Esempio n. 1
0
def render_bmf_space_webpage(field_label, level_label):
    info = {}
    t = "Bianchi modular forms of level %s over %s" % (level_label,
                                                       field_label)
    credit = bianchi_credit
    bread = [('Bianchi modular forms', url_for(".index")),
             (field_pretty(field_label),
              url_for(".render_bmf_field_dim_table_gl2",
                      field_label=field_label)), (level_label, '')]
    friends = []
    properties2 = []

    if not field_label_regex.match(field_label):
        info[
            'err'] = "%s is not a valid label for an imaginary quadratic field" % field_label
    else:
        pretty_field_label = field_pretty(field_label)
        if not db_dims().find({'field_label': field_label}):
            info[
                'err'] = "no dimension information exists in the database for field %s" % pretty_field_label
        else:
            t = "Bianchi Modular Forms of level %s over %s" % (
                level_label, pretty_field_label)
            data = db_dims().find({
                'field_label': field_label,
                'level_label': level_label
            })
            nres = data.count()
            if nres == 0:
                info[
                    'err'] = "no dimension information exists in the database for level %s and field %s" % (
                        level_label, pretty_field_label)
            else:
                data = data.next()
                info['label'] = data['label']
                nf = WebNumberField(field_label)
                info['base_galois_group'] = nf.galois_string()
                info['field_label'] = field_label
                info['pretty_field_label'] = pretty_field_label
                info['level_label'] = level_label
                info['level_norm'] = data['level_norm']
                info['field_degree'] = nf.degree()
                info['field_classno'] = nf.class_number()
                info['field_disc'] = str(nf.disc())
                info['field_poly'] = teXify_pol(str(nf.poly()))
                info['field_knowl'] = nf_display_knowl(field_label,
                                                       getDBConnection(),
                                                       pretty_field_label)
                w = 'i' if nf.disc() == -4 else 'a'
                L = nf.K().change_names(w)
                alpha = L.gen()
                info['field_gen'] = latex(alpha)
                I = ideal_from_label(L, level_label)
                info['level_gen'] = latex(I.gens_reduced()[0])
                info['level_fact'] = latex(I.factor())
                dim_data = data['gl2_dims']
                weights = dim_data.keys()
                weights.sort(key=lambda w: int(w))
                for w in weights:
                    dim_data[w]['dim'] = dim_data[w]['cuspidal_dim']
                info['dim_data'] = dim_data
                info['weights'] = weights
                info['nweights'] = len(weights)

                newdim = data['gl2_dims']['2']['new_dim']
                newforms = db_forms().find({
                    'field_label': field_label,
                    'level_label': level_label
                }).sort('label_suffix', ASCENDING)
                info['nfdata'] = [{
                    'label':
                    f['short_label'],
                    'url':
                    url_for(".render_bmf_webpage",
                            field_label=f['field_label'],
                            level_label=f['level_label'],
                            label_suffix=f['label_suffix']),
                    'wt':
                    f['weight'],
                    'dim':
                    f['dimension'],
                    'sfe':
                    "+1" if f['sfe'] == 1 else "-1",
                    'bc':
                    bc_info(f['bc']),
                    'cm':
                    cm_info(f['CM']),
                } for f in newforms]
                info['nnewforms'] = len(info['nfdata'])
                properties2 = [('Base field', pretty_field_label),
                               ('Level', info['level_label']),
                               ('Norm', str(info['level_norm'])),
                               ('New dimension', str(newdim))]
                friends = [('Newform {}'.format(f['label']), f['url'])
                           for f in info['nfdata']]

    return render_template("bmf-space.html",
                           info=info,
                           credit=credit,
                           title=t,
                           bread=bread,
                           properties2=properties2,
                           friends=friends)
Esempio n. 2
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def FIELD(label):
    nf = WebNumberField(label, gen_name=special_names.get(label, 'a'))
    nf.parse_NFelt = lambda s: nf.K()([QQ(c.encode()) for c in s.split(",")])
    nf.latex_poly = web_latex(nf.poly())
    return nf
Esempio n. 3
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def bmf_field_dim_table(**args):
    argsdict = to_dict(args)
    argsdict.update(to_dict(request.args))
    gl_or_sl = argsdict['gl_or_sl']

