def show_ecnf1(nf):
if nf == "1.1.1.1":
return redirect(url_for("ec.rational_elliptic_curves", **request.args))
if request.args:
return elliptic_curve_search(data=request.args)
start = 0
count = 50
try:
nf_label = nf_string_to_label(nf)
except ValueError:
return search_input_error()
query = {'field_label': nf_label}
cursor = db_ecnf().find(query)
nres = cursor.count()
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC),
('conductor_label', ASC), ('iso_nlabel', ASC),
('number', ASC)]).skip(start).limit(count)
bread = [('Elliptic Curves', url_for(".index")),
(nf_label, url_for('.show_ecnf1', nf=nf_label))]
res = list(res)
for e in res:
e['field_knowl'] = nf_display_knowl(e['field_label'],
getDBConnection(),
field_pretty(e['field_label']))
info = {}
info['field'] = nf_label
info['query'] = query
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
#don't risk recomputing all the ecnf stats just to show curves for a single number field
#if nf_label:
#info['stats'] = ecnf_field_summary(nf_label)
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curves over %s' % field_pretty(nf_label)
return render_template("ecnf-search-results.html",
info=info,
credit=ecnf_credit,
bread=bread,
title=t,
learnmore=learnmore_list())
def show_ecnf1(nf):
if nf == "1.1.1.1":
return redirect(url_for("ec.rational_elliptic_curves", **request.args))
if request.args:
return elliptic_curve_search(data=request.args)
start = 0
count = 50
try:
nf_label = nf_string_to_label(nf)
except ValueError:
return search_input_error()
query = {'field_label': nf_label}
cursor = db_ecnf().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC), ('conductor_label', ASC), ('iso_nlabel', ASC), ('number', ASC)]).skip(start).limit(count)
bread = [('Elliptic Curves', url_for(".index")),
(nf_label, url_for('.show_ecnf1', nf=nf_label))]
res = list(res)
for e in res:
e['field_knowl'] = nf_display_knowl(e['field_label'], getDBConnection(), field_pretty(e['field_label']))
info = {}
info['field'] = nf_label
info['query'] = query
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
#don't risk recomputing all the ecnf stats just to show curves for a single number field
#if nf_label:
#info['stats'] = ecnf_field_summary(nf_label)
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curves over %s' % field_pretty(nf_label)
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t, learnmore=learnmore_list())
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 field_display_gen(label, poly, disc=None, self_dual=None, truncate=0):
"""
This function is used to display a number field knowl. When the field
is not in the LMFDB, it uses a dynamic knowl displaying the polynomial
and discriminant. Otherwise, it uses the standard LMFDB number field knowl.
INPUT:
- ``label`` -- the LMFDB label for the field (``None`` if not in the LMFDB)
- ``poly`` -- the defining polynomial for the field as a list
- ``disc`` -- the discriminant of the field, as a list of (p, e) pairs
- ``truncate`` -- an integer, the maximum length of the field label before truncation.
If 0, no truncation will occur.
"""
if label is None:
if poly:
if self_dual:
unit = ZZ(1)
else:
unit = ZZ(-1)**((len(poly)-1)//2)
return polyquo_knowl(poly, disc, unit, 12)
else:
return ''
else:
name = field_pretty(label)
if truncate and name == label and len(name) > truncate:
parts = label.split('.')
parts[2] = r'\(\cdots\)'
name = '.'.join(parts)
return nf_display_knowl(label, name)
def field_label(F, pretty=True, check=False):
r"""
Returns the LMFDB label of the field F.
"""
if F.absolute_degree() == 1:
p = 'x'
else:
pp = F.absolute_polynomial()
x = pp.parent().gen()
p = str(pp).replace(str(x), 'x')
l = poly_to_field_label(p)
if l is None:
if check:
return False
else:
if pretty:
return web_latex_split_on_pm(pp)
else:
return pp
else:
if check:
return True
if pretty:
return field_pretty(l)
else:
return l
def field_label(F, pretty = True, check=False):
r"""
Returns the LMFDB label of the field F.
"""
if F.absolute_degree() == 1:
p = 'x'
else:
pp = F.absolute_polynomial()
x = pp.parent().gen()
p = str(pp).replace(str(x), 'x')
l = poly_to_field_label(p)
if l is None:
if check:
return False
else:
if pretty:
return web_latex_split_on_pm(pp)
else:
return pp
else:
if check:
return True
if pretty:
return field_pretty(l)
else:
return l
def get_nf_info(lab):
r""" extract number field label from string and pretty"""
try:
label = nf_string_to_label(lab)
pretty = field_pretty (label)
except ValueError as err:
raise ValueError(Markup("<span style='color:black'>%s</span> is not a valid number field label. %s" % (escape(lab),err)))
return label, pretty
def get_nf_info(lab):
r""" extract number field label from string and pretty"""
try:
label = nf_string_to_label(lab)
pretty = field_pretty (label)
except ValueError as err:
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid number field. %s" % (lab,err)), "error")
raise ValueError
return label, pretty
def extend_from_db(self):
setattr(self._value, "lmfdb_label", self._db_value)
if not self._db_value is None and self._db_value != '':
try:
url = url_for("number_fields.by_label", label=self._db_value)
except RuntimeError:
emf_logger.critical("could not set url for the label")
url = ''
setattr(self._value, "lmfdb_url",url)
setattr(self._value, "lmfdb_pretty", field_pretty(self._db_value))
else:
if hasattr(self._value,'absolute_polynomial'):
setattr(self._value, "lmfdb_pretty", web_latex_split_on_pm(self._value.absolute_polynomial()))
elif self._value.absolute_degree()==1:
setattr(self._value, "lmfdb_pretty", field_pretty('1.1.1.1'))
setattr(self._value, "lmfdb_label", '1.1.1.1')
else:
emf_logger.critical("could not set lmfdb_pretty for the label")
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
if len(request.args) > 0:
return elliptic_curve_search(data=request.args)
bread = get_bread()
# the dict data will hold additional information to be displayed on
# the main browse and search page
data = {}
# data['fields'] holds data for a sample of number fields of different
# signatures for a general browse:
data['fields'] = []
# Rationals
data['fields'].append(['the rational field', (('1.1.1.1', [url_for('ec.rational_elliptic_curves'), '$\Q$']),)])
# Real quadratics (only a sample)
rqfs = ['2.2.%s.1' % str(d) for d in [5, 89, 229, 497]]
data['fields'].append(['real quadratic fields', ((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)]) for nf in rqfs)])
# Imaginary quadratics
iqfs = ['2.0.%s.1' % str(d) for d in [4, 8, 3, 7, 11]]
data['fields'].append(['imaginary quadratic fields', ((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)]) for nf in iqfs)])
# Cubics
cubics = ['3.1.23.1']
data['fields'].append(['cubic fields', ((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)]) for nf in cubics)])
# data['highlights'] holds data (URL and descriptive text) for a
# sample of elliptic curves with interesting features:
data['highlights'] = []
data['highlights'].append(
['A curve with $C_3\\times C_3$ torsion',
url_for('.show_ecnf', nf='2.0.3.1', class_label='a', conductor_label='[2268,36,18]', number=int(1))]
)
data['highlights'].append(
['A curve with $C_4\\times C_4$ torsion',
url_for('.show_ecnf', nf='2.0.4.1', class_label='b', conductor_label='[5525,870,5]', number=int(9))]
)
data['highlights'].append(
['A curve with CM by $\\sqrt{-267}$',
url_for('.show_ecnf', nf='2.2.89.1', class_label='a', conductor_label='81.1', number=int(1))]
)
data['highlights'].append(
['An isogeny class with isogenies of degree $3$ and $89$ (and $267$)',
url_for('.show_ecnf_isoclass', nf='2.2.89.1', class_label='a', conductor_label='81.1')]
)
data['highlights'].append(
['A curve with everywhere good reduction, but no global minimal model',
url_for('.show_ecnf', nf='2.2.229.1', class_label='a', conductor_label='1.1', number=int(1))]
)
return render_template("ecnf-index.html",
title="Elliptic Curves over Number Fields",
data=data,
bread=bread)
def get_nf_info(lab):
