def download_hmf_magma(**args):
label = str(args['label'])
f = get_hmf(label)
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = get_hmf_field(f['field_label'])
hecke_pol = f['hecke_polynomial']
hecke_eigs = map(str, f['hecke_eigenvalues'])
AL_eigs = f['AL_eigenvalues']
outstr = 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<w> := NumberField(g);\n'
outstr += 'ZF := Integers(F);\n\n'
# outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
# outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'
outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'
outstr += 'primesArray := [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if hecke_pol != 'x':
outstr += 'heckePol := ' + hecke_pol + ';\n'
outstr += 'K<e> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
outstr += '\nheckeEigenvaluesArray := [' + ', '.join([st for st in hecke_eigs]) + '];'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i := 1 to #heckeEigenvaluesArray do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'
outstr += 'ALEigenvalues := AssociativeArray();\n'
for s in AL_eigs:
outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + str(s[1]) + ';\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):\n'
outstr += 'M := HilbertCuspForms(F, NN);\n'
outstr += 'S := NewSubspace(M);\n'
outstr += '// SetVerbose("ModFrmHil", 1);\n'
outstr += 'newspaces := NewformDecomposition(S);\n'
outstr += 'newforms := [Eigenform(U) : U in newspaces];\n'
outstr += 'ppind := 0;\n'
outstr += 'while #newforms gt 1 do\n'
outstr += ' pp := primes[ppind];\n'
outstr += ' newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];\n'
outstr += 'end while;\n'
outstr += 'f := newforms[1];\n'
outstr += '// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;\n'
outstr += '*/\n'
return outstr
def download_hmf_sage(**args):
label = str(args['label'])
f = get_hmf(label)
if f is None:
return "No such form"
hecke_pol = f['hecke_polynomial']
hecke_eigs = map(str, f['hecke_eigenvalues'])
AL_eigs = f['AL_eigenvalues']
F = WebNumberField(f['field_label'])
F_hmf = get_hmf_field(f['field_label'])
outstr = '/*\n This code can be loaded, or copied and paste using cpaste, into Sage.\n'
outstr += ' It will load the data associated to the HMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += '*/\n\n'
outstr += 'P.<x> = PolynomialRing(QQ)\n'
outstr += 'g = P(' + str(F.coeffs()) + ')\n'
outstr += 'F.<w> = NumberField(g)\n'
outstr += 'ZF = F.ring_of_integers()\n\n'
outstr += 'NN = ZF.ideal(' + f["level_ideal"] + ')\n\n'
outstr += 'primes_array = [\n' + ','.join(
[st for st in F_hmf["primes"]]).replace('],[', '],\\\n[') + ']\n'
outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'
if hecke_pol != 'x':
outstr += 'heckePol = ' + hecke_pol + '\n'
outstr += 'K.<e> = NumberField(heckePol)\n'
else:
outstr += 'heckePol = x\nK = QQ\ne = 1\n'
outstr += '\nhecke_eigenvalues_array = [' + ', '.join(
[st for st in hecke_eigs]) + ']'
outstr += '\nhecke_eigenvalues = {}\n'
outstr += 'for i in range(len(hecke_eigenvalues_array)):\n hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'
outstr += 'AL_eigenvalues = {}\n'
for s in AL_eigs:
outstr += 'AL_eigenvalues[ZF.ideal(%s)] = %s\n' % (s[0], s[1])
outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'
return outstr
def download_hmf_sage(**args):
label = str(args['label'])
f = get_hmf(label)
if f is None:
return "No such form"
hecke_pol = f['hecke_polynomial']
hecke_eigs = map(str, f['hecke_eigenvalues'])
AL_eigs = f['AL_eigenvalues']
F = WebNumberField(f['field_label'])
F_hmf = get_hmf_field(f['field_label'])
outstr = '/*\n This code can be loaded, or copied and paste using cpaste, into Sage.\n'
outstr += ' It will load the data associated to the HMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += '*/\n\n'