    field_label = argsdict['field_label']
    field_label = nf_string_to_label(field_label)

    start = 0
    if 'start' in argsdict:
        start = int(argsdict['start'])

    info = {}
    info['gl_or_sl'] = gl_or_sl
    # level_flag controls whether to list all levels ('all'), only
    # those with positive cuspidal dimension ('cusp'), or only those
    # with positive new dimension ('new').  Default is 'cusp'.
    level_flag = argsdict.get('level_flag', 'cusp')
    info['level_flag'] = level_flag
    count = 50
    if 'count' in argsdict:
        count = int(argsdict['count'])

    pretty_field_label = field_pretty(field_label)
    bread = [('Bianchi Modular Forms', url_for(".index")),
             (pretty_field_label, ' ')]
    properties = []
    if gl_or_sl == 'gl2_dims':
        info['group'] = 'GL(2)'
        info['bgroup'] = '\GL(2,\mathcal{O}_K)'
    else:
        info['group'] = 'SL(2)'
        info['bgroup'] = '\SL(2,\mathcal{O}_K)'
    t = ' '.join([
        'Dimensions of spaces of {} Bianchi modular forms over'.format(
            info['group']), pretty_field_label
    ])
    query = {}
    query['field_label'] = field_label
    query[gl_or_sl] = {'$exists': True}
    if level_flag != 'all':
        # find which weights are present (TODO: get this from a stats collection)
        wts = list(
            sum((Set(d.keys()) for d in db_dims().distinct(gl_or_sl)), Set()))
    if level_flag == 'cusp':
        # restrict the query to only return levels where at least one
        # cuspidal dimension is positive:
        query.update({
            '$or': [{
                gl_or_sl + '.{}.cuspidal_dim'.format(w): {
                    '$gt': 0
                }
            } for w in wts]
        })
    if level_flag == 'new':
        # restrict the query to only return levels where at least one
        # new dimension is positive:
        query.update({
            '$or': [{
                gl_or_sl + '.{}.new_dim'.format(w): {
                    '$gt': 0
                }
            } for w in wts]
        })
    data = db_dims().find(query)
    data = data.sort([('level_norm', ASCENDING)])
    info['number'] = nres = data.count()
    if nres > count or start != 0:
        info['report'] = 'Displaying items %s-%s of %s levels,' % (
            start + 1, min(nres, start + count), nres)
    else:
        info['report'] = 'Displaying all %s levels,' % nres

    data = list(data.skip(start).limit(count))
    info['field'] = field_label
    info['field_pretty'] = pretty_field_label
    nf = WebNumberField(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()))
    weights = set()
    for dat in data:
        weights = weights.union(set(dat[gl_or_sl].keys()))
    weights = list([int(w) for w in weights])
    weights.sort()
    info['weights'] = weights
    info['nweights'] = len(weights)
    info['count'] = count
    info['start'] = start
    info['more'] = int(start + count < nres)

    dims = {}
    for dat in data:
        dims[dat['level_label']] = d = {}
        for w in weights:
            sw = str(w)
            if sw in dat[gl_or_sl]:
                d[w] = {
                    'd': dat[gl_or_sl][sw]['cuspidal_dim'],
                    'n': dat[gl_or_sl][sw]['new_dim']
                }
            else:
                d[w] = {'d': '?', 'n': '?'}
    info['nlevels'] = len(data)
    dimtable = [{
        'level_label':
        dat['level_label'],
        'level_norm':
        dat['level_norm'],
        'level_space':
        url_for(".render_bmf_space_webpage",
                field_label=field_label,
                level_label=dat['level_label'])
        if gl_or_sl == 'gl2_dims' else "",
        'dims':
        dims[dat['level_label']]
    } for dat in data]
    print("Length of dimtable = {}".format(len(dimtable)))
    info['dimtable'] = dimtable
    return render_template("bmf-field_dim_table.html",
                           info=info,
                           title=t,
                           properties=properties,
                           bread=bread)
Esempio n. 4
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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)
Esempio n. 5
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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)
Esempio n. 6
0
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['hmf_field'] = hmf_field
    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 and nf.disc()**2 * data['level_norm'] < 40000:
        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.search.find(
        {'field_label': data['field_label'], 'level_ideal': data['level_ideal']},{'dimension':True})
    dim_space = 0
    for v in forms_space:
        dim_space += v['dimension']