r""" extract number field label from string and pretty"""
try:
label = nf_string_to_label(lab)
pretty = field_pretty(label)
except ValueError as err:
flash_error("%s is not a valid number field. %s", lab, err)
raise ValueError
return label, pretty
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
if len(request.args) > 0:
return elliptic_curve_search(data=request.args)
bread = get_bread()
# the dict data will hold additional information to be displayed on
# the main browse and search page
data = {}
# data['fields'] holds data for a sample of number fields of different
# signatures for a general browse:
data['fields'] = []
# Rationals
data['fields'].append(['the rational field', (('1.1.1.1', [url_for('ec.rational_elliptic_curves'), '$\Q$']),)])
# Real quadratics (only a sample)
rqfs = ['2.2.%s.1' % str(d) for d in [5, 89, 229, 497]]
data['fields'].append(['real quadratic fields', ((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)]) for nf in rqfs)])
# Imaginary quadratics
iqfs = ['2.0.%s.1' % str(d) for d in [4, 8, 3, 7, 11]]
data['fields'].append(['imaginary quadratic fields', ((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)]) for nf in iqfs)])
# Cubics
cubics = ['3.1.23.1']
data['fields'].append(['cubic fields', ((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)]) for nf in cubics)])
# data['highlights'] holds data (URL and descriptive text) for a
# sample of elliptic curves with interesting features:
data['highlights'] = []
data['highlights'].append(
['A curve with $C_3\\times C_3$ torsion',
url_for('.show_ecnf', nf='2.0.3.1', class_label='a', conductor_label='[2268,36,18]', number=int(1))]
)
data['highlights'].append(
['A curve with $C_4\\times C_4$ torsion',
url_for('.show_ecnf', nf='2.0.4.1', class_label='b', conductor_label='[5525,870,5]', number=int(9))]
)
data['highlights'].append(
['A curve with CM by $\\sqrt{-267}$',
url_for('.show_ecnf', nf='2.2.89.1', class_label='a', conductor_label='81.1', number=int(1))]
)
data['highlights'].append(
['An isogeny class with isogenies of degree $3$ and $89$ (and $267$)',
url_for('.show_ecnf_isoclass', nf='2.2.89.1', class_label='a', conductor_label='81.1')]
)
data['highlights'].append(
['A curve with everywhere good reduction, but no global minimal model',
url_for('.show_ecnf', nf='2.2.229.1', class_label='a', conductor_label='1.1', number=int(1))]
)
return render_template("ecnf-index.html",
title="Elliptic Curves over Number Fields",
data=data,
bread=bread)
def get_nf_info(lab):
r""" extract number field label from string and pretty"""
try:
label = nf_string_to_label(lab)
pretty = field_pretty (label)
except ValueError as err:
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid number field. %s" % (lab,err)), "error")
raise ValueError
return label, pretty
def field_display_gen(self, label, poly):
if label is None:
if poly:
return polyquo_knowl(poly)
else:
return 'Unknown'
elif label == u'1.1.1.1': # rationals, special case
return nf_display_knowl(label, name=r"\(\Q\)")
else:
return nf_display_knowl(label, field_pretty(label))
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 end_field_statement(field_label, poly):
if field_label == '1.1.1.1':
return """All endomorphisms of the Jacobian are defined over \(\Q\)"""
elif field_label != '':
pretty = field_pretty(field_label)
url = url_for("number_fields.by_label", label=field_label)
return """Smallest field over which all endomorphisms are defined:<br>
Galois number field \(K = \Q (a) \simeq \) <a href=%s>%s</a> with defining polynomial \(%s\)""" % (url, pretty, poly)
else:
return """Smallest field over which all endomorphisms are defined:<br>
Galois number field \(K = \Q (a)\) with defining polynomial \(%s\)""" % poly
def end_field_statement(field_label, poly):
if field_label == '1.1.1.1':
return """All endomorphisms of the Jacobian are defined over \(\Q\)"""
elif field_label != '':
pretty = field_pretty(field_label)
url = url_for("number_fields.by_label", label=field_label)
return """Smallest field over which all endomorphisms are defined:<br>
Galois number field \(K = \Q (a) \simeq \) <a href=%s>%s</a> with defining polynomial \(%s\)""" % (url, pretty, poly)
else:
return """Smallest field over which all endomorphisms are defined:<br>
Galois number field \(K = \Q (a)\) with defining polynomial \(%s\)""" % poly
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
if len(request.args) > 0:
return elliptic_curve_search(data=request.args)
bread = get_bread()
data = {}
nfs = db_ecnf().distinct("field_label")
nfs = ["2.0.4.1", "2.2.5.1", "3.1.23.1"]
data["fields"] = [(nf, field_pretty(nf)) for nf in nfs if int(nf.split(".")[2]) < 200]
return render_template("ecnf-index.html", title="Elliptic Curves over Number Fields", data=data, bread=bread)
def show_ecnf1(nf):
if nf == "1.1.1.1":
return redirect(url_for("ec.rational_elliptic_curves", **request.args))
if request.args:
return elliptic_curve_search(data=request.args)
start = 0
count = 50
nf_label = parse_field_string(nf)
query = {"field_label": nf_label}
cursor = db_ecnf().find(query)
nres = cursor.count()
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
res = (
cursor.sort(
[
("field_label", ASC),
("conductor_norm", ASC),
("conductor_label", ASC),
("iso_label", ASC),
("number", ASC),
]
)
.skip(start)
.limit(count)
)
bread = [("Elliptic Curves", url_for(".index")), (nf_label, url_for(".show_ecnf1", nf=nf_label))]
res = list(res)
for e in res:
e["field_knowl"] = nf_display_knowl(e["field_label"], getDBConnection(), e["field_label"])
info = {}
info["field"] = nf_label
info["query"] = query
info["curves"] = res # [ECNF(e) for e in res]
info["number"] = nres
info["start"] = start
info["count"] = count
info["more"] = int(start + count < nres)
info["field_pretty"] = field_pretty
info["web_ainvs"] = web_ainvs
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
t = "Elliptic Curves over %s" % field_pretty(nf_label)
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
def extend_from_db(self):
setattr(self._value, "lmfdb_label", self._db_value)
if not self._db_value is None and self._db_value != "":
try:
url = url_for("number_fields.by_label", label=self._db_value)
except RuntimeError:
emf_logger.critical("could not set url for the label")
url = ""
setattr(self._value, "lmfdb_url", url)
setattr(self._value, "lmfdb_pretty", field_pretty(self._db_value))
else:
setattr(self._value, "lmfdb_pretty", web_latex_split_on_pm(self._value.absolute_polynomial()))
def split_field_statement(is_simple_geom, field_label, poly):
if is_simple_geom:
return """Simple over \(\overline{\Q}\)"""
elif field_label == '1.1.1.1':
return """Splits over \(\Q\)"""
elif field_label != '':
pretty = field_pretty(field_label)
url = url_for("number_fields.by_label", label=field_label)
return """Splits over the number field \(\Q (b) \simeq \) <a href=%s>%s</a> with defining polynomial:<br> \(%s\)"""\
% (url, pretty, poly)
else:
return """Splits over the number field \(\Q (b)\) with defining polynomial:<br> \(%s\)""" % poly
def split_field_statement(is_simple_geom, field_label, poly):
if is_simple_geom:
return """Simple over \(\overline{\Q}\)"""
elif field_label == '1.1.1.1':
return """Splits over \(\Q\)"""
elif field_label != '':
pretty = field_pretty(field_label)
url = url_for("number_fields.by_label", label=field_label)
return """Splits over the number field \(\Q (b) \simeq \) <a href=%s>%s</a> with defining polynomial:<br> \(%s\)"""\
% (url, pretty, poly)
else:
return """Splits over the number field \(\Q (b)\) with defining polynomial:<br> \(%s\)""" % poly
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
info = to_dict(request.args, search_array=ECNFSearchArray(), stats=ECNF_stats())
if request.args:
return elliptic_curve_search(info)
bread = get_bread()
# the dict data will hold additional information to be displayed on
# the main browse and search page
# info['fields'] holds data for a sample of number fields of different
# signatures for a general browse:
info['fields'] = []