outstr += 'P.<x> = PolynomialRing(QQ)\n'
outstr += 'g = P(' + str(F.coeffs()) + ')\n'
outstr += 'F.<w> = NumberField(g)\n'
outstr += 'ZF = F.ring_of_integers()\n\n'
outstr += 'NN = ZF.ideal(' + f["level_ideal"] + ')\n\n'
outstr += 'primes_array = [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[',
'],\\\n[') + ']\n'
outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'
if hecke_pol != 'x':
outstr += 'heckePol = ' + hecke_pol + '\n'
outstr += 'K.<e> = NumberField(heckePol)\n'
else:
outstr += 'heckePol = x\nK = QQ\ne = 1\n'
outstr += '\nhecke_eigenvalues_array = [' + ', '.join([st for st in hecke_eigs]) + ']'
outstr += '\nhecke_eigenvalues = {}\n'
outstr += 'for i in range(len(hecke_eigenvalues_array)):\n hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'
outstr += 'AL_eigenvalues = {}\n'
for s in AL_eigs:
outstr += 'AL_eigenvalues[ZF.ideal(%s)] = %s\n' % (s[0],s[1])
outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'
return outstr
def download_hmf_magma(**args):
label = str(args['label'])
f = get_hmf(label)
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = get_hmf_field(f['field_label'])
hecke_pol = f['hecke_polynomial']
hecke_eigs = map(str, f['hecke_eigenvalues'])
AL_eigs = f['AL_eigenvalues']
outstr = '/*\n This code can be loaded, or copied and pasted, into Magma.\n'
outstr += ' It will load the data associated to the HMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += ' At the *bottom* of the file, there is code to recreate the\n'
outstr += ' Hilbert modular form in Magma, by creating the HMF space\n'
outstr += ' and cutting out the corresponding Hecke irreducible subspace.\n'
outstr += ' From there, you can ask for more eigenvalues or modify as desired.\n'
outstr += ' It is commented out, as this computation may be lengthy.\n'
outstr += '*/\n\n'
outstr += 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<w> := NumberField(g);\n'
outstr += 'ZF := Integers(F);\n\n'
# outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
# outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'
outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'
outstr += 'primesArray := [\n' + ','.join(
[st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if hecke_pol != 'x':
outstr += 'heckePol := ' + hecke_pol + ';\n'
outstr += 'K<e> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
outstr += '\nheckeEigenvaluesArray := [' + ', '.join(
[st for st in hecke_eigs]) + '];'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i := 1 to #heckeEigenvaluesArray do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'
outstr += 'ALEigenvalues := AssociativeArray();\n'
for s in AL_eigs:
outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + str(
s[1]) + ';\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):\n'
outstr += 'M := HilbertCuspForms(F, NN);\n'
outstr += 'S := NewSubspace(M);\n'
outstr += '// SetVerbose("ModFrmHil", 1);\n'
outstr += 'newspaces := NewformDecomposition(S);\n'
outstr += 'newforms := [Eigenform(U) : U in newspaces];\n'
outstr += 'ppind := 0;\n'
outstr += 'while #newforms gt 1 do\n'
outstr += ' pp := primes[ppind];\n'
outstr += ' newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];\n'
outstr += 'end while;\n'
outstr += 'f := newforms[1];\n'
outstr += '// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;\n'
outstr += '*/\n'
return outstr
def 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,·)$</a>' %
(url_for('characters.render_Dirichletwebpage',
modulus=data['conductor'],
number=j), data['conductor'], j)
for j in dirichlet_chars
]
if len(data['dirichlet_group']) == 1:
data[
'dirichlet_group'] = r'<span style="white-space:nowrap">$\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)
def download_hmf_magma(**args):