    info['newspace_dimension'] = dim_space

    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']])
    display_eigs = display_eigs or (max_eig_len<=300)
    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'] = '...'

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

    if 'is_CM' in data:
        is_CM = data['is_CM']
    else:
        is_CM = '?'
    info['is_CM'] = is_CM

    if 'is_base_change' in data:
        is_base_change = data['is_base_change']
    else:
        is_base_change = '?'
    info['is_base_change'] = is_base_change

    if 'q_expansions' in data:
        info['q_expansions'] = data['q_expansions']

    properties2 = [('Base field', '%s' % info['field'].field_pretty()),
                   ('Weight', '%s' % data['weight']),
                   ('Level norm', '%s' % data['level_norm']),
                   ('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
                   ('Label', '%s' % data['label']),
                   ('Dimension', '%s' % data['dimension']),
                   ('CM', is_CM),
                   ('Base change', is_base_change)
                   ]

    return render_template("hilbert_modular_form.html", downloads=info["downloads"], info=info, properties2=properties2, credit=hmf_credit, title=t, bread=bread, friends=info['friends'], learnmore=learnmore_list())
def bmf_field_dim_table(**args):
    argsdict = to_dict(args)
    argsdict.update(to_dict(request.args))
    gl_or_sl = argsdict['gl_or_sl']

    field_label=argsdict['field_label']
    field_label = nf_string_to_label(field_label)

    start = 0
    if 'start' in argsdict:
        start = int(argsdict['start'])

    info={}
    info['gl_or_sl'] = gl_or_sl
    # level_flag controls whether to list all levels ('all'), only
    # those with positive cuspidal dimension ('cusp'), or only those
    # with positive new dimension ('new').  Default is 'cusp'.
    level_flag = argsdict.get('level_flag', 'cusp')
    info['level_flag'] = level_flag
    count = 50
    if 'count' in argsdict:
        count = int(argsdict['count'])

    pretty_field_label = field_pretty(field_label)
    bread = [('Bianchi Modular Forms', url_for(".index")), (
        pretty_field_label, ' ')]
    properties = []
    if gl_or_sl=='gl2_dims':
        info['group'] = 'GL(2)'
        info['bgroup'] = '\GL(2,\mathcal{O}_K)'
    else:
        info['group'] = 'SL(2)'
        info['bgroup'] = '\SL(2,\mathcal{O}_K)'
    t = ' '.join(['Dimensions of Spaces of {} Bianchi Modular Forms over'.format(info['group']), pretty_field_label])
    query = {}
    query['field_label'] = field_label
    query[gl_or_sl] = {'$exists': True}
    data = db_dims().find(query)
    data = data.sort([('level_norm', ASCENDING)])
    info['number'] = nres = data.count()
    if nres > count or start != 0:
        info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
    else:
        info['report'] = 'Displaying all %s levels,' % nres