# Rationals
# info['fields'].append(['the rational field', (('1.1.1.1', [url_for('ec.rational_elliptic_curves'), '$\Q$']),)]) # Removed due to ambiguity
# Real quadratics (sample)
rqfs = ['2.2.{}.1'.format(d) for d in [8, 12, 5, 24, 28, 40, 44, 13, 56, 60]]
info['fields'].append(['By <a href="{}">real quadratic field</a>'.format(url_for('.statistics_by_signature', d=2, r=2)),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in rqfs)])
# Imaginary quadratics (sample)
iqfs = ['2.0.{}.1'.format(d) for d in [4, 8, 3, 7, 11, 19, 43]]
info['fields'].append(['By <a href="{}">imaginary quadratic field</a>'.format(url_for('.statistics_by_signature', d=2, r=0)),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in iqfs)])
# Cubics (sample)
cubics = ['3.1.23.1'] + ['3.3.{}.1'.format(d) for d in [49,81,148,169,229,257,316]]
info['fields'].append(['By <a href="{}">cubic field</a>'.format(url_for('.statistics_by_degree', d=3)),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in cubics)])
# Quartics (sample)
quartics = ['4.4.{}.1'.format(d) for d in [725,1125,1600,1957,2000,2048,2225,2304]]
info['fields'].append(['By <a href="{}">totally real quartic field</a>'.format(url_for('.statistics_by_degree', d=4)),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in quartics)])
# Quintics (sample)
quintics = ['5.5.{}.1'.format(d) for d in [14641, 24217, 36497, 38569, 65657, 70601, 81509]]
info['fields'].append(['By <a href="{}">totally real quintic field</a>'.format(url_for('.statistics_by_degree', d=5)),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in quintics)])
# Sextics (sample)
sextics = ['6.6.{}.1'.format(d) for d in [300125, 371293, 434581, 453789, 485125, 592661, 703493]]
info['fields'].append(['By <a href="{}">totally real sextic field</a>'.format(url_for('.statistics_by_degree', d=6)),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in sextics)])
return render_template("ecnf-index.html",
title="Elliptic curves over number fields",
info=info,
bread=bread, learnmore=learnmore_list())
def extend_from_db(self):
setattr(self._value, "lmfdb_label", self._db_value)
if not self._db_value is None and self._db_value != '':
try:
url = url_for("number_fields.by_label", label=self._db_value)
except RuntimeError:
emf_logger.critical("could not set url for the label")
url = ''
setattr(self._value, "lmfdb_url", url)
setattr(self._value, "lmfdb_pretty", field_pretty(self._db_value))
else:
setattr(self._value, "lmfdb_pretty",
web_latex_split_on_pm(self._value.absolute_polynomial()))
def spl_fod_statement(is_simple_geom, spl_fod_label, spl_fod_poly):
if is_simple_geom:
return """Simple over \(\overline{\Q}\)"""
elif spl_fod_label == '1.1.1.1':
return """Splits over \(\Q\)"""
elif spl_fod_label != '':
spl_fod_pretty = field_pretty(spl_fod_label)
spl_fod_url = url_for("number_fields.by_label", label=spl_fod_label)
return """Splits over the number field \(\Q (b) \cong \) <a href=%s>%s</a> with defining polynomial:<br> \(%s\)"""\
% (spl_fod_url, spl_fod_pretty, spl_fod_poly)
else:
return """Splits over the number field \(\Q (b)\) with defining polynomialL<br>  %s"""\
% spl_fod_poly
def spl_fod_statement(is_simple_geom, spl_fod_label, spl_fod_poly):
if is_simple_geom:
return """Simple over \(\overline{\Q}\)"""
elif spl_fod_label == '1.1.1.1':
return """Splits over \(\Q\)"""
elif spl_fod_label != '':
spl_fod_pretty = field_pretty(spl_fod_label)
spl_fod_url = url_for("number_fields.by_label", label=spl_fod_label)
return """Splits over the number field \(\Q (b) \cong \) <a href=%s>%s</a> with defining polynomial:<br> \(%s\)"""\
% (spl_fod_url, spl_fod_pretty, spl_fod_poly)
else:
return """Splits over the number field \(\Q (b)\) with defining polynomialL<br>  %s"""\
% spl_fod_poly
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 stats_for_field(self, F):
forms = self.forms
fields = self.fields
# pipeline = [{"$match": {'field_label':F}},
# {"$project" : { 'level_norm' : 1 }},
# {"$group":{"_id":"level_norm", "nforms": {"$sum": 2}, "maxnorm" : {"$max": '$level_norm'}}}]
# res = forms.aggregate(pipeline).next()
res = [f['level_norm'] for f in forms.find({'field_label':F}, ['level_norm'])]
stats = {}
stats['nforms'] = len(res) # res['nforms']
stats['maxnorm'] = max(res+[0]) # res['maxnorm']
stats['field_knowl'] = nf_display_knowl(F, lmfdb.base.getDBConnection(), field_pretty(F))
stats['forms'] = url_for('hmf.hilbert_modular_form_render_webpage', field_label=F)
return stats
def field_summary(self, field):
stats = self.field_normstats[field]
ncurves = stats['ncurves']
nclasses = stats['nclasses']
max_norm = stats['max_norm']
ec_knowl = self.ec_knowl if ncurves == 1 else self.ec_knowls
iso_knowl = self.iso_knowl if ncurves == 1 else self.iso_knowls
nf_knowl = self.nf_knowl if ncurves == 1 else self.nf_knowls
cond_knowl = self.cond_knowl if ncurves == 1 else self.cond_knowls
s = '' if max_norm == 1 else 'up to '
norm_phrase = ' of norm {}{}.'.format(s, max_norm)
return ''.join([
r'The database currently contains {} '.format(ncurves), ec_knowl,
r' defined over the ', nf_knowl,
r' {}, in {} '.format(field_pretty(field), nclasses), iso_knowl,
r', with ', cond_knowl, norm_phrase
])
def ecnf_field_summary(field):
stats = get_field_stats(field)
s = '' if stats['ncurves'] == 1 else 's'
ec_knowl = '<a knowl="ec">elliptic curve%s</a>' % s
s = '' if stats['nclasses'] == 1 else 'es'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny class%s</a>' % s
nf_knowl = '<a knowl="nf">number field</a>'
s = '' if stats['maxnorm'] == 1 else 's'
cond_knowl = '<a knowl="ec.conductor">conductor%s</a>' % s
s = '' if stats['maxnorm'] == 1 else 'up to '
return ''.join([
r'The database currently contains %s ' % stats['ncurves'], ec_knowl,
r' defined over the ', nf_knowl,
r' %s, in %s ' % (field_pretty(field), stats['nclasses']), iso_knowl,
r', with ', cond_knowl,
r' of norm %s %s.' % (s, stats['maxnorm'])
])
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 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 ecnf_field_summary(field):
stats = get_field_stats(field)
s = '' if stats['ncurves']==1 else 's'
ec_knowl = '<a knowl="ec">elliptic curve%s</a>' % s
s = '' if stats['nclasses']==1 else 'es'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny class%s</a>' % s
nf_knowl = '<a knowl="nf">number field</a>'
s = '' if stats['maxnorm']==1 else 's'
cond_knowl = '<a knowl="ec.conductor">conductor%s</a>' % s
s = '' if stats['maxnorm']==1 else 'up to '
return ''.join([r'The database currently contains %s ' % stats['ncurves'],
ec_knowl,
r' defined over the ',
nf_knowl,
r' %s, in %s ' % (field_pretty(field), stats['nclasses']),
iso_knowl,
r', with ',
cond_knowl,
r' of norm %s %s.' % (s,stats['maxnorm'])])
def end_lattice_statement(lattice):
statement = ''
for ED in lattice:
if ED[0][0]:
# Add link and prettify if available:
statement += """Over subfield \(F \simeq \) <a href=%s>%s</a> with generator \(%s\) with minimal polynomial \(%s\)"""\
% (url_for("number_fields.by_label", label=ED[0][0]),
field_pretty(ED[0][0]), strlist_to_nfelt(ED[0][2], 'a'),
intlist_to_poly(ED[0][1]))
else:
statement += """Over subfield \(F\) with generator \(%s\) with minimal polynomial \(%s\)"""\
% (strlist_to_nfelt(ED[0][2], 'a'), intlist_to_poly(ED[0][1]))
statement += """:<br>"""
statement += end_statement(ED[1], ED[2], field=r'F', ring=ED[3])
statement += """Sato Tate group: %s""" % st_link_by_name(1,4,ED[4])
statement += """<br>"""
statement += gl2_simple_statement(ED[1], ED[2])
statement += """<p></p>"""
return statement
def ecnf_field_summary(field):
data = db.ec_nfcurves.stats.get_oldstat('conductor_norm_by_field')[field]
ncurves = data['ncurves']
s = '' if ncurves == 1 else 's'
ec_knowl = '<a knowl="ec">elliptic curve{}</a>'.format(s)
nclasses = data['nclasses']
s = '' if nclasses == 1 else 'es'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny class{}</a>'.format(s)