label = str(args['label'])
f = get_hmf(label)
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = get_hmf_field(f['field_label'])
hecke_pol = f['hecke_polynomial']
hecke_eigs = [str(eig) for eig in f['hecke_eigenvalues']]
AL_eigs = f['AL_eigenvalues']
outstr = '/*\n This code can be loaded, or copied and pasted, into Magma.\n'
outstr += ' It will load the data associated to the HMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += ' At the *bottom* of the file, there is code to recreate the\n'
outstr += ' Hilbert modular form in Magma, by creating the HMF space\n'
outstr += ' and cutting out the corresponding Hecke irreducible subspace.\n'
outstr += ' From there, you can ask for more eigenvalues or modify as desired.\n'
outstr += ' It is commented out, as this computation may be lengthy.\n'
outstr += '*/\n\n'
outstr += 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<w> := NumberField(g);\n'
outstr += 'ZF := Integers(F);\n\n'
# outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
# outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'
outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'
outstr += 'primesArray := [\n' + ','.join(
[st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if hecke_pol != 'x':
outstr += 'heckePol := ' + hecke_pol + ';\n'
outstr += 'K<e> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
outstr += '\nheckeEigenvaluesArray := [' + ', '.join(
[st for st in hecke_eigs]) + '];'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i := 1 to #heckeEigenvaluesArray do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'
outstr += 'ALEigenvalues := AssociativeArray();\n'
for s in AL_eigs:
outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + str(
s[1]) + ';\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '\n'.join([
'print "To reconstruct the Hilbert newform f, type',
' f, iso := Explode(make_newform());";', '',
'function make_newform();', ' M := HilbertCuspForms(F, NN);',
' S := NewSubspace(M);', ' // SetVerbose("ModFrmHil", 1);',
' NFD := NewformDecomposition(S);',
' newforms := [* Eigenform(U) : U in NFD *];', '',
' if #newforms eq 0 then;',
' print "No Hilbert newforms at this level";', ' return 0;',
' end if;', '', ' print "Testing ", #newforms, " possible newforms";',
' newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];',
' print #newforms, " newforms have the correct Hecke field";', '',
' if #newforms eq 0 then;',
' print "No Hilbert newform found with the correct Hecke field";',
' return 0;', ' end if;', '', ' autos := Automorphisms(K);',
' xnewforms := [* *];', ' for f in newforms do;',
' if K eq RationalField() then;',
' Append(~xnewforms, [* f, autos[1] *]);', ' else;',
' flag, iso := IsIsomorphic(K,BaseField(f));',
' for a in autos do;', ' Append(~xnewforms, [* f, a*iso *]);',
' end for;', ' end if;', ' end for;', ' newforms := xnewforms;', '',
' for P in primes do;', ' xnewforms := [* *];',
' for f_iso in newforms do;', ' f, iso := Explode(f_iso);',
' if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;',
' Append(~xnewforms, f_iso);', ' end if;', ' end for;',
' newforms := xnewforms;', ' if #newforms eq 0 then;',
' print "No Hilbert newform found which matches the Hecke eigenvalues";',
' return 0;', ' else if #newforms eq 1 then;',
' print "success: unique match";', ' return newforms[1];',
' end if;', ' end if;', ' end for;',
' print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";',
' return newforms[1];', '', 'end function;'
])
return outstr
def download_bmf_sage(**args):
"""Generates the sage code for the user to obtain the BMF eigenvalues.
As in the HMF case, and unlike the website, we export *all* eigenvalues in
the database, not just 50, and not just those away from the level."""
