Ejemplo n.º 1
0
def render_field_webpage(args):
    data = None
    info = {}
    bread = bread_prefix()

    # This function should not be called unless label is set.
    label = clean_input(args['label'])
    nf = WebNumberField(label)
    data = {}
    if nf.is_null():
        if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
            flash_error("Number field %s was not found in the database.",
                        label)
        else:
            flash_error("%s is not a valid label for a number field.", label)
        return redirect(url_for(".number_field_render_webpage"))

    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['autstring'] = r'\Gal' if data['is_galois'] else r'\Aut'
    data['is_abelian'] = nf.is_abelian()
    if nf.is_abelian():
        conductor = nf.conductor()
        data['conductor'] = conductor
        dirichlet_chars = nf.dirichlet_group()
        if dirichlet_chars:
            data['dirichlet_group'] = [
                r'<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
            ]
            if len(data['dirichlet_group']) == 1:
                data[
                    'dirichlet_group'] = r'<span style="white-space:nowrap">$\lbrace$' + data[
                        'dirichlet_group'][0] + r'$\rbrace$</span>'
            else:
                data['dirichlet_group'] = r'$\lbrace$' + ', '.join(
                    data['dirichlet_group']
                    [:-1]) + '<span style="white-space:nowrap">' + data[
                        'dirichlet_group'][-1] + r'$\rbrace$</span>'
        if data['conductor'].is_prime() or data['conductor'] == 1:
            data['conductor'] = r"\(%s\)" % str(data['conductor'])
        else:
            factored_conductor = factor_base_factor(data['conductor'],
                                                    ram_primes)
            factored_conductor = factor_base_factorization_latex(
                factored_conductor)
            data['conductor'] = r"\(%s=%s\)" % (str(
                data['conductor']), factored_conductor)
    data['galois_group'] = group_pretty_and_nTj(n, t, True)
    data['auts'] = db.gps_transitive.lookup(r'{}T{}'.format(n, t))['auts']
    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'] = bigint_knowl(D, cutoff=60, sides=3)
    else:
        data['discriminant'] = bigint_knowl(
            D, cutoff=60,
            sides=3) + r"\(\medspace = %s\)" % data['disc_factor']
    if nf.frobs():
        data['frob_data'], data['seeram'] = see_frobs(nf.frobs())
    else:  # fallback in case we haven't computed them in a case
        data['frob_data'], data['seeram'] = frobs(nf)
    # This could put commas in the rd, we don't want to trigger spaces
    data['rd'] = ('$%s$' % fixed_prec(nf.rd(), 2)).replace(',', '{,}')
    # 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]).rstrip('L')
            else:
                from lmfdb.local_fields.main import show_slope_content
                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]
                    myurl = url_for('local_fields.by_label', label=lab)
                    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]
    # Get rid of python L for big numbers
    ram_primes = ram_primes.replace('L', '')
    if not 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 'computed' 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)
    rootofunity = '%s (order $%d$)' % (nf.root_of_1_gen(),
                                       nf.root_of_1_order())

    info.update({
        'label': pretty_label,
        'label_raw': label,
        'polynomial': web_latex(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_safe(),
        'cnf': nf.cnf(),
        'grh_label': grh_label,
        'loc_alg': loc_alg
    })

    bread.append(('%s' % nf_label_pretty(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 galois_closure[1]:
            resinfo.append(('gc', galois_closure[1]))
            if galois_closure[2]:
                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 sextic_twins[1]:
            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 siblings[0]:
        for sibdeg in siblings[0]:
            if not sibdeg[2]:
                sibdeg[2] = dnc
            else:
                nsibs = len(sibdeg[2])
                sibdeg[2] = ', '.join(sibdeg[2])
                if nsibs < 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 arith_equiv[1]:
            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()
    title = "Number field %s" % info['label']

    if npr == 1:
        primes = 'prime'
    else:
        primes = 'primes'
    if len(ram_primes) > 30:
        ram_primes = 'see page'
    else:
        ram_primes = '$%s$' % ram_primes

    properties = [('Label', nf_label_pretty(label)),
                  ('Degree', prop_int_pretty(data['degree'])),
                  ('Signature', '$%s$' % data['signature']),
                  ('Discriminant', prop_int_pretty(D)),
                  ('Root discriminant', '%s' % data['rd']),
                  ('Ramified ' + primes + '', ram_primes),
                  ('Class number', '%s %s' % (data['class_number'], grh_lab)),
                  ('Class group',
                   '%s %s' % (data['class_group_invs'], grh_lab)),
                  ('Galois group', group_pretty_and_nTj(data['degree'], t))]
    downloads = [('Stored data to gp',
                  url_for('.nf_download', nf=label, download_type='data'))]
    for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
        downloads.append(('Download {} code'.format(lang[0]),
                          url_for(".nf_download",
                                  nf=label,
                                  download_type=lang[1])))
    from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
    from lmfdb.artin_representations.math_classes import artin_label_pretty
    try:
        info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
        arts = [
            z.label()
            for z in info["tim_number_field"].artin_representations()
        ]
        #print arts
        for ar in arts:
            info['friends'].append((
                'Artin representation ' + artin_label_pretty(ar),
                url_for(
                    "artin_representations.render_artin_representation_webpage",
                    label=ar)))
        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("nf-show-field.html",
                           properties=properties,
                           credit=NF_credit,
                           title=title,
                           bread=bread,
                           code=nf.code,
                           friends=info.pop('friends'),
                           downloads=downloads,
                           learnmore=learnmore,
                           info=info,
                           KNOWL_ID="nf.%s" % label)
Ejemplo n.º 2
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():
        if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
            flash_error("Number field %s was not found in the database.", label)
        else:
            flash_error("%s is not a valid label for a global number field.", label)
        return redirect(url_for(".number_field_render_webpage"))

    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_pretty_and_nTj(n,t,True)
    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)
    # This could put commas in the rd, we don't want to trigger spaces
    data['rd'] = ('$%s$' % fixed_prec(nf.rd(),2)).replace(',','{,}')
    # 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]).rstrip('L')
            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]
    # Get rid of python L for big numbers
    ram_primes = ram_primes.replace('L', '')
    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()
    title = "Global Number Field %s" % info['label']

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

    if len(label)>25:
        label = label[:16]+'...'+label[-6:]
    properties2 = [('Label', label),
                   ('Degree', '$%s$' % data['degree']),
                   ('Signature', '$%s$' % data['signature']),
                   ('Discriminant', '$%s$' % data['disc_factor']),
                   ('Root discriminant', '%s' % data['rd']),
                   ('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_pretty_and_nTj(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, KNOWL_ID="nf.%s"%label)