nf_knowl = '<a knowl="nf">number field</a>'
max_norm = data['max_norm']
s = '' if max_norm == 1 else 's'
cond_knowl = '<a knowl="ec.conductor">conductor{}</a>'.format(s)
s = '' if max_norm == 1 else 'up to '
return ''.join([
r'The database currently contains {} '.format(ncurves), ec_knowl,
r' defined over the ', nf_knowl,
r' {}, in {} '.format(field_pretty(field), nclasses), iso_knowl,
r', with ', cond_knowl, r' of norm {} {}.'.format(s, data['max_norm'])
])
def end_lattice_statement(lattice):
statement = ''
for ED in lattice:
if ED[0][0]:
# Add link and prettify if available:
statement += """Over subfield \(F \simeq \) <a href=%s>%s</a> with generator \(%s\) with minimal polynomial \(%s\)"""\
% (url_for("number_fields.by_label", label=ED[0][0]),
field_pretty(ED[0][0]), strlist_to_nfelt(ED[0][2], 'a'),
intlist_to_poly(ED[0][1]))
else:
statement += """Over subfield \(F\) with generator \(%s\) with minimal polynomial \(%s\)"""\
% (strlist_to_nfelt(ED[0][2], 'a'), intlist_to_poly(ED[0][1]))
statement += """:<br>"""
statement += end_statement(ED[1], ED[2], field=r'F', ring=ED[3])
statement += """Sato Tate group: %s""" % st_link_by_name(1, 4, ED[4])
statement += """<br>"""
statement += gl2_simple_statement(ED[1], ED[2])
statement += """<p></p>"""
return statement
def field_summary(self, field):
stats = self.field_normstats[field]
ncurves = stats['ncurves']
nclasses = stats['nclasses']
max_norm = stats['max_norm']
ec_knowl = self.ec_knowl if ncurves==1 else self.ec_knowls
iso_knowl = self.iso_knowl if ncurves==1 else self.iso_knowls
nf_knowl = self.nf_knowl if ncurves==1 else self.nf_knowls
cond_knowl = self.cond_knowl if ncurves==1 else self.cond_knowls
s = '' if max_norm==1 else 'up to '
norm_phrase = ' of norm {}{}.'.format(s, max_norm)
return ''.join([r'The database currently contains {} '.format(ncurves),
ec_knowl,
r' defined over the ',
nf_knowl,
r' {}, in {} '.format(field_pretty(field), nclasses),
iso_knowl,
r', with ',
cond_knowl,
norm_phrase])
def ecnf_field_summary(field):
data = db_ecnfstats().find_one({'_id':'conductor_norm_by_field'})[field]
ncurves = data['ncurves']
s = '' if ncurves==1 else 's'
ec_knowl = '<a knowl="ec">elliptic curve{}</a>'.format(s)
nclasses = data['nclasses']
s = '' if nclasses==1 else 'es'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny class{}</a>'.format(s)
nf_knowl = '<a knowl="nf">number field</a>'
max_norm = data['max_norm']
s = '' if max_norm==1 else 's'
cond_knowl = '<a knowl="ec.conductor">conductor{}</a>'.format(s)
s = '' if max_norm==1 else 'up to '
return ''.join([r'The database currently contains {} '.format(ncurves),
ec_knowl,
r' defined over the ',
nf_knowl,
r' {}, in {} '.format(field_pretty(field), nclasses),
iso_knowl,
r', with ',
cond_knowl,
r' of norm {} {}.'.format(s,data['max_norm'])])
def show_ecnf1(nf):
if request.args:
return elliptic_curve_search(data=request.args)
start = 0
count = 20
query = {'field_label' : nf}
cursor = db_ecnf().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC), ('conductor_label', ASC), ('iso_label', ASC), ('number', ASC)]).skip(start).limit(count)
bread = [('Elliptic Curves', url_for(".index")),
(nf, url_for('.show_ecnf1', nf=nf))]
res = list(res)
for e in res:
e['field_knowl'] = nf_display_knowl(e['field_label'], getDBConnection(), e['field_label'])
info = {}
info['field'] = nf
info['query'] = query
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curves over %s' % field_pretty(nf)
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
@search_wrap(template="hilbert_modular_form_search.html",
table=db.hmf_forms,
title='Hilbert modular form search results',
err_title='Hilbert modular form search error',
per_page=50,
shortcuts={'jump': hilbert_modular_form_jump},
projection=[
'field_label', 'short_label', 'label', 'level_ideal',
'dimension'
],
cleaners={
"level_ideal":
lambda v: teXify_pol(v['level_ideal']),
"field_knowl":
lambda e: nf_display_knowl(e['field_label'],
field_pretty(e['field_label']))
},
bread=lambda: get_bread("Search results"),
learnmore=learnmore_list,
url_for_label=url_for_label,
credit=lambda: hmf_credit,
properties=lambda: [])
def hilbert_modular_form_search(info, query):
parse_nf_string(info, query, 'field_label', name="Field")
parse_ints(info, query, 'deg', name='Field degree')
parse_ints(info, query, 'disc', name="Field discriminant")
parse_ints(info, query, 'dimension')
parse_ints(info, query, 'level_norm', name="Level norm")
parse_hmf_weight(info,
query,
'weight',
except ValueError:
return redirect(url_for(".hilbert_modular_form_render_webpage"))
hmf_columns = SearchColumns([
MultiProcessedCol(
"label",
"mf.hilbert.label",
"Label", ["field_label", "label", "short_label"],
lambda fld, label, short: '<a href="%s">%s</a>' % (url_for(
'hmf.render_hmf_webpage', field_label=fld, label=label), short),
default=True),
ProcessedCol("field_label",
"nf",
"Base field",
lambda fld: nf_display_knowl(fld, field_pretty(fld)),
default=True),
MathCol("deg", "nf.degree", "Field degree"),
MathCol("disc", "nf.discriminant", "Field discriminant"),
ProcessedCol("level_ideal",
"mf.hilbert.level_norm",
"Level",
teXify_pol,
mathmode=True,
default=True),
MathCol("level_norm", "mf.level_norm", "Level norm"),
MathCol("weight", "mf.hilbert.weight_vector", "Weight"),
MathCol("dimension", "mf.hilbert.dimension", "Dimension", default=True),
ProcessedCol("is_CM",
"mf.cm",
"CM",
def elliptic_curve_search(**args):
info = to_dict(args['data'])
if 'download' in info and info['download'] != 0:
return download_search(info)
if not 'query' in info:
info['query'] = {}
bread = [('Elliptic Curves', url_for(".index")),
('Search Results', '.')]
if 'jump' in info:
label = info.get('label', '').replace(" ", "")
# This label should be a full isogeny class label or a full
# curve label (including the field_label component)
try:
nf, cond_label, iso_label, number = split_full_label(label.strip())
except ValueError:
info['err'] = ''
return search_input_error(info, bread)
return show_ecnf(nf, cond_label, iso_label, number)
query = {}
try:
parse_ints(info,query,'conductor_norm')
parse_noop(info,query,'conductor_label')
parse_nf_string(info,query,'field',name="base number field",qfield='field_label')
parse_nf_elt(info,query,'jinv',name='j-invariant')
parse_ints(info,query,'torsion',name='Torsion order',qfield='torsion_order')
parse_bracketed_posints(info,query,'torsion_structure',maxlength=2)
if 'torsion_structure' in query and not 'torsion_order' in query:
query['torsion_order'] = reduce(mul,[int(n) for n in query['torsion_structure']],1)
except ValueError:
return search_input_error(info, bread)
if 'include_isogenous' in info and info['include_isogenous'] == 'off':
info['number'] = 1
query['number'] = 1
if 'include_base_change' in info and info['include_base_change'] == 'off':
query['base_change'] = []
else:
info['include_base_change'] = "on"
if 'include_Q_curves' in info:
if info['include_Q_curves'] == 'exclude':
query['q_curve'] = False
elif info['include_Q_curves'] == 'only':
query['q_curve'] = True
if 'include_cm' in info:
if info['include_cm'] == 'exclude':
query['cm'] = 0
elif info['include_cm'] == 'only':
query['cm'] = {'$ne' : 0}
info['query'] = query
count = parse_count(info, 50)
start = parse_start(info)
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC), ('conductor_label', ASC), ('iso_nlabel', ASC), ('number', ASC)]).skip(start).limit(count)
res = list(res)
for e in res:
e['numb'] = str(e['number'])
e['field_knowl'] = nf_display_knowl(e['field_label'], getDBConnection(), field_pretty(e['field_label']))
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curve search results'
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
def elliptic_curve_search(info):
if info.get('download') == '1' and info.get('Submit') and info.get(
'query'):
return download_search(info)
if not 'query' in info:
info['query'] = {}
bread = info.get('bread', [('Elliptic Curves', url_for(".index")),
('Search Results', '.')])