label = "-".join([args['field_label'], args['level_label'], args['label_suffix']])
try:
f = WebBMF.by_label(label)
except ValueError:
return "Bianchi newform not found"
hecke_pol = f.hecke_poly_obj
hecke_eigs = f.hecke_eigs
F = WebNumberField(f.field_label)
K = f.field.K()
primes_in_K = [p for p,_ in zip(primes_iter(K),hecke_eigs)]
prime_gens = [p.gens_reduced() for p in primes_in_K]
outstr = '"""\n This code can be loaded, or copied and paste using cpaste, into Sage.\n'
outstr += ' It will load the data associated to the BMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data (if known).\n'
outstr += '"""\n\n'
outstr += 'P = PolynomialRing(QQ, "x")\nx = P.gen()\n'
outstr += 'g = P(' + str(F.coeffs()) + ')\n'
outstr += 'F = NumberField(g, "{}")\n'.format(K.gen())
outstr += '{} = F.gen()\n'.format(K.gen())
outstr += 'ZF = F.ring_of_integers()\n\n'
outstr += 'NN = ZF.ideal({})\n\n'.format(f.level.gens())
outstr += 'primes_array = [\n' + ','.join([str(st).replace(' ', '') for st in prime_gens]).replace('],[',
'],\\\n[') + ']\n'
outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'
Qx = PolynomialRing(QQ,'x')
if hecke_pol != 'x':
outstr += 'heckePol = P({})\n'.format(str((Qx(hecke_pol)).list()))
outstr += 'K = NumberField(heckePol, "z")\nz = K.gen()\n'
else:
outstr += 'heckePol = x\nK = QQ\ne = 1\n'
hecke_eigs_processed = [str(st).replace(' ', '') if st != 'not known' else '"not known"' for st in hecke_eigs]
outstr += '\nhecke_eigenvalues_array = [' + ', '.join(hecke_eigs_processed) + ']'
outstr += '\nhecke_eigenvalues = {}\n'
outstr += 'for i in range(len(hecke_eigenvalues_array)):\n hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'
if f.have_AL:
AL_eigs = f.AL_table_data
outstr += 'AL_eigenvalues = {}\n'
for s in AL_eigs:
outstr += 'AL_eigenvalues[ZF.ideal(%s)] = %s\n' % (s[0],s[1])
else:
outstr += 'AL_eigenvalues ="not known"\n'
outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'
return outstr
def download_bmf_magma(**args):
label = "-".join([args['field_label'], args['level_label'], args['label_suffix']])
try:
f = WebBMF.by_label(label)
except ValueError:
return "Bianchi newform not found"
hecke_pol = f.hecke_poly_obj
hecke_eigs = f.hecke_eigs
F = WebNumberField(f.field_label)
K = f.field.K()
primes_in_K = [p for p,_ in zip(primes_iter(K),hecke_eigs)]
prime_gens = [list(p.gens()) for p in primes_in_K]
outstr = '/*\n This code can be loaded, or copied and pasted, into Magma.\n'
outstr += ' It will load the data associated to the BMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += ' At the *bottom* of the file, there is code to recreate the\n'
outstr += ' Bianchi modular form in Magma, by creating the BMF space\n'
outstr += ' and cutting out the corresponding Hecke irreducible subspace.\n'
outstr += ' From there, you can ask for more eigenvalues or modify as desired.\n'
outstr += ' It is commented out, as this computation may be lengthy.\n'
outstr += '*/\n\n'
outstr += 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<{}> := NumberField(g);\n'.format(K.gen())
outstr += 'ZF := Integers(F);\n\n'
outstr += 'NN := ideal<ZF | {}>;\n\n'.format(set(f.level.gens()))
outstr += 'primesArray := [\n' + ','.join([str(st).replace(' ', '') for st in prime_gens]).replace('],[',
'],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if hecke_pol != 'x':
outstr += 'heckePol := ' + hecke_pol + ';\n'
outstr += 'K<z> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
hecke_eigs_processed = [str(st).replace(' ', '') if st != 'not known' else '"not known"' for st in hecke_eigs]
outstr += '\nheckeEigenvaluesList := [*\n'+ ',\n'.join(hecke_eigs_processed) + '\n*];\n'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i in [1..#heckeEigenvaluesList] do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesList[i];\nend for;\n'
if f.have_AL:
AL_eigs = f.AL_table_data
outstr += '\nALEigenvalues := AssociativeArray();\n'
for s in AL_eigs:
outstr += 'ALEigenvalues[ideal<ZF | {}>] := {};\n'.format(set(s[0]), s[1])
else:
outstr += '\nALEigenvalues := "not known";\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '\n'.join([
'print "To reconstruct the Bianchi newform f, type',
' f, iso := Explode(make_newform());";',