    # convert data to a list and eliminate levels where all
    # new/cuspidal dimensions are 0.  (This could be done at the
    # search stage, but that requires adding new fields to each
    # record.)
    def filter(dat, flag):
        dat1 = dat[gl_or_sl]
        return any([int(dat1[w][flag])>0 for w in dat1])
    flag = 'cuspidal_dim' if level_flag=='cusp' else 'new_dim'
    data = [dat for dat in data if level_flag == 'all' or filter(dat, flag)]

    data = data[start:start+count]
    info['field'] = field_label
    info['field_pretty'] = pretty_field_label
    nf = WebNumberField(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()))
    weights = set()
    for dat in data:
        weights = weights.union(set(dat[gl_or_sl].keys()))
    weights = list([int(w) for w in weights])
    weights.sort()
    info['weights'] = weights
    info['nweights'] = len(weights)
    info['count'] = count
    info['start'] = start
    info['more'] = int(start + count < nres)

    data.sort(key = lambda x: [int(y) for y in x['level_label'].split(".")])
    dims = {}
    for dat in data:
        dims[dat['level_label']] = d = {}
        for w in weights:
            sw = str(w)
            if sw in dat[gl_or_sl]:
                d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
                        'n': dat[gl_or_sl][sw]['new_dim']}
            else:
                d[w] = {'d': '?', 'n': '?'}
    info['nlevels'] = len(data)
    dimtable = [{'level_label': dat['level_label'],
                 'level_norm': dat['level_norm'],
                 'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
                  'dims': dims[dat['level_label']]} for dat in data]
    info['dimtable'] = dimtable
    return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
Esempio n. 8
0
def FIELD(label):
    nf = WebNumberField(label, gen_name=special_names.get(label, 'a'))
    nf.parse_NFelt = lambda s: nf.K()([QQ(c.encode()) for c in s.split(",")])
    nf.latex_poly = web_latex(nf.poly())
    return nf
Esempio n. 9
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)
Esempio n. 10
0
def bmf_field_dim_table(**args):
    argsdict = to_dict(args)
    argsdict.update(to_dict(request.args))
    gl_or_sl = argsdict['gl_or_sl']

    field_label=argsdict['field_label']
    field_label = nf_string_to_label(field_label)

    start = 0
    if 'start' in argsdict:
        start = int(argsdict['start'])

    info={}
    info['gl_or_sl'] = gl_or_sl
    # level_flag controls whether to list all levels ('all'), only
    # those with positive cuspidal dimension ('cusp'), or only those
    # with positive new dimension ('new').  Default is 'cusp'.
    level_flag = argsdict.get('level_flag', 'cusp')
    info['level_flag'] = level_flag
    count = 50
    if 'count' in argsdict:
        count = int(argsdict['count'])

    pretty_field_label = field_pretty(field_label)
    bread = [('Bianchi Modular Forms', url_for(".index")), (
        pretty_field_label, ' ')]
    properties = []
    if gl_or_sl=='gl2_dims':
        info['group'] = 'GL(2)'
        info['bgroup'] = '\GL(2,\mathcal{O}_K)'
    else:
        info['group'] = 'SL(2)'
        info['bgroup'] = '\SL(2,\mathcal{O}_K)'
    t = ' '.join(['Dimensions of spaces of {} Bianchi modular forms over'.format(info['group']), pretty_field_label])
    query = {}
    query['field_label'] = field_label
    query[gl_or_sl] = {'$exists': True}
    data = db_dims().find(query)
    data = data.sort([('level_norm', ASCENDING)])
    info['number'] = nres = data.count()
    if nres > count or start != 0:
        info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
    else:
        info['report'] = 'Displaying all %s levels,' % nres

    # convert data to a list and eliminate levels where all
    # new/cuspidal dimensions are 0.  (This could be done at the
    # search stage, but that requires adding new fields to each
    # record.)
    def filter(dat, flag):
        dat1 = dat[gl_or_sl]
        return any([int(dat1[w][flag])>0 for w in dat1])
    flag = 'cuspidal_dim' if level_flag=='cusp' else 'new_dim'
    data = [dat for dat in data if level_flag == 'all' or filter(dat, flag)]