if 'jump' in info:
label = info.get('label', '').replace(" ", "")
# This label should be a full isogeny class label or a full
# curve label (including the field_label component)
try:
nf, cond_label, iso_label, number = split_full_label(label.strip())
except ValueError:
info['err'] = ''
return search_input_error(info, bread)
return redirect(
url_for(".show_ecnf",
nf=nf,
conductor_label=cond_label,
class_label=iso_label,
number=number), 301)
query = {}
if 'jinv' in info:
if info.get('field', '').strip() == '2.2.5.1':
info['jinv'] = info['jinv'].replace('phi', 'a')
if info.get('field', '').strip() == '2.0.4.1':
info['jinv'] = info['jinv'].replace('i', 'a')
try:
parse_ints(info, query, 'conductor_norm')
parse_noop(info, query, 'conductor_label')
parse_nf_string(info,
query,
'field',
name="base number field",
qfield='field_label')
parse_nf_elt(info, query, 'jinv', name='j-invariant')
parse_ints(info,
query,
'torsion',
name='Torsion order',
qfield='torsion_order')
parse_bracketed_posints(info, query, 'torsion_structure', maxlength=2)
if 'torsion_structure' in query and not 'torsion_order' in query:
query['torsion_order'] = reduce(
mul, [int(n) for n in query['torsion_structure']], 1)
parse_ints(info, query, field='isodeg', qfield='isogeny_degrees')
except (TypeError, ValueError):
return search_input_error(info, bread)
if query.get('jinv'):
query['jinv'] = ','.join(query['jinv'])
if query.get('field_label') == '1.1.1.1':
return redirect(url_for("ec.rational_elliptic_curves", **request.args),
301)
if 'include_isogenous' in info and info['include_isogenous'] == 'off':
info['number'] = 1
query['number'] = 1
if 'include_base_change' in info and info['include_base_change'] == 'off':
query['base_change'] = []
else:
info['include_base_change'] = "on"
if 'include_Q_curves' in info:
if info['include_Q_curves'] == 'exclude':
query['q_curve'] = False
elif info['include_Q_curves'] == 'only':
query['q_curve'] = True
if 'include_cm' in info:
if info['include_cm'] == 'exclude':
query['cm'] = 0
elif info['include_cm'] == 'only':
query['cm'] = {'$ne': 0}
info['query'] = query
count = parse_count(info, 50)
start = parse_start(info)
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC),
('conductor_label', ASC), ('iso_nlabel', ASC),
('number', ASC)]).skip(start).limit(count)
res = list(res)
for e in res:
e['numb'] = str(e['number'])
e['field_knowl'] = nf_display_knowl(e['field_label'],
getDBConnection(),
field_pretty(e['field_label']))
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = info.get('title', 'Elliptic Curve search results')
return render_template("ecnf-search-results.html",
info=info,
credit=ecnf_credit,
bread=bread,
title=t)
raise ValueError("A CM discriminant must be a fundamental discriminant of an imaginary quadratic field.")
cm_list += [-el for el in cm_list]
return cm_list
@search_parser
def parse_cm_list(inp, query, qfield):
query[qfield] = {'$in': make_cm_query(inp)}
ecnf_columns = SearchColumns([
MultiProcessedCol("label", "ec.curve_label", "Label",
["short_label", "field_label", "conductor_label", "iso_label", "number"],
lambda label, field, conductor, iso, number: '<a href="%s">%s</a>' % (
url_for('.show_ecnf', nf=field, conductor_label=conductor, class_label=iso, number=number),
label),
default=True, align="center"),
ProcessedCol("field_label", "nf", "Base field", lambda field: nf_display_knowl(field, field_pretty(field)), default=True, align="center"),
MultiProcessedCol("conductor", "ec.conductor_label", "Conductor",
["field_label", "conductor_label"],
lambda field, conductor: '<a href="%s">%s</a>' %(
url_for('.show_ecnf_conductor', nf=field, conductor_label=conductor),
conductor),
default=True, align="center"),
MultiProcessedCol("iso_class", "ec.isogeny_class", "Isogeny class",
["field_label", "conductor_label", "iso_label", "short_class_label"],
lambda field, conductor, iso, short_class_label: '<a href="%s">%s</a>' % (
url_for('.show_ecnf_isoclass', nf=field, conductor_label=conductor, class_label=iso),
short_class_label),
default=True, align="center"),
MultiProcessedCol("ainvs", "ec.weierstrass_coeffs", "Weierstrass coefficients",
["field_label", "conductor_label", "iso_label", "number", "ainvs"],
lambda field, conductor, iso, number, ainvs: '<a href="%s">%s</a>' % (
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
if len(request.args) > 0:
return elliptic_curve_search(to_dict(request.args))
bread = get_bread()
# the dict data will hold additional information to be displayed on
# the main browse and search page
data = {}
# data['fields'] holds data for a sample of number fields of different
# signatures for a general browse:
ecnfstats = db_ecnfstats()
fields_by_deg = ecnfstats.find_one({'_id':'fields_by_degree'})
fields_by_sig = ecnfstats.find_one({'_id':'fields_by_signature'})
data['fields'] = []
# Rationals
data['fields'].append(['the rational field', (('1.1.1.1', [url_for('ec.rational_elliptic_curves'), '$\Q$']),)])
# Real quadratics (sample)
rqfs = ['2.2.{}.1'.format(d) for d in [5, 89, 229, 497]]
niqfs = len(fields_by_sig['0,1'])
nrqfs = len(fields_by_sig['2,0'])
data['fields'].append(['{} real quadratic fields, including'.format(nrqfs),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in rqfs)])
# Imaginary quadratics (sample)
iqfs = ['2.0.{}.1'.format(d) for d in [4, 8, 3, 7, 11]]
data['fields'].append(['{} imaginary quadratic fields, including'.format(niqfs),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in iqfs)])
# Cubics (sample)
cubics = ['3.1.23.1'] + ['3.3.{}.1'.format(d) for d in [49,148,1957]]
ncubics = len(fields_by_deg['3'])
data['fields'].append(['{} cubic fields, including'.format(ncubics),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in cubics)])
# Quartics (sample)
quartics = ['4.4.{}.1'.format(d) for d in [725,2777,9909,19821]]
nquartics = len(fields_by_deg['4'])
data['fields'].append(['{} totally real quartic fields, including'.format(nquartics),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in quartics)])
# Quintics (sample)
quintics = ['5.5.{}.1'.format(d) for d in [14641, 24217, 36497, 38569, 65657]]
nquintics = len(fields_by_deg['5'])
data['fields'].append(['{} totally real quintic fields, including'.format(nquintics),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in quintics)])
# Sextics (sample)
sextics = ['6.6.{}.1'.format(d) for d in [300125, 371293, 434581, 453789, 485125]]
nsextics = len(fields_by_deg['6'])
data['fields'].append(['{} totally real sextic fields, including'.format(nsextics),
((nf, [url_for('.show_ecnf1', nf=nf), field_pretty(nf)])
for nf in sextics)])
data['degrees'] = sorted([int(d) for d in fields_by_deg.keys() if d!='_id'])
# data['highlights'] holds data (URL and descriptive text) for a
# sample of elliptic curves with interesting features:
data['highlights'] = []
data['highlights'].append(
['A curve with $C_3\\times C_3$ torsion',
url_for('.show_ecnf', nf='2.0.3.1', class_label='a', conductor_label='2268.36.18', number=int(1))]
)
data['highlights'].append(
['A curve with $C_4\\times C_4$ torsion',
url_for('.show_ecnf', nf='2.0.4.1', class_label='b', conductor_label='5525.870.5', number=int(9))]
)
data['highlights'].append(
['A curve with CM by $\\sqrt{-267}$',
url_for('.show_ecnf', nf='2.2.89.1', class_label='a', conductor_label='81.1', number=int(1))]
)
data['highlights'].append(
['An isogeny class with isogenies of degree $3$ and $89$ (and $267$)',
url_for('.show_ecnf_isoclass', nf='2.2.89.1', class_label='a', conductor_label='81.1')]
)
data['highlights'].append(
['A curve with everywhere good reduction, but no global minimal model',
url_for('.show_ecnf', nf='2.2.229.1', class_label='a', conductor_label='1.1', number=int(1))]
)
return render_template("ecnf-index.html",
title="Elliptic Curves over Number Fields",
data=data,
bread=bread, learnmore=learnmore_list_remove('Completeness'))
def end_statement(factorsQQ, factorsRR, field='', ring=None):
