(str(nf.poly()))

    info.update(data)

    info['downloads'] = [
        ('Download to Magma', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='magma')),
        ('Download to Sage', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='sage'))
        ]
    if hmf_field['narrow_class_no'] == 1:
        info['friends'] = [('L-function',
                            url_for("l_functions.l_function_hmf_page", field=info['field_label'], label=info['label'], character='0', number='0'))]
    else:
        info['friends'] = [('L-function not available', "")]
    if data['dimension'] == 1:   # Try to attach associated elliptic curve
        lab = split_class_label(info['label'])
        ec_from_hmf = db_ecnf().find_one({"label": label + '1'})
        if ec_from_hmf == None:
            info['friends'] += [('Elliptic curve not available', "")]
        else:
            info['friends'] += [('Isogeny class ' + info['label'], url_for("ecnf.show_ecnf_isoclass", nf=lab[0], conductor_label=lab[1], class_label=lab[2]))]

    bread = [('Hilbert Modular Forms', url_for(".hilbert_modular_form_render_webpage")), ('%s' % data[
                                                                                         'label'], ' ')]

    t = "Hilbert Cusp Form %s" % info['label']

    forms_space = C.hmfs.forms.find(
        {'field_label': data['field_label'], 'level_ideal': data['level_ideal']})
    dim_space = 0
    for v in forms_space:
        dim_space += v['dimension']

    info['newspace_dimension'] = dim_space

    w = polygen(QQ, 'w')
    e = polygen(QQ, 'e')
    eigs = data['hecke_eigenvalues']
    eigs = eigs[:min(len(eigs), numeigs)]

    primes = hmf_field['primes']
    n = min(len(eigs), len(primes))
    info['eigs'] = [{'eigenvalue': teXify_pol(eigs[i]),
                     'prime_ideal': teXify_pol(primes[i]),
                     'prime_norm': primes[i][1:primes[i].index(',')]} for i in range(n)]

    try:
        display_eigs = request.args['display_eigs']
        if display_eigs in ['True', 'true', '1', 'yes']:
            display_eigs = True
        else:
            display_eigs = False
    except KeyError:
        display_eigs = False

    if 'numeigs' in request.args:
        display_eigs = True

    info['hecke_polynomial'] = teXify_pol(info['hecke_polynomial'])

    if 'AL_eigenvalues_fixed' in data:
        if data['AL_eigenvalues_fixed'] == 'done':
            info['AL_eigs'] = [{'eigenvalue': teXify_pol(al[1]),
                                'prime_ideal': teXify_pol(al[0]),
                                'prime_norm': al[0][1:al[0].index(',')]} for al in data['AL_eigenvalues']]
        else:
            info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
    else:
        info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
    info['AL_eigs_count'] = len(info['AL_eigs']) != 0

    max_eig_len = max([len(eig['eigenvalue']) for eig in info['eigs']])
#.........这里部分代码省略.........

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' % label2
        info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
        return search_input_error(info, bread)

    info['wnf'] = nf
    from lmfdb.WebNumberField import nf_display_knowl
    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()
    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.K())
    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)
    info.update({
        'label': pretty_label,
        'label_raw': label,
        'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
        'ram_primes': ram_primes,
        'integral_basis': zk,
        'regulator': web_latex(nf.regulator()),
        'unit_rank': nf.unit_rank(),
        'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
        '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'])))
#.........这里部分代码省略.........

def render_hmf_webpage(**args):
    C = getDBConnection()
    data = None
    if "label" in args:
        label = str(args["label"])
        data = C.hmfs.forms.find_one({"label": label})
    if data is None:
        return "No such field"
    info = {}
    try:
        info["count"] = args["count"]
    except KeyError:
        info["count"] = 10

    try:
        numeigs = request.args["numeigs"]
        numeigs = int(numeigs)
    except:
        numeigs = 20

    hmf_field = C.hmfs.fields.find_one({"label": data["field_label"]})
    nf = WebNumberField(data["field_label"])
    info["base_galois_group"] = nf.galois_string()
    info["field_degree"] = nf.degree()
    info["field_disc"] = str(nf.disc())
    info["field_poly"] = teXify_pol(str(nf.poly()))

    info.update(data)

    info["downloads"] = [
        (
            "Download to Magma",
            url_for(
                ".render_hmf_webpage_download",
                field_label=info["field_label"],
                label=info["label"],
                download_type="magma",
            ),
        ),
        (
            "Download to Sage",
            url_for(
                ".render_hmf_webpage_download",
                field_label=info["field_label"],
                label=info["label"],
                download_type="sage",
            ),
        ),
    ]
    info["friends"] = []
    info["friends"] = [
        (
            "L-function",
            url_for(
                "l_functions.l_function_hmf_page",
                field=info["field_label"],
                label=info["label"],
                character="0",
                number="0",
            ),
        )
    ]

