def myhelper(self, coefmult):
coef = string2list(coefmult[0])
subfield = self.from_coeffs(coef)
if subfield._data is None:
deg = len(coef) - 1
mypol = latex(coeff_to_poly(coef))
mypol = mypol.replace(' ','').replace('+','%2B').replace('{', '%7B').replace('}','%7d')
mypol = '<a title = "Field missing" knowl="nf.field.missing" kwargs="poly=%s">Deg %d</a>' % (mypol,deg)
return [mypol, coefmult[1]]
return [nf_display_knowl(subfield.get_label(),subfield.field_pretty()), coefmult[1]]
def formatfield(coef):
coef = string2list(coef)
thefield = WebNumberField.from_coeffs(coef)
if thefield._data is None:
deg = len(coef) - 1
mypol = latex(coeff_to_poly(coef))
mypol = mypol.replace(' ','').replace('+','%2B').replace('{', '%7B').replace('}','%7d')
mypol = '<a title = "Field missing" knowl="nf.field.missing" kwargs="poly=%s">Deg %d</a>' % (mypol,deg)
return mypol
return nf_display_knowl(thefield.get_label(),thefield.field_pretty())
def _order_basis_forward(self):
basis = []
for i, (num, den) in enumerate(zip(self.hecke_ring_numerators, self.hecke_ring_denominators)):
numsize = sum(len(str(c)) for c in num if c)
if numsize > 80:
num = web_latex_poly(num, self._nu_latex, superscript=True)
else:
num = web_latex(coeff_to_poly(num, self._nu_var))
betai = r'\(\beta_{%s}\)'%i
basis.append((betai, self._make_frac(num, den)))
return self._make_table(basis)
def lf_knowl_guts(label, C):
f = C.localfields.fields.find_one({"label": label})
ans = "Local number field %s<br><br>" % label
ans += "Extension of $\Q_{%s}$ defined by %s<br>" % (str(f["p"]), web_latex(coeff_to_poly(f["coeffs"])))
GG = f["gal"]
ans += "Degree: %s<br>" % str(GG[0])
ans += "Ramification index $e$: %s<br>" % str(f["e"])
ans += "Residue field degree $f$: %s<br>" % str(f["f"])
ans += "Discriminant ideal: $(p^{%s})$ <br>" % str(f["c"])
ans += "Galois group $G$: %s<br>" % group_display_knowl(GG[0], GG[1], C)
ans += '<div align="right">'
ans += '<a href="%s">%s home page</a>' % (str(url_for("local_fields.by_label", label=label)), label)
ans += "</div>"
return ans
def make_torsion_growth(self):
if self.tor_gro is None:
self.torsion_growth_data_exists = False
return
tor_gro = self.tor_gro
self.torsion_growth_data_exists = True
self.tg = tg = {}
tg['data'] = tgextra = []
# find all base-changes of this curve in the database, if any
bcs = list(db.ec_nfcurves.search({'base_change': {'$contains': [self.lmfdb_label]}}, projection='label'))
bcfs = [lab.split("-")[0] for lab in bcs]
for F, T in tor_gro.items():
tg1 = {}
tg1['bc'] = "Not in database"
if ":" in F:
F = F.replace(":",".")
field_data = nf_display_knowl(F, field_pretty(F))
deg = int(F.split(".")[0])
bcc = [x for x,y in zip(bcs, bcfs) if y==F]
if bcc:
from lmfdb.ecnf.main import split_full_label
F, NN, I, C = split_full_label(bcc[0])
tg1['bc'] = bcc[0]
tg1['bc_url'] = url_for('ecnf.show_ecnf', nf=F, conductor_label=NN, class_label=I, number=C)
else:
field_data = web_latex_split_on_pm(coeff_to_poly(string2list(F)))
deg = F.count(",")
tg1['d'] = deg
tg1['f'] = field_data
tg1['t'] = '\(' + ' \\times '.join(['\Z/{}\Z'.format(n) for n in T.split(",")]) + '\)'
tg1['m'] = 0
tgextra.append(tg1)
tgextra.sort(key = lambda x: x['d'])
tg['n'] = len(tgextra)
lastd = 1
for tg1 in tgextra:
d = tg1['d']
if d!=lastd:
tg1['m'] = len([x for x in tgextra if x['d']==d])