# field is a latex string describing the basechange field (default is empty)
# ring is optional, if unspecified only endomorphism algebra is described
statement = '<table style="margin-left: 10px; margin-top: -12px">'
factorsQQ_number = len(factorsQQ)
factorsQQ_pretty = [field_pretty(fac[0]) for fac in factorsQQ if fac[0]]
# endomorphism ring is an invariant of the curve but not the isogeny class, so we make it optional
if ring:
# First row: description of the endomorphism ring as an order in the endomorphism algebra
statement += r"<tr><td>\(\End (J_{%s})\)</td><td>\(\simeq\)</td><td>" % field
# First the case of a maximal order:
if ring[0] == 1:
# Single factor:
if factorsQQ_number == 1:
# Number field or not:
if factorsQQ[0][2] == -1:
# Prettify in quadratic case:
if len(factorsQQ[0][1]) in [2, 3]:
statement += r"\(%s\)" % ring_pretty(
factorsQQ[0][1], 1)
else:
statement += r"the maximal order of \(\End (J_{%s}) \otimes \Q\)" % field
else:
# Use M_2 over integers if this applies:
if factorsQQ[0][2] == 1 and factorsQQ[0][0] == '1.1.1.1':
statement += r"\(\mathrm{M}_2 (\Z)\)"
# TODO: Add flag that indicates whether we are over a PID, in
# which case we can use the following lines:
#if factorsQQ[0][2] == 1:
# statement += (r"\(\mathrm{M}_2 (%s)\)"
# % ring_pretty(factorsQQ[0][1], 1))
else:
statement += r"a maximal order of \(\End (J_{%s}) \otimes \Q\)" % field
# If there are two factors, then they are both at most quadratic
# and we can prettify them
else:
statement += r'\(' + r' \times '.join(
[ring_pretty(factorQQ[1], 1)
for factorQQ in factorsQQ]) + r'\)'
# Then the case where there is still a single factor:
elif factorsQQ_number == 1:
# Number field case:
if factorsQQ[0][2] == -1:
# Prettify in quadratic case:
if len(factorsQQ[0][1]) in [2, 3]:
statement += r"\(%s\)" % ring_pretty(
factorsQQ[0][1], ring[0])
else:
statement += r"an order of conductor of norm \(%s\) in \(\End (J_{%s}) \otimes \Q\)" % (
ring[0], field)
# Otherwise mention whether the order is Eichler:
elif ring[1] == 1:
statement += r"an Eichler order of index \(%s\) in a maximal order of \(\End (J_{%s}) \otimes \Q\)" % (
ring[0], field)
else:
statement += r"a non-Eichler order of index \(%s\) in a maximal order of \(\End (J_{%s}) \otimes \Q\)" % (
ring[0], field)
# Finally the case of two factors. We can prettify to some extent, since we
# can describe the maximal order here
else:
statement += r"an order of index \(%s\) in \(%s\)" % (
ring[0], r' \times '.join(
[ring_pretty(factorQQ[1], 1) for factorQQ in factorsQQ]))
# End of first row:
statement += "</td></tr>"
# Second row: description of endomorphism algebra factors (this is the first row if ring=None)
statement += r"<tr><td>\(\End (J_{%s}) \otimes \Q \)</td><td>\(\simeq\)</td><td>" % field
# In the case of only one factor we either get a number field or a
# quaternion algebra:
if factorsQQ_number == 1:
# First we deal with the number field case,
# in which we have set the discriminant to be -1
if factorsQQ[0][2] == -1:
# Prettify if labels available, otherwise return defining polynomial:
if factorsQQ_pretty:
statement += "<a href=%s>%s</a>" % (url_for(
"number_fields.by_label",
label=factorsQQ[0][0]), factorsQQ_pretty[0])
else:
statement += r"the number field with defining polynomial \(%s\)" % intlist_to_poly(
factorsQQ[0][1])
# Detect CM by presence of a quartic polynomial:
if len(factorsQQ[0][1]) == 5:
statement += " (CM)"
# TODO: Get the following line to work
#statement += " ({{ KNOWL('ag.complex_multiplication', title='CM') }})"
# Up next is the case of a matrix ring (trivial disciminant), with
# labels and full prettification always available:
elif factorsQQ[0][2] == 1:
statement += r"\(\mathrm{M}_2(\)<a href=%s>%s</a>\()\)" % (url_for(
"number_fields.by_label",
label=factorsQQ[0][0]), factorsQQ_pretty[0])
# And finally we deal with quaternion algebras over the rationals:
else:
statement += (
"the quaternion algebra over <a href=%s>%s</a> of discriminant %s"
% (url_for("number_fields.by_label", label=factorsQQ[0][0]),
factorsQQ_pretty[0], factorsQQ[0][2]))
# If there are two factors, then we get two at most quadratic fields:
else:
statement += (r"<a href=%s>%s</a> \(\times\) <a href=%s>%s</a>" %
(url_for("number_fields.by_label",
label=factorsQQ[0][0]), factorsQQ_pretty[0],
url_for("number_fields.by_label",
label=factorsQQ[1][0]), factorsQQ_pretty[1]))
# End of second row:
statement += "</td></tr>"
# Third row: description of algebra tensored with RR (this is the second row if ring=None)
statement += r"<tr><td>\(\End (J_{%s}) \otimes \R\)</td><td>\(\simeq\)</td> <td>\(%s\)</td></tr>" % (
field, factorsRR_raw_to_pretty(factorsRR))
# End of statement:
statement += "</table>"
return statement
def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = curve['Lhash']
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(data['cond'])))
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1,4,data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page", cond=data['slabel'][0], x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['disc_key'][3:]) # use disc_key rather than abs_disc (will work when abs_disc > 2^63)
data['disc'] = curve['disc_sign'] * curve['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = list_to_min_eqn(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(data['disc']))
data['igusa_clebsch'] = [ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [web_latex(zfactor(i)) for i in data['igusa_clebsch']]
data['igusa_factor_latex'] = [ web_latex(zfactor(j)) for j in data['igusa'] ]
data['aut_grp_id'] = curve['aut_grp_id']
data['geom_aut_grp_id'] = curve['geom_aut_grp_id']
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['has_square_sha'] = "square" if curve['has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",",1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['torsion_order'] = curve['torsion_order']
data['torsion_factors'] = [ ZZ(a) for a in literal_eval(curve['torsion_subgroup']) ]
if len(data['torsion_factors']) == 0:
data['torsion_subgroup'] = '\mathrm{trivial}'
else:
data['torsion_subgroup'] = ' \\times '.join([ '\Z/{%s}\Z' % n for n in data['torsion_factors'] ])
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['tama'] = ''
for i in range(tama.count()):
item = tama.next()
if item['tamagawa_number'] > 0:
tamgwnr = str(item['tamagawa_number'])
else:
tamgwnr = 'N/A'
data['tama'] += tamgwnr + ' (p = ' + str(item['p']) + ')'
if (i+1 < tama.count()):
data['tama'] += ', '
if ratpts:
if len(ratpts['rat_pts']):
data['rat_pts'] = ', '.join(web_latex('(' +' : '.join(P) + ')') for P in ratpts['rat_pts'])
data['rat_pts_v'] = 2 if ratpts['rat_pts_v'] else 1
# data['mw_rank'] = ratpts['mw_rank']
# data['mw_rank_v'] = ratpts['mw_rank_v']
else:
data['rat_pts_v'] = 0
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl (curve['two_torsion_field'][0], getDBConnection(), field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data['two_torsion_field_knowl'] = """splitting field of \(%s\) with Galois group %s"""%(intlist_to_poly(t[1]),group_display_knowl(t[2][0],t[2][1],getDBConnection()))
else:
# invariants specific to isogeny class
curves_data = g2c_db_curves().find({"class" : curve['class']},{'_id':int(0),'label':int(1),'eqn':int(1),'disc_key':int(1)}).sort([("disc_key", ASCENDING), ("label", ASCENDING)])
if not curves_data:
raise KeyError("No curves found in database for isogeny class %s of genus 2 curve %s." %(curve['class'],curve['label']))
data['curves'] = [ {"label" : c['label'], "equation_formatted" : list_to_min_eqn(literal_eval(c['eqn'])), "url": url_for_curve_label(c['label'])} for c in curves_data ]
lfunc_data = g2c_db_lfunction_by_hash(curve['Lhash'])
if not lfunc_data:
raise KeyError("No Lfunction found in database for isogeny class of genus 2 curve %s." %curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [[nth_prime(n+1),lfunc_data['euler_factors'][n]] for n in range(len(lfunc_data['euler_factors'])) if nth_prime(n+1) < 30 and (data['cond'] % nth_prime(n+1))]