    # info['learnmore'] = [('Number Field labels',
    # url_for("render_labels_page")), ('Galois group
    # labels',url_for("render_groups_page")), ('Discriminant
    # ranges',url_for("render_discriminants_page"))]

    bread = [("Hilbert Modular Forms", url_for(".hilbert_modular_form_render_webpage")), ("%s" % data["label"], " ")]

    t = "Hilbert Cusp Form %s" % info["label"]
    credit = 'Lassina Dembele, Steve Donnelly and <A HREF="http://www.cems.uvm.edu/~voight/">John Voight</A>'

    forms_space = C.hmfs.forms.find({"field_label": data["field_label"], "level_ideal": data["level_ideal"]})
    dim_space = 0
    for v in forms_space:
        dim_space += v["dimension"]

    info["newspace_dimension"] = dim_space

    w = polygen(QQ, "w")
    e = polygen(QQ, "e")
    eigs = data["hecke_eigenvalues"]
    eigs = eigs[: min(len(eigs), numeigs)]

    primes = hmf_field["primes"]
    n = min(len(eigs), len(primes))
    info["eigs"] = [
        {
            "eigenvalue": teXify_pol(eigs[i]),
            "prime_ideal": teXify_pol(primes[i]),
            "prime_norm": primes[i][1 : primes[i].index(",")],
        }
        for i in range(n)
    ]

    try:
        display_eigs = request.args["display_eigs"]
        if display_eigs in ["True", "true", "1", "yes"]:
            display_eigs = True
#.........这里部分代码省略.........

def render_hmf_webpage(**args):
    C = getDBConnection()
    data = None
    if 'label' in args:
        label = str(args['label'])
        data = C.hmfs.forms.find_one({'label': label})
    if data is None:
        return "No such field"
    info = {}
    try:
        info['count'] = args['count']
    except KeyError:
        info['count'] = 10

    try:
        numeigs = request.args['numeigs']
        numeigs = int(numeigs)
    except:
        numeigs = 20

    hmf_field = C.hmfs.fields.find_one({'label': data['field_label']})
    nf = WebNumberField(data['field_label'])
    info['base_galois_group'] = nf.galois_string()
    info['field_degree'] = nf.degree()
    info['field_disc'] = str(nf.disc())
    info['field_poly'] = teXify_pol(str(nf.poly()))

    info.update(data)

    info['downloads'] = [
        ('Download to Magma', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='magma')),
        ('Download to Sage', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='sage'))
        ]
    info['friends'] = []
    info['friends'] = [('L-function',
                        url_for("l_functions.l_function_hmf_page", field=info['field_label'], label=info['label'], character='0', number='0'))]

# info['learnmore'] = [('Number Field labels',
# url_for("render_labels_page")), ('Galois group
# labels',url_for("render_groups_page")), ('Discriminant
# ranges',url_for("render_discriminants_page"))]

    bread = [('Hilbert Modular Forms', url_for(".hilbert_modular_form_render_webpage")), ('%s' % data[
                                                                                         'label'], ' ')]

    t = "Hilbert Cusp Form %s" % info['label']
    credit = 'Lassina Dembele, Steve Donnelly and <A HREF="http://www.cems.uvm.edu/~voight/">John Voight</A>'

    forms_space = C.hmfs.forms.find(
        {'field_label': data['field_label'], 'level_ideal': data['level_ideal']})
    dim_space = 0
    for v in forms_space:
        dim_space += v['dimension']

    info['newspace_dimension'] = dim_space

    w = polygen(QQ, 'w')
    e = polygen(QQ, 'e')
    eigs = data['hecke_eigenvalues']
    eigs = eigs[:min(len(eigs), numeigs)]

    primes = hmf_field['primes']
    n = min(len(eigs), len(primes))
    info['eigs'] = [{'eigenvalue': teXify_pol(eigs[i]),
                     'prime_ideal': teXify_pol(primes[i]),
                     'prime_norm': primes[i][1:primes[i].index(',')]} for i in range(n)]

    try:
        display_eigs = request.args['display_eigs']
        if display_eigs in ['True', 'true', '1', 'yes']:
            display_eigs = True
        else:
            display_eigs = False
    except KeyError:
        display_eigs = False

    if 'numeigs' in request.args:
        display_eigs = True

    info['hecke_polynomial'] = teXify_pol(info['hecke_polynomial'])

    if 'AL_eigenvalues_fixed' in data:
        if data['AL_eigenvalues_fixed'] == 'done':
            info['AL_eigs'] = [{'eigenvalue': teXify_pol(al[1]),
                                'prime_ideal': teXify_pol(al[0]),
                                'prime_norm': al[0][1:al[0].index(',')]} for al in data['AL_eigenvalues']]
        else:
            info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
    else:
        info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
    info['AL_eigs_count'] = len(info['AL_eigs']) != 0

    if not display_eigs:
        for eig in info['eigs']:
            if len(eig['eigenvalue']) > 300:
                eig['eigenvalue'] = '...'
        for eig in info['AL_eigs']:
            if len(eig['eigenvalue']) > 300:
                eig['eigenvalue'] = '...'

#.........这里部分代码省略.........

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" % label2
        info["label"] = args["label_orig"] if "label_orig" in args else args["label"]
        return search_input_error(info, bread)

    info["wnf"] = nf
    from lmfdb.WebNumberField import nf_display_knowl

    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.K())
    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)
    info.update(
        {
            "label": pretty_label,
            "label_raw": label,
            "polynomial": web_latex_split_on_pm(nf.K().defining_polynomial()),
            "ram_primes": ram_primes,
            "integral_basis": zk,
            "regulator": web_latex(nf.regulator()),
            "unit_rank": nf.unit_rank(),
            "root_of_unity": web_latex(nf.K().primitive_root_of_unity()),
            "fund_units": nf.units(),
            "grh_label": grh_label,
        }
    )

    bread.append(("%s" % info["label_raw"], " "))
    info["downloads_visible"] = True
#.........这里部分代码省略.........