lastd = d
## Hard-code this for now. While something like
## max(db.ec_curves.search({},projection='tor_degs')) might
## work, since 'tor_degs' is in the extra table it is very
## slow. Note that the *only* place where this number is used
## is in the ec-curve template where it says "The number
## fields ... of degree up to {{data.tg.maxd}} such that...".
tg['maxd'] = 7
def lf_knowl_guts(label, C):
f = C.localfields.fields.find_one({'label':label})
ans = 'Local number field %s<br><br>'% label
ans += 'Extension of $\Q_{%s}$ defined by %s<br>'%(str(f['p']),web_latex(coeff_to_poly(f['coeffs'])))
GG = f['gal']
ans += 'Degree: %s<br>' % str(GG[0])
ans += 'Ramification index $e$: %s<br>' % str(f['e'])
ans += 'Residue field degree $f$: %s<br>' % str(f['f'])
ans += 'Discriminant ideal: $(p^{%s})$ <br>' % str(f['c'])
ans += 'Galois group $G$: %s<br>' % group_display_knowl(GG[0], GG[1], C)
ans += '<div align="right">'
ans += '<a href="%s">%s home page</a>' % (str(url_for("local_fields.by_label", label=label)),label)
ans += '</div>'
return ans
def display_hecke_cutters(self):
polynomials = []
truncated = False
for p,F in self.hecke_cutters:
cut = len(F) - 1
count = 0
while cut >= 0 and count < 8:
if F[cut]:
count += 1
cut -= 1
if count < 8 or cut == 0 and abs(F[0]) < 100:
F = latex(coeff_to_poly(F, 'T%s'%p))
else:
# truncate to the first 8 nonzero coefficients
F = [0]*(cut+1) + F[cut+1:]
F = latex(coeff_to_poly(F, 'T%s'%p)) + r' + \cdots'
truncated = True
polynomials.append(web_latex_split_on_pm(F))
title = 'linear operator'
if len(polynomials) > 1:
title += 's'
knowl = display_knowl('mf.elliptic.hecke_cutter', title=title)
desc = "<p>This newform can be constructed as the "
if truncated or len(polynomials) > 1:
if len(polynomials) > 1:
desc += "intersection of the kernels "
else:
desc += "kernel "
desc += "of the following %s acting on %s:</p>\n<table>"
desc = desc % (knowl, self.display_newspace())
desc += "\n".join("<tr><td>%s</td></tr>" % F for F in polynomials) + "\n</table>"
elif len(polynomials) == 1:
desc += "kernel of the %s %s acting on %s."
desc = desc % (knowl, polynomials[0], self.display_newspace())
else:
desc = r"<p>There are no other newforms in %s.</p>"%(self.display_newspace())
return desc
def make_torsion_growth(self):
try:
tor_gro = self.tor_gro
self.torsion_growth_data_exists = True
except AttributeError:
self.torsion_growth_data_exists = False
return
self.tg = tg = {}
tg['data'] = tgextra = []
# find all base-changes of this curve in the database, if any
from lmfdb.ecnf.WebEllipticCurve import db_ecnf
bcs = [res['label'] for res in db_ecnf().find({'base_change': self.lmfdb_label}, projection={'label': True, '_id': False})]
bcfs = [lab.split("-")[0] for lab in bcs]
for F, T in tor_gro.items():
tg1 = {}
tg1['bc'] = "Not in database"
if ":" in F:
F = F.replace(":",".")