'',
'function make_newform();',
' M := BianchiCuspForms(F, NN);',
' S := NewSubspace(M);',
' // SetVerbose("Bianchi", 1);',
' NFD := NewformDecomposition(S);',
' newforms := [* Eigenform(U) : U in NFD *];',
'',
' if #newforms eq 0 then;',
' print "No Bianchi newforms at this level";',
' return 0;',
' end if;',
'',
' print "Testing ", #newforms, " possible newforms";',
' newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];',
' print #newforms, " newforms have the correct Hecke field";',
'',
' if #newforms eq 0 then;',
' print "No Bianchi newform found with the correct Hecke field";',
' return 0;',
' end if;',
'',
' autos := Automorphisms(K);',
' xnewforms := [* *];',
' for f in newforms do;',
' if K eq RationalField() then;',
' Append(~xnewforms, [* f, autos[1] *]);',
' else;',
' flag, iso := IsIsomorphic(K,BaseField(f));',
' for a in autos do;',
' Append(~xnewforms, [* f, a*iso *]);',
' end for;',
' end if;',
' end for;',
' newforms := xnewforms;',
'',
' for P in primes do;',
' if Valuation(NN,P) eq 0 then;',
' xnewforms := [* *];',
' for f_iso in newforms do;',
' f, iso := Explode(f_iso);',
' if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;',
' Append(~xnewforms, f_iso);',
' end if;',
' end for;',
' newforms := xnewforms;',
' if #newforms eq 0 then;',
' print "No Bianchi newform found which matches the Hecke eigenvalues";',
' return 0;',
' else if #newforms eq 1 then;',
' print "success: unique match";',
' return newforms[1];',
' end if;',
' end if;',
' end if;',
' end for;',
' print #newforms, "Bianchi newforms found which match the Hecke eigenvalues";',
' return newforms[1];',
'',
'end function;'])
return outstr
def download_hmf_magma(**args):
label = str(args['label'])
f = get_hmf(label)
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = get_hmf_field(f['field_label'])
hecke_pol = f['hecke_polynomial']
hecke_eigs = map(str, f['hecke_eigenvalues'])
AL_eigs = f['AL_eigenvalues']
outstr = '/*\n This code can be loaded, or copied and pasted, into Magma.\n'
outstr += ' It will load the data associated to the HMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += ' At the *bottom* of the file, there is code to recreate the\n'
outstr += ' Hilbert modular form in Magma, by creating the HMF space\n'
outstr += ' and cutting out the corresponding Hecke irreducible subspace.\n'
outstr += ' From there, you can ask for more eigenvalues or modify as desired.\n'
outstr += ' It is commented out, as this computation may be lengthy.\n'
outstr += '*/\n\n'
outstr += 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<w> := NumberField(g);\n'
outstr += 'ZF := Integers(F);\n\n'
# outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
# outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'
outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'
outstr += 'primesArray := [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if hecke_pol != 'x':
outstr += 'heckePol := ' + hecke_pol + ';\n'
outstr += 'K<e> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
outstr += '\nheckeEigenvaluesArray := [' + ', '.join([st for st in hecke_eigs]) + '];'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i := 1 to #heckeEigenvaluesArray do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'
outstr += 'ALEigenvalues := AssociativeArray();\n'
for s in AL_eigs:
outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + str(s[1]) + ';\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):\n'
outstr += 'M := HilbertCuspForms(F, NN);\n'
outstr += 'S := NewSubspace(M);\n'
outstr += '// SetVerbose("ModFrmHil", 1);\n'
outstr += 'newspaces := NewformDecomposition(S);\n'
outstr += 'newforms := [Eigenform(U) : U in newspaces];\n'
outstr += 'ppind := 0;\n'
outstr += 'while #newforms gt 1 do\n'
outstr += ' pp := primes[ppind];\n'
outstr += ' newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];\n'
outstr += 'end while;\n'
outstr += 'f := newforms[1];\n'
outstr += '// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;\n'
outstr += '*/\n'
return outstr
def 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,·)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\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)