    data = data[start:start+count]
    info['field'] = field_label
    info['field_pretty'] = pretty_field_label
    nf = WebNumberField(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()))
    weights = set()
    for dat in data:
        weights = weights.union(set(dat[gl_or_sl].keys()))
    weights = list([int(w) for w in weights])
    weights.sort()
    info['weights'] = weights
    info['nweights'] = len(weights)
    info['count'] = count
    info['start'] = start
    info['more'] = int(start + count < nres)

    data.sort(key = lambda x: [int(y) for y in x['level_label'].split(".")])
    dims = {}
    for dat in data:
        dims[dat['level_label']] = d = {}
        for w in weights:
            sw = str(w)
            if sw in dat[gl_or_sl]:
                d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
                        'n': dat[gl_or_sl][sw]['new_dim']}
            else:
                d[w] = {'d': '?', 'n': '?'}
    info['nlevels'] = len(data)
    dimtable = [{'level_label': dat['level_label'],
                 'level_norm': dat['level_norm'],
                 'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
                  'dims': dims[dat['level_label']]} for dat in data]
    info['dimtable'] = dimtable
    return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
Esempio n. 11
0
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['hmf_field'] = hmf_field
    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 and nf.disc(
    )**2 * data['level_norm'] < 40000:
        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.search.find(
        {
            'field_label': data['field_label'],
            'level_ideal': data['level_ideal']
        }, {'dimension': True})
    dim_space = 0
    for v in forms_space:
        dim_space += v['dimension']

    info['newspace_dimension'] = dim_space

    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']])
    display_eigs = display_eigs or (max_eig_len <= 300)
    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'] = '...'

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

    if 'is_CM' in data:
        is_CM = data['is_CM']
    else:
        is_CM = '?'
    info['is_CM'] = is_CM

    if 'is_base_change' in data:
        is_base_change = data['is_base_change']
    else:
        is_base_change = '?'
    info['is_base_change'] = is_base_change

    if 'q_expansions' in data:
        info['q_expansions'] = data['q_expansions']

    properties2 = [('Base field', '%s' % info['field'].field_pretty()),
                   ('Weight', '%s' % data['weight']),
                   ('Level norm', '%s' % data['level_norm']),
                   ('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
                   ('Label', '%s' % data['label']),
                   ('Dimension', '%s' % data['dimension']), ('CM', is_CM),
                   ('Base change', is_base_change)]

    return render_template("hilbert_modular_form.html",
                           downloads=info["downloads"],
                           info=info,
                           properties2=properties2,
                           credit=hmf_credit,
                           title=t,
                           bread=bread,
                           friends=info['friends'],
                           learnmore=learnmore_list())
Esempio n. 12
0
def render_hmf_webpage(**args):
    if 'data' in args:
        data = args['data']
        label = data['label']
    else:
        label = str(args['label'])
        data = get_hmf(label)
    if data is None:
        flash(
            Markup(
                "Error: <span style='color:black'>%s</span> is not a valid Hilbert modular form label. It must be of the form (number field label) - (level label) - (orbit label) separated by dashes, such as 2.2.5.1-31.1-a"
                % args['label']), "error")
        return search_input_error()
    info = {}
    try:
        info['count'] = args['count']
    except KeyError:
        info['count'] = 10

    hmf_field = get_hmf_field(data['field_label'])
    gen_name = findvar(hmf_field['ideals'])
    nf = WebNumberField(data['field_label'], gen_name=gen_name)
    info['hmf_field'] = hmf_field
    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 and nf.disc(
    )**2 * data['level_norm'] < 40000:
        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 = db_search().find(
        {
            'field_label': data['field_label'],
            'level_ideal': data['level_ideal']
        }, {'dimension': True})
    dim_space = 0
    for v in forms_space:
        dim_space += v['dimension']

    info['newspace_dimension'] = dim_space

    # Get hecke_polynomial, hecke_eigenvalues and AL_eigenvalues
    try:
        numeigs = request.args['numeigs']
        numeigs = int(numeigs)
    except:
        numeigs = 20
    info['numeigs'] = numeigs

    hecke_pol = data['hecke_polynomial']
    eigs = data['hecke_eigenvalues']
    eigs = eigs[:min(len(eigs), numeigs)]
    AL_eigs = data['AL_eigenvalues']