field_data = nf_display_knowl(F, getDBConnection(), field_pretty(F))
deg = int(F.split(".")[0])
bcc = [x for x,y in zip(bcs, bcfs) if y==F]
if bcc:
from lmfdb.ecnf.main import split_full_label
F, NN, I, C = split_full_label(bcc[0])
tg1['bc'] = bcc[0]
tg1['bc_url'] = url_for('ecnf.show_ecnf', nf=F, conductor_label=NN, class_label=I, number=C)
else:
field_data = web_latex_split_on_pm(coeff_to_poly(string2list(F)))
deg = F.count(",")
tg1['d'] = deg
tg1['f'] = field_data
tg1['t'] = '\(' + ' \\times '.join(['\Z/{}\Z'.format(n) for n in T.split(",")]) + '\)'
tg1['m'] = 0
tgextra.append(tg1)
tgextra.sort(key = lambda x: x['d'])
tg['n'] = len(tgextra)
lastd = 1
for tg1 in tgextra:
d = tg1['d']
if d!=lastd:
tg1['m'] = len([x for x in tgextra if x['d']==d])
lastd = d
tg['maxd'] = max(db_ecstats().find_one({'_id': 'torsion_growth'})['degrees'])
def local_field_data(label):
f = db.lf_fields.lookup(label)
nicename = ''
if f['n'] < 3:
nicename = ' = '+ prettyname(f)
ans = 'Local number field %s%s<br><br>'% (label, nicename)
ans += 'Extension of $\Q_{%s}$ defined by %s<br>'%(str(f['p']),web_latex(coeff_to_poly(f['coeffs'])))
gt = f['gal']
gn = f['n']
ans += 'Degree: %s<br>' % str(gt)
ans += 'Ramification index $e$: %s<br>' % str(f['e'])
ans += 'Residue field degree $f$: %s<br>' % str(f['f'])
ans += 'Discriminant ideal: $(p^{%s})$ <br>' % str(f['c'])
ans += 'Galois group $G$: %s<br>' % group_pretty_and_nTj(gn, gt, True)
ans += '<div align="right">'
ans += '<a href="%s">%s home page</a>' % (str(url_for("local_fields.by_label", label=label)),label)
ans += '</div>'
return ans
def make_torsion_growth(self):
if self.tor_gro is None:
self.torsion_growth_data_exists = False
return
tor_gro = self.tor_gro
self.torsion_growth_data_exists = True
self.tg = tg = {}
tg['data'] = tgextra = []
# find all base-changes of this curve in the database, if any
bcs = list(db.ec_nfcurves.search({'base_change': {'$contains': [self.lmfdb_label]}}, projection='label'))
bcfs = [lab.split("-")[0] for lab in bcs]
for F, T in tor_gro.items():
tg1 = {}
tg1['bc'] = "Not in database"
if ":" in F:
F = F.replace(":",".")
field_data = nf_display_knowl(F, field_pretty(F))
deg = int(F.split(".")[0])
bcc = [x for x,y in zip(bcs, bcfs) if y==F]
if bcc:
from lmfdb.ecnf.main import split_full_label
F, NN, I, C = split_full_label(bcc[0])
tg1['bc'] = bcc[0]
tg1['bc_url'] = url_for('ecnf.show_ecnf', nf=F, conductor_label=NN, class_label=I, number=C)
else:
field_data = web_latex_split_on_pm(coeff_to_poly(string2list(F)))
deg = F.count(",")
tg1['d'] = deg
tg1['f'] = field_data
tg1['t'] = '\(' + ' \\times '.join(['\Z/{}\Z'.format(n) for n in T.split(",")]) + '\)'
tg1['m'] = 0
tgextra.append(tg1)
tgextra.sort(key = lambda x: x['d'])
tg['n'] = len(tgextra)
lastd = 1
for tg1 in tgextra:
d = tg1['d']
if d!=lastd:
tg1['m'] = len([x for x in tgextra if x['d']==d])
lastd = d
## Hard code for now
#tg['maxd'] = max(db.ec_curves.stats.get_oldstat('torsion_growth')['degrees'])
tg['maxd'] = 7
def polynomial_latex(self):
return web_latex(coeff_to_poly(self.polynomial()), enclose=False)
def poly(self):
return coeff_to_poly(string2list(self._data['coeffs']))
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = db.lf_fields.lookup(label)
if data is None:
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Field <span style='color:black'>%s</span> was not found in the database.", label)
else:
flash_error("<span style='color:black'>%s</span> is not a valid label for a local number field.", label)
return redirect(url_for(".index"))
title = 'Local Number Field ' + prettyname(data)
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
Qp = r'\Q_{%d}' % p
e = data['e']
f = data['f']
cc = data['c']
gt = data['gal']
gn = data['n']
the_gal = WebGaloisGroup.from_nt(gn,gt)
isgal = ' Galois' if the_gal.order() == gn else ' not Galois'
abelian = ' and abelian' if the_gal.is_abelian() else ''
galphrase = 'This field is'+isgal+abelian+' over $\Q_{%d}$.'%p