data['good_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['good_lfactors']]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = """Endomorphism %s over \(\Q\):<br>""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_base'], endo['factorsRR_base'], ring=data['end_ring_base'] if is_curve else None)
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = """Endomorphism %s over \(\overline{\Q}\):""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_geom'], data['factorsRR_geom'], field=r'\overline{\Q}', ring=data['end_ring_geom'] if is_curve else None)
data['real_geom_end_alg_name'] = end_alg_name(curve['real_geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(data['is_simple_geom'], data['split_field_label'], data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'], data.get('split_labels'), data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
self.plot = encode_plot(eqn_list_to_curve_plot(data['min_eqn'], data['rat_pts'].split(',') if 'rat_pts' in data else []))
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(self.plot)
properties += [
(None, plot_link),
('Conductor',str(data['cond'])),
('Discriminant', str(data['disc'])),
]
properties += [
('Sato-Tate group', data['st_group_link']),
('\(\\End(J_{\\overline{\\Q}}) \\otimes \\R\)', '\(%s\)' % data['real_geom_end_alg_name']),
('\(\\overline{\\Q}\)-simple', bool_pretty(data['is_simple_geom'])),
('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = [('L-function', data['lfunc_url'])]
if is_curve:
friends.append(('Isogeny class %s.%s' % (data['slabel'][0], data['slabel'][1]), url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1])))
for friend in g2c_db_lfunction_instances().find({'Lhash':data['Lhash']},{'_id':False,'url':True}):
if 'url' in friend:
add_friend (friends, lfunction_friend_from_url(friend['url']))
if 'urls' in friend:
for url in friends['urls']:
add_friend (friends, lfunction_friend_from_url(friend['url']))
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
add_friend (friends, ("Elliptic curve " + friend_label, url_for_ec(friend_label)))
else:
add_friend (friends, ("EC isogeny class " + ec_label_class(friend_label), url_for_ec_class(friend_label)))
if is_curve:
friends.append(('Twists', url_for(".index_Q", g20 = str(data['g2'][0]), g21 = str(data['g2'][1]), g22 = str(data['g2'][2]))))
# Breadcrumbs
self.bread = bread = [
('Genus 2 Curves', url_for(".index")),
('$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0], url_for(".by_conductor", cond=data['slabel'][0])),
('%s' % data['slabel'][1], url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1]))
]
if is_curve:
bread += [
('%s' % data['slabel'][2], url_for(".by_url_isogeny_class_discriminant", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2])),
('%s' % data['slabel'][3], url_for(".by_url_curve_label", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2], num=data['slabel'][3]))
]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else "Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage':'','magma':''} # use default show names
code['curve'] = {'sage':'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s))'%(data['min_eqn'][0],data['min_eqn'][1]),
'magma':'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'%(data['min_eqn'][0],data['min_eqn'][1])}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation('+str(data['cond'])+',2),R!'+str(bad2)+'>*]'
else:
magma_cond_option = ''
code['cond'] = {'magma': 'Conductor(LSeries(C%s)); Factorization($1);'% magma_cond_option}
code['disc'] = {'magma':'Discriminant(C); Factorization(Integers()!$1);'}
code['igusa_clebsch'] = {'sage':'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':'IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];'}
code['igusa'] = {'magma':'IgusaInvariants(C); [Factorization(Integers()!a): a in $1];'}
code['g2'] = {'magma':'G2Invariants(C);'}
code['aut'] = {'magma':'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {'magma':'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'}
code['num_rat_wpts'] = {'magma':'#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'}
if ratpts:
code['rat_pts'] = {'magma': '[' + ','.join(["C![%s,%s,%s]"%(p[0],p[1],p[2]) for p in ratpts['rat_pts']]) + '];' }
code['two_selmer'] = {'magma':'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'}
code['has_square_sha'] = {'magma':'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {'magma':'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'}
code['torsion_subgroup'] = {'magma':'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'}
def quad_field_knowl(disc):
r = 2 if disc > 0 else 0
field_label = "2.%d.%d.1" % (r, abs(disc))
field_name = field_pretty(field_label)
return nf_display_knowl(field_label, field_name)
return redirect(url_for("ecnf.index"))
def url_for_label(label):
if label == 'random':
return url_for(".random")
nf, cond_label, iso_label, number = split_full_label(label.strip())
return url_for(".show_ecnf", nf=nf, conductor_label=cond_label, class_label=iso_label, number=number)
@search_wrap(template="ecnf-search-results.html",
table=db.ec_nfcurves,
title='Elliptic curve search results',
err_title='Elliptic curve search input error',
shortcuts={'jump':elliptic_curve_jump,
'download':download_search},
cleaners={'numb':lambda e: str(e['number']),
'field_knowl':lambda e: nf_display_knowl(e['field_label'], field_pretty(e['field_label']))},
url_for_label=url_for_label,
learnmore=learnmore_list,
bread=lambda:[('Elliptic curves', url_for(".index")), ('Search results', '.')])
def elliptic_curve_search(info, query):
parse_nf_string(info,query,'field',name="base number field",qfield='field_label')
if query.get('field_label') == '1.1.1.1':
return redirect(url_for("ec.rational_elliptic_curves", **request.args), 301)
parse_ints(info,query,'conductor_norm')
parse_noop(info,query,'conductor_label')
parse_ints(info,query,'rank')
parse_ints(info,query,'torsion',name='Torsion order',qfield='torsion_order')
parse_bracketed_posints(info,query,'torsion_structure',maxlength=2)
if 'torsion_structure' in query and not 'torsion_order' in query:
t_o = 1
def set_info_for_web_newform(level=None,
weight=None,
character=None,
label=None,
**kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info['level'] = level
info['weight'] = weight
info['character'] = character
info['label'] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(
level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, 'prec', default_prec, int)
bprec = my_get(info, 'bprec', default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level,
weight=weight,
character=character,
label=label,
prec=prec)
emf_logger.debug("defined webnewform for rendering!")
except IndexError as e:
WNF = None
info['error'] = e.message
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms",
level=level,
weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms",
level=level,
weight=weight,
character=character)
bread = [(EMF_TOP, url1)]
bread.append(("Level %s" % level, url2))
bread.append(("Weight %s" % weight, url3))
if int(character) == 0:
bread.append(("Trivial Character", url4))
else:
bread.append(("Character \( %s \)" % (WNF.character.latex_name), url4))
bread.append(
("Newform %d.%d.%d.%s" % (level, weight, int(character), label), ''))
info['bread'] = bread
properties2 = list()
friends = list()
space_url = url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character)
friends.append(
('\( S_{%s}(%s, %s)\)' %
(WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url:
friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty,
WNF.base_ring.lmfdb_url))
if hasattr(WNF.coefficient_field,
"lmfdb_url") and WNF.coefficient_field.lmfdb_label:
friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty,
WNF.coefficient_field.lmfdb_url))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)",
WNF.character.url()))
if WNF.dimension == 0:
info['error'] = "This space is empty!"