    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'] = web_latex_split_on_pm(teXify_pol(hecke_pol))

    if not AL_eigs:  # empty list
        if data['level_norm'] == 1:  # OK, no bad primes
            info['AL_eigs'] = 'none'
        else:  # not OK, AL eigs are missing
            info['AL_eigs'] = 'missing'
    else:
        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']]

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

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

    if 'is_CM' in data:
        is_CM = data['is_CM']
    else:
        is_CM = '?'
    info['is_CM'] = is_CM

    if 'is_base_change' in data:
        is_base_change = data['is_base_change']
    else:
        is_base_change = '?'
    info['is_base_change'] = is_base_change

    if 'q_expansions' in data:
        info['q_expansions'] = data['q_expansions']

    properties2 = [('Base field', '%s' % info['field'].field_pretty()),
                   ('Weight', '%s' % data['weight']),
                   ('Level norm', '%s' % data['level_norm']),
                   ('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
                   ('Label', '%s' % data['label']),
                   ('Dimension', '%s' % data['dimension']), ('CM', is_CM),
                   ('Base change', is_base_change)]

    return render_template("hilbert_modular_form.html",
                           downloads=info["downloads"],
                           info=info,
                           properties2=properties2,
                           credit=hmf_credit,
                           title=t,
                           bread=bread,
                           friends=info['friends'],
                           learnmore=learnmore_list())
Esempio n. 13
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def FIELD(label):
    nf = WebNumberField(label, gen_name=special_names.get(label, "a"))
    nf.parse_NFelt = lambda s: nf.K()([QQ(str(c)) for c in s])
    nf.latex_poly = web_latex(nf.poly())
    return nf
Esempio n. 14
0
def render_hmf_webpage(**args):
    if 'data' in args:
        data = args['data']
        label = data['label']
    else:
        label = str(args['label'])
        data = get_hmf(label)
    if data is None:
        flash(Markup("Error: <span style='color:black'>%s</span> is not a valid Hilbert modular form label. It must be of the form (number field label) - (level label) - (orbit label) separated by dashes, such as 2.2.5.1-31.1-a" % args['label']), "error")
        return search_input_error()
    info = {}
    try:
        info['count'] = args['count']
    except KeyError:
        info['count'] = 10

    hmf_field = get_hmf_field(data['field_label'])
    gen_name = findvar(hmf_field['ideals'])
    nf = WebNumberField(data['field_label'], gen_name=gen_name)
    info['hmf_field'] = hmf_field
    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 and nf.disc()**2 * data['level_norm'] < 40000:
        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 = [("Modular Forms", url_for('mf.modular_form_main_page')), ('Hilbert Modular Forms', url_for(".hilbert_modular_form_render_webpage")),
        ('%s' % data['label'], ' ')]

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

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

    info['newspace_dimension'] = dim_space

    # Get hecke_polynomial, hecke_eigenvalues and AL_eigenvalues
    try:
        numeigs = request.args['numeigs']
        numeigs = int(numeigs)
    except:
        numeigs = 20
    info['numeigs'] = numeigs

    hecke_pol  = data['hecke_polynomial']
    eigs       = data['hecke_eigenvalues']
    eigs = eigs[:min(len(eigs), numeigs)]
    AL_eigs    = data['AL_eigenvalues']

    primes = hmf_field['primes']
    n = min(len(eigs), len(primes))
    info['eigs'] = [{'eigenvalue': add_space_if_positive(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'] = web_latex_split_on_pm(teXify_pol(hecke_pol))

    if not AL_eigs: # empty list
        if data['level_norm']==1: # OK, no bad primes
            info['AL_eigs'] = 'none'
        else:                     # not OK, AL eigs are missing
            info['AL_eigs'] = 'missing'
    else:
        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']]