autstring = r'\Gal' if the_gal.order() == gn else r'\Aut'
prop2 = [
('Label', label),
('Base', '\(%s\)' % Qp),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_pretty_and_nTj(gn, gt)),
]
# Look up the unram poly so we can link to it
unramlabel = db.lf_fields.lucky({'p': p, 'n': f, 'c': 0}, projection=0)
if unramlabel is None:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
else:
unramfriend = "/LocalNumberField/%s" % unramlabel
unramdata = db.lf_fields.lookup(unramlabel)
Px = PolynomialRing(QQ, 'x')
Pxt=PolynomialRing(Px,'t')
Pt = PolynomialRing(QQ, 't')
Ptx = PolynomialRing(Pt, 'x')
if data['f'] == 1:
unramp = r'$%s$' % Qp
# Eliminate t from the eisenstein polynomial
eisenp = Pxt(str(data['eisen']).replace('y','x'))
eisenp = Pt(str(data['unram'])).resultant(eisenp)
eisenp = web_latex(eisenp)
else:
unramp = data['unram'].replace('t','x')
unramp = web_latex(Px(str(unramp)))
unramp = prettyname(unramdata)+' $\\cong '+Qp+'(t)$ where $t$ is a root of '+unramp
eisenp = Ptx(str(data['eisen']).replace('y','x'))
eisenp = '$'+web_latex(eisenp, False)+'\\in'+Qp+'(t)[x]$'
rflabel = db.lf_fields.lucky({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']}, projection=0)
if rflabel is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rflabel
gsm = data['gsm']
if gsm == [0]:
gsm = 'Not computed'
elif gsm == [-1]:
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(gsm))
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': p,
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': lf_display_knowl( rflabel, name=printquad(data['rf'], p)),
'base': lf_display_knowl(str(p)+'.1.0.1', name='$%s$'%Qp),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_pretty_and_nTj(gn, gt, True),
'gt': gt,
'inertia': group_display_inertia(data['inertia']),
'unram': unramp,
'eisen': eisenp,
'gms': data['gms'],
'gsm': gsm,
'galphrase': galphrase,
'autstring': autstring,
'subfields': format_subfields(data['subfields'],p),
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())
def computation_minimal_polynomial_latex(self):
pol = coeff_to_poly(self._data["QpRts-minpoly"])
return web_latex(pol, enclose=False)
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = lfdb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field ' + label
polynomial = coeff_to_poly(string2list(data['coeffs']))
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
the_gal = WebGaloisGroup.from_nt(gn,gt)
isgal = ' Galois' if the_gal.order() == gn else ' not Galois'
abelian = ' and abelian' if the_gal.is_abelian() else ''
galphrase = 'This field is'+isgal+abelian+' over $\Q_{%d}$.'%p
autstring = r'\Gal' if the_gal.order() == gn else r'\Aut'
prop2 = [
('Label', label),
('Base', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, db())),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = lfdb().find_one({'p': p, 'n': f, 'c': 0})
if unramdata is not None:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = lfdb().find_one({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
gsm = data['gsm']
if gsm == '0':
gsm = 'Not computed'
elif gsm == '-1':
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(string2list(gsm)))
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': p,
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, db()),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], db()),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'gsm': gsm,
'galphrase': galphrase,
'autstring': autstring,
'subfields': format_subfields(data['subfields'],p),
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def coeff_to_nf(c):
return NumberField(coeff_to_poly(c), 'a')
def display_poly(coeffs):
return web_latex(coeff_to_poly(string2list(coeffs)))
def defining_polynomial(self):
if self.field_poly:
return web_latex_split_on_pm(web_latex(coeff_to_poly(self.field_poly), enclose=False))
return None
def computation_minimal_polynomial_raw_typeset(self):
pol = coeff_to_poly(self._data["QpRts-minpoly"])
return raw_typeset_poly(pol)
def format_coeffs(coeffs):
return pol_to_html(str(coeff_to_poly(string2list(coeffs))))
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = lfdb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field ' + label
polynomial = coeff_to_poly(string2list(data['coeffs']))
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
the_gal = WebGaloisGroup.from_nt(gn, gt)
isgal = ' Galois' if the_gal.order() == gn else ' not Galois'
abelian = ' and abelian' if the_gal.is_abelian() else ''
galphrase = 'This field is' + isgal + abelian + ' over $\Q_{%d}$.' % p
autstring = r'\Gal' if the_gal.order() == gn else r'\Aut'
prop2 = [
('Label', label),
('Base', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, db())),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = lfdb().find_one({'p': p, 'n': f, 'c': 0})
if unramdata is not None:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = lfdb().find_one({
'p': p,
'n': {
'$in': [1, 2]
},
'rf': data['rf']
})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
gsm = data['gsm']
if gsm == '0':
gsm = 'Not computed'
elif gsm == '-1':
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(string2list(gsm)))
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': p,
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, db()),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], db()),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'gsm': gsm,
'galphrase': galphrase,
'autstring': autstring,
'subfields': format_subfields(data['subfields'], p),
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore)
def display_poly(coeffs):
return web_latex(coeff_to_poly(string2list(coeffs)))
def test_coeff_to_poly(self):
r"""
Checking utility: coeff_to_poly
"""
x = var('x')
self.assertEqual(coeff_to_poly("1 - 3x + x^2"), x**2 - 3 * x + 1)
def polynomial_raw_typeset(self):
return raw_typeset_poly(coeff_to_poly(self.polynomial()))
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = db.lf_fields.lookup(label)
if data is None:
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Field %s was not found in the database.", label)
else:
flash_error("%s is not a valid label for a $p$-adic field.",
label)
return redirect(url_for(".index"))
title = '$p$-adic field ' + prettyname(data)
titletag = 'p-adic field ' + prettyname(data)
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
Qp = r'\Q_{%d}' % p
e = data['e']
f = data['f']
cc = data['c']
gt = int(data['galois_label'].split('T')[1])
gn = data['n']
the_gal = WebGaloisGroup.from_nt(gn, gt)
isgal = ' Galois' if the_gal.order() == gn else ' not Galois'
abelian = ' and abelian' if the_gal.is_abelian() else ''
galphrase = 'This field is' + isgal + abelian + r' over $\Q_{%d}.$' % p
autstring = r'\Gal' if the_gal.order() == gn else r'\Aut'
prop2 = [
('Label', label),
('Base', r'\(%s\)' % Qp),
('Degree', r'\(%s\)' % data['n']),
('e', r'\(%s\)' % e),
('f', r'\(%s\)' % f),
('c', r'\(%s\)' % cc),
('Galois group', group_pretty_and_nTj(gn, gt)),
]
# Look up the unram poly so we can link to it
unramlabel = db.lf_fields.lucky({'p': p, 'n': f, 'c': 0}, projection=0)
if unramlabel is None:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
else:
unramfriend = "/LocalNumberField/%s" % unramlabel
unramdata = db.lf_fields.lookup(unramlabel)
Px = PolynomialRing(QQ, 'x')
Pt = PolynomialRing(QQ, 't')
Ptx = PolynomialRing(Pt, 'x')
if data['f'] == 1:
unramp = r'$%s$' % Qp
eisenp = Ptx(str(data['eisen']).replace('y', 'x'))
eisenp = web_latex(eisenp)
else:
unramp = data['unram'].replace('t', 'x')
unramp = web_latex(Px(str(unramp)))
unramp = prettyname(
unramdata
) + ' $\\cong ' + Qp + '(t)$ where $t$ is a root of ' + unramp
eisenp = Ptx(str(data['eisen']).replace('y', 'x'))
eisenp = '$' + web_latex(eisenp, False) + '\\in' + Qp + '(t)[x]$'
rflabel = db.lf_fields.lucky(
{
'p': p,
'n': {
'$in': [1, 2]
},
'rf': data['rf']
}, projection=0)
if rflabel is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rflabel
gsm = data['gsm']
if gsm == [0]:
gsm = 'Not computed'