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

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

    if 'is_CM' in data:
        is_CM = data['is_CM']
    else:
        is_CM = '?'
    info['is_CM'] = is_CM

    if 'is_base_change' in data:
        is_base_change = data['is_base_change']
    else:
        is_base_change = '?'
    info['is_base_change'] = is_base_change

    if 'q_expansions' in data:
        info['q_expansions'] = data['q_expansions']

    properties2 = [('Base field', '%s' % info['field'].field_pretty()),
                   ('Weight', '%s' % data['weight']),
                   ('Level norm', '%s' % data['level_norm']),
                   ('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
                   ('Label', '%s' % data['label']),
                   ('Dimension', '%s' % data['dimension']),
                   ('CM', is_CM),
                   ('Base change', is_base_change)
                   ]

    return render_template("hilbert_modular_form.html", downloads=info["downloads"], info=info, properties2=properties2, credit=hmf_credit, title=t, bread=bread, friends=info['friends'], learnmore=learnmore_list())
Esempio n. 15
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def render_bmf_space_webpage(field_label, level_label):
    info = {}
    t = "Bianchi modular forms of level %s over %s" % (level_label, field_label)
    credit = bianchi_credit
    bread = [('Bianchi modular forms', url_for(".index")),
             (field_pretty(field_label), url_for(".render_bmf_field_dim_table_gl2", field_label=field_label)),
             (level_label, '')]
    friends = []
    properties2 = []

    if not field_label_regex.match(field_label):
        info['err'] = "%s is not a valid label for an imaginary quadratic field" % field_label
    else:
        pretty_field_label = field_pretty(field_label)
        if not db_dims().find({'field_label': field_label}):
            info['err'] = "no dimension information exists in the database for field %s" % pretty_field_label
        else:
            t = "Bianchi Modular Forms of level %s over %s" % (level_label, pretty_field_label)
            data = db_dims().find({'field_label': field_label, 'level_label': level_label})
            nres = data.count()
            if nres==0:
                info['err'] = "no dimension information exists in the database for level %s and field %s" % (level_label, pretty_field_label)
            else:
                data = data.next()
                info['label'] = data['label']
                nf = WebNumberField(field_label)
                info['base_galois_group'] = nf.galois_string()
                info['field_label'] = field_label
                info['pretty_field_label'] = pretty_field_label
                info['level_label'] = level_label
                info['level_norm'] = data['level_norm']
                info['field_degree'] = nf.degree()
                info['field_classno'] = nf.class_number()
                info['field_disc'] = str(nf.disc())
                info['field_poly'] = teXify_pol(str(nf.poly()))
                info['field_knowl'] = nf_display_knowl(field_label, getDBConnection(), pretty_field_label)
                w = 'i' if nf.disc()==-4 else 'a'
                L = nf.K().change_names(w)
                alpha = L.gen()
                info['field_gen'] = latex(alpha)
                I = ideal_from_label(L,level_label)
                info['level_gen'] = latex(I.gens_reduced()[0])
                info['level_fact'] = latex(I.factor())
                dim_data = data['gl2_dims']
                weights = dim_data.keys()
                weights.sort(key=lambda w: int(w))
                for w in weights:
                    dim_data[w]['dim']=dim_data[w]['cuspidal_dim']
                info['dim_data'] = dim_data
                info['weights'] = weights
                info['nweights'] = len(weights)

                newdim = data['gl2_dims']['2']['new_dim']
                newforms = db_forms().find({'field_label':field_label, 'level_label':level_label}).sort('label_suffix', ASCENDING)
                info['nfdata'] = [{
                    'label': f['short_label'],
                    'url': url_for(".render_bmf_webpage",field_label=f['field_label'], level_label=f['level_label'], label_suffix=f['label_suffix']),
                    'wt': f['weight'],
                    'dim': f['dimension'],
                    'sfe': "+1" if f['sfe']==1 else "-1",
                    'bc': bc_info(f['bc']),
                    'cm': cm_info(f['CM']),
                    } for f in newforms]
                info['nnewforms'] = len(info['nfdata'])
                properties2 = [('Base field', pretty_field_label), ('Level',info['level_label']), ('Norm',str(info['level_norm'])), ('New dimension',str(newdim))]
                friends = [('Newform {}'.format(f['label']), f['url']) for f in info['nfdata'] ]