elif gsm == [-1]:
gsm = 'Does not exist'
else:
gsm = lf_formatfield(','.join([str(b) for b in gsm]))
info.update({
'polynomial':
web_latex(polynomial),
'n':
data['n'],
'p':
p,
'c':
data['c'],
'e':
data['e'],
'f':
data['f'],
't':
data['t'],
'u':
data['u'],
'rf':
lf_display_knowl(rflabel, name=printquad(data['rf'], p)),
'base':
lf_display_knowl(str(p) + '.1.0.1', name='$%s$' % Qp),
'hw':
data['hw'],
'slopes':
show_slopes(data['slopes']),
'gal':
group_pretty_and_nTj(gn, gt, True),
'gt':
gt,
'inertia':
group_display_inertia(data['inertia']),
'unram':
unramp,
'eisen':
eisenp,
'gms':
data['gms'],
'gsm':
gsm,
'galphrase':
galphrase,
'autstring':
autstring,
'subfields':
format_subfields(data['subfields'], p),
'aut':
data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template(
"lf-show-field.html",
credit=LF_credit,
title=title,
titletag=titletag,
bread=bread,
info=info,
properties=prop2,
friends=friends,
learnmore=learnmore_list(),
KNOWL_ID="lf.%s" % label,
)
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.localfields.fields.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field ' + label
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
prop2 = [
('Label', label),
('Base', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, C)),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = C.localfields.fields.find_one({'p': p, 'n': f, 'c': 0})
if len(unramdata) > 0:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = C.localfields.fields.find_one({
'p': p,
'n': {
'$in': [1, 2]
},
'rf': data['rf']
})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': data['p'],
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, C),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], C),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore)
def display_poly(coeffs):
return web_latex(coeff_to_poly(coeffs))
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.localfields.fields.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field:' + label
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
prop2 = [
('Base', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, C)),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = C.localfields.fields.find_one({'p': p, 'n': f, 'c': 0})
if len(unramdata) > 0:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = C.localfields.fields.find_one({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': data['p'],
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, C),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], C),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def poly(self, var="x"):
return coeff_to_poly(self._data['coeffs'], var=var)
def test_coeff_to_poly(self):
r"""
Checking utility: coeff_to_poly
"""
x = var('x')
self.assertEqual(coeff_to_poly("1 - 3x + x^2"), x**2 - 3*x + 1)
def coeff_to_nf(c):
return NumberField(coeff_to_poly(c), 'a')
def display_poly(coeffs):
return web_latex(coeff_to_poly(coeffs))
def poly(self):
return coeff_to_poly(self._data['coeffs'])
def format_coeffs(coeffs):
return pol_to_html(str(coeff_to_poly(coeffs)))
def web_poly(self):
return pol_to_html(str(coeff_to_poly(self.coeffs())))
def poly(self):
return coeff_to_poly(self._data['coeffs'])
def make_torsion_growth(self):
try:
tor_gro = self.tor_gro
except AttributeError: # for curves with norsion growth data
tor_gro = None
if tor_gro is None:
self.torsion_growth_data_exists = False
return
self.torsion_growth_data_exists = True
self.tg = tg = {}
tg['data'] = tgextra = []
# find all base-changes of this curve in the database, if any
bcs = list(
db.ec_nfcurves.search(
{'base_change': {
'$contains': [self.lmfdb_label]
}},
projection='label'))
bcfs = [lab.split("-")[0] for lab in bcs]
for F, T in tor_gro.items():
tg1 = {}
tg1['bc'] = "Not in database"
# mongo did not allow "." in a dict key so we changed (e.g.) '3.1.44.1' to '3:1:44:1'
# Here we change it back (but this code also works in case the fields already use ".")
F = F.replace(":", ".")