    return render_template("bmf-space.html", info=info, credit=credit, title=t, bread=bread, properties2=properties2, friends=friends)
Esempio n. 16
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def bmf_field_dim_table(**args):
    argsdict = to_dict(args)
    argsdict.update(to_dict(request.args))
    gl_or_sl = argsdict['gl_or_sl']

    field_label=argsdict['field_label']
    field_label = nf_string_to_label(field_label)

    start = 0
    if 'start' in argsdict:
        start = int(argsdict['start'])

    info={}
    info['gl_or_sl'] = gl_or_sl
    # level_flag controls whether to list all levels ('all'), only
    # those with positive cuspidal dimension ('cusp'), or only those
    # with positive new dimension ('new').  Default is 'cusp'.
    level_flag = argsdict.get('level_flag', 'cusp')
    info['level_flag'] = level_flag
    count = 50
    if 'count' in argsdict:
        count = int(argsdict['count'])

    pretty_field_label = field_pretty(field_label)
    bread = [('Bianchi Modular Forms', url_for(".index")), (
        pretty_field_label, ' ')]
    properties = []
    if gl_or_sl=='gl2_dims':
        info['group'] = 'GL(2)'
        info['bgroup'] = '\GL(2,\mathcal{O}_K)'
    else:
        info['group'] = 'SL(2)'
        info['bgroup'] = '\SL(2,\mathcal{O}_K)'
    t = ' '.join(['Dimensions of spaces of {} Bianchi modular forms over'.format(info['group']), pretty_field_label])
    query = {}
    query['field_label'] = field_label
    query[gl_or_sl] = {'$exists': True}
    if level_flag != 'all':
        # find which weights are present (TODO: get this from a stats collection)
        wts = list(sum((Set(d.keys()) for d in db_dims().distinct(gl_or_sl)),Set()))
    if level_flag == 'cusp':
        # restrict the query to only return levels where at least one
        # cuspidal dimension is positive:
        query.update(
            {'$or':[{gl_or_sl+'.{}.cuspidal_dim'.format(w):{'$gt':0}} for w in wts]}
        )
    if level_flag == 'new':
        # restrict the query to only return levels where at least one
        # new dimension is positive:
        query.update(
            {'$or':[{gl_or_sl+'.{}.new_dim'.format(w):{'$gt':0}} for w in wts]}
        )
    data = db_dims().find(query)
    data = data.sort([('level_norm', ASCENDING)])
    info['number'] = nres = data.count()
    if nres > count or start != 0:
        info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
    else:
        info['report'] = 'Displaying all %s levels,' % nres

    data = list(data.skip(start).limit(count))
    info['field'] = field_label
    info['field_pretty'] = pretty_field_label
    nf = WebNumberField(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()))
    weights = set()
    for dat in data:
        weights = weights.union(set(dat[gl_or_sl].keys()))
    weights = list([int(w) for w in weights])
    weights.sort()
    info['weights'] = weights
    info['nweights'] = len(weights)
    info['count'] = count
    info['start'] = start
    info['more'] = int(start + count < nres)

    dims = {}
    for dat in data:
        dims[dat['level_label']] = d = {}
        for w in weights:
            sw = str(w)
            if sw in dat[gl_or_sl]:
                d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
                        'n': dat[gl_or_sl][sw]['new_dim']}
            else:
                d[w] = {'d': '?', 'n': '?'}
    info['nlevels'] = len(data)
    dimtable = [{'level_label': dat['level_label'],
                 'level_norm': dat['level_norm'],
                 'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
                  'dims': dims[dat['level_label']]} for dat in data]
    print("Length of dimtable = {}".format(len(dimtable)))
    info['dimtable'] = dimtable
    return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)