if "." in F:
field_data = nf_display_knowl(F, field_pretty(F))
deg = int(F.split(".")[0])
bcc = [x for x, y in zip(bcs, bcfs) if y == F]
if bcc:
from lmfdb.ecnf.main import split_full_label
F, NN, I, C = split_full_label(bcc[0])
tg1['bc'] = bcc[0]
tg1['bc_url'] = url_for('ecnf.show_ecnf',
nf=F,
conductor_label=NN,
class_label=I,
number=C)
else:
field_data = web_latex(coeff_to_poly(string2list(F)))
deg = F.count(",")
tg1['d'] = deg
tg1['f'] = field_data
tg1['t'] = r'\(' + r' \times '.join(
[r'\Z/{}\Z'.format(n) for n in T.split(",")]) + r'\)'
tg1['m'] = 0
tgextra.append(tg1)
tgextra.sort(key=lambda x: x['d'])
tg['n'] = len(tgextra)
lastd = 1
for tg1 in tgextra:
d = tg1['d']
if d != lastd:
tg1['m'] = len([x for x in tgextra if x['d'] == d])
lastd = d
## Hard-code this for now. While something like
## max(db.ec_curves.search({},projection='tor_degs')) might
## work, since 'tor_degs' is in the extra table it is very
## slow. Note that the *only* place where this number is used
## is in the ec-curve template where it says "The number
## fields ... of degree up to {{data.tg.maxd}} such that...".
tg['maxd'] = 7
def web_poly(self):
return pol_to_html(str(coeff_to_poly(self.coeffs())))
def render_field_webpage(args):
data = None
info = {}
if "label" in args:
label = clean_input(args["label"])
C = base.getDBConnection()
data = C.localfields.fields.find_one({"label": label})
if data is None:
bread = get_bread([("Search error", " ")])
info["err"] = "Field " + label + " was not found in the database."
info["label"] = label
return search_input_error(info, bread)
title = "Local Number Field " + label
polynomial = coeff_to_poly(data["coeffs"])
p = data["p"]
e = data["e"]
f = data["f"]
cc = data["c"]
GG = data["gal"]
gn = GG[0]
gt = GG[1]
prop2 = [
("Label", label),
("Base", "\(\Q_{%s}\)" % p),
("Degree", "\(%s\)" % data["n"]),
("e", "\(%s\)" % e),
("f", "\(%s\)" % f),
("c", "\(%s\)" % cc),
("Galois group", group_display_short(gn, gt, C)),
]
Pt = PolynomialRing(QQ, "t")
Pyt = PolynomialRing(Pt, "y")
eisenp = Pyt(str(data["eisen"]))
unramp = Pyt(str(data["unram"]))
# Look up the unram poly so we can link to it
unramdata = C.localfields.fields.find_one({"p": p, "n": f, "c": 0})
if len(unramdata) > 0:
unramfriend = "/LocalNumberField/%s" % unramdata["label"]
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ""
rfdata = C.localfields.fields.find_one({"p": p, "n": {"$in": [1, 2]}, "rf": data["rf"]})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ""
else:
rffriend = "/LocalNumberField/%s" % rfdata["label"]
info.update(
{
"polynomial": web_latex(polynomial),
"n": data["n"],
"p": data["p"],
"c": data["c"],
"e": data["e"],
"f": data["f"],
"t": data["t"],
"u": data["u"],
"rf": printquad(data["rf"], p),
"hw": data["hw"],
"slopes": show_slopes(data["slopes"]),
"gal": group_display_knowl(gn, gt, C),
"gt": gt,
"inertia": group_display_inertia(data["inertia"], C),
"unram": web_latex(unramp),
"eisen": web_latex(eisenp),
"gms": data["gms"],
"aut": data["aut"],
}
)
friends = [("Galois group", "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != "":
friends.append(("Unramified subfield", unramfriend))
if rffriend != "":
friends.append(("Discriminant root field", rffriend))
bread = get_bread([(label, " ")])
learnmore = [
("Completeness of the data", url_for(".completeness_page")),
("Source of the data", url_for(".how_computed_page")),
("Local field labels", url_for(".labels_page")),
]
return render_template(
"lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore,
)