info['title'] = 'Newform ' + WNF.hecke_orbit_label
info['learnmore'] = [('History of Modular forms',
url_for('holomorphic_mf_history'))]
if 'error' in info:
return info
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
## Example to illustrate the different cases
## base = CyclotomicField(n) -- of degree phi(n)
## coefficient_field = NumberField( p(x)) for some p in base['x'] of degree m
## we would then have cdeg = m*phi(n) and bdeg = phi(n)
## and rdeg = m
## Unfortunately, for e.g. base = coefficient_field = CyclotomicField(6)
## we get coefficient_field.relative_degree() == 2 although it should be 1
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if cdeg == 1:
rdeg = 1
else:
## just setting rdeg = WNF.coefficient_field.relative_degree() does not give correct result...
##
rdeg = QQ(cdeg) / QQ(bdeg)
cf_is_QQ = (cdeg == 1)
br_is_QQ = (bdeg == 1)
if cf_is_QQ:
info['satake'] = WNF.satake
if WNF.complexity_of_first_nonvanishing_coefficients(
) > default_max_height:
info['qexp'] = ""
info['qexp_display'] = ''
info['hide_qexp'] = True
n, c = WNF.first_nonvanishing_coefficient()
info['trace_nv'] = latex(c.trace())
info['norm_nv'] = '\\approx ' + latex(c.norm().n())
info['index_nv'] = n
else:
info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ')
info['qexp_display'] = url_for(".get_qexp_latex",
level=level,
weight=weight,
character=character,
label=label)
info["hide_qexp"] = False
info['max_cn_qexp'] = WNF.q_expansion.prec()
## All combinations should be tested...
## 13/4/4/a -> base ring = coefficient_field = QQ(zeta_6)
## 13/3/8/a -> base_ring = QQ(zeta_4), coefficient_field has poly x^2+(2\zeta_4+2x-3\zeta_$ over base_ring
## 13/4/3/a -> base_ring = coefficient_field = QQ(zeta_3)
## 13/4/1/a -> all rational
## 13/6/1/a/ -> base_ring = QQ, coefficient_field = Q(sqrt(17))
## These are variables which needs to be set properly below
info['polvars'] = {'base_ring': 'x', 'coefficient_field': '\\alpha'}
if not cf_is_QQ:
if rdeg > 1: # not WNF.coefficient_field == WNF.base_ring:
## Here WNF.base_ring should be some cyclotomic field and we have an extension over this.
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = web_latex_poly(p1,
'\\alpha') # make the variable \alpha
c_pol_ltx_x = web_latex_poly(p1, 'x')
zeta = p1.base_ring().gens()[0]
# p2 = zeta.minpoly() #this is not used anymore
# b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore
z1 = zeta.multiplicative_order()
info['coeff_field'] = [
WNF.coefficient_field.absolute_polynomial_latex('x'),
c_pol_ltx_x, z1
]
if hasattr(WNF.coefficient_field,
"lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [
WNF.coefficient_field.lmfdb_url,
WNF.coefficient_field.lmfdb_pretty,
WNF.coefficient_field.lmfdb_label
]
if z1 == 4:
info[
'polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(
c_pol_ltx)
info['polvars']['base_ring'] = 'i'
elif z1 <= 2:
info[
'polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(
c_pol_ltx)
else:
info[
'polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\) ' % (
c_pol_ltx, z1, z1)
info['polvars']['base_ring'] = '\zeta_{{ {0} }}'.format(z1)
if z1 == 3:
info[
'polynomial_st'] += 'is a primitive cube root of unity.'
else:
info[
'polynomial_st'] += 'is a primitive {0}-th root of unity.'.format(
z1)
elif not br_is_QQ:
## Now we have base and coefficient field being equal, meaning that since the coefficient field is not QQ it is some cyclotomic field
## generated by some \zeta_n
p1 = WNF.coefficient_field.absolute_polynomial()
z1 = WNF.coefficient_field.gens()[0].multiplicative_order()
c_pol_ltx = web_latex_poly(p1, '\\zeta_{{{0}}}'.format(z1))
c_pol_ltx_x = web_latex_poly(p1, 'x')
info['coeff_field'] = [
WNF.coefficient_field.absolute_polynomial_latex('x'),
c_pol_ltx_x
]
if hasattr(WNF.coefficient_field,
"lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [
WNF.coefficient_field.lmfdb_url,
WNF.coefficient_field.lmfdb_pretty,
WNF.coefficient_field.lmfdb_label
]
if z1 == 4:
info[
'polynomial_st'] = '<div class="where">where \(\zeta_4=e^{{\\frac{{\\pi i}}{{ 2 }} }}=i \).</div>'.format(
c_pol_ltx)
info['polvars']['coefficient_field'] = 'i'
elif z1 <= 2:
info['polynomial_st'] = ''
else:
info[
'polynomial_st'] = '<div class="where">where \(\zeta_{{{0}}}=e^{{\\frac{{2\\pi i}}{{ {0} }} }}\) '.format(
z1)
info['polvars']['coefficient_field'] = '\zeta_{{{0}}}'.format(
z1)
if z1 == 3:
info[
'polynomial_st'] += 'is a primitive cube root of unity.</div>'
else:
info[
'polynomial_st'] += 'is a primitive {0}-th root of unity.</div>'.format(
z1)
else:
info['polynomial_st'] = ''
if info["hide_qexp"]:
info['polynomial_st'] = ''
info['degree'] = int(cdeg)
if cdeg == 1:
info['is_rational'] = 1
info['coeff_field_pretty'] = [
WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty
]
else:
info['is_rational'] = 0
emf_logger.debug("PREC2: {0}".format(prec))
info['embeddings'] = WNF._embeddings[
'values'] #q_expansion_embeddings(prec, bprec,format='latex')
info['embeddings_len'] = len(info['embeddings'])
properties2 = [('Level', str(level)), ('Weight', str(weight)),
('Character', '$' + WNF.character.latex_name + '$'),
('Label', WNF.hecke_orbit_label),
('Dimension of Galois orbit', str(WNF.dimension))]
if (ZZ(level)).is_squarefree():
info['twist_info'] = WNF.twist_info
if isinstance(info['twist_info'],
list) and len(info['twist_info']) > 0:
info['is_minimal'] = info['twist_info'][0]
if (info['twist_info'][0]):
s = 'Is minimal<br>'
else:
s = 'Is a twist of lower level<br>'
properties2 += [('Twist info', s)]
else:
info['twist_info'] = 'Twist info currently not available.'
properties2 += [('Twist info', 'not available')]
args = list()
for x in range(5, 200, 10):
args.append({'digits': x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key('tau') and len(CM['tau']) != 0:
info['CM_values'] = CM
info['is_cm'] = WNF.is_cm
if WNF.is_cm == 1:
info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc)
info['cm_disc'] = WNF.cm_disc
info['cm_field_knowl'] = nf_display_knowl(
info['cm_field'], getDBConnection(),
field_pretty(info['cm_field']))
info['cm_field_url'] = url_for("number_fields.by_label",
label=info["cm_field"])
if WNF.is_cm is None or WNF.is_cm == -1:
s = '- Unknown (insufficient data)<br>'
elif WNF.is_cm == 1:
s = 'Yes<br>'
else:
s = 'No<br>'
properties2.append(('CM', s))
alev = WNF.atkin_lehner_eigenvalues()
info['atkinlehner'] = None
if isinstance(alev, dict) and len(alev.keys()) > 0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if (level == 1):
poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6', '')
if poly != '':
d, monom, coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info['explicit_formulas'] = '\('
for i in range(len(coeffs)):
c = QQ(coeffs[i])
s = ""
if d > 1 and i > 0 and c > 0:
s = "+"
if c < 0:
s = "-"
if c.denominator() > 1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(
abs(c.numerator()), c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]
b = monom[i][1]
if a == 0 and b != 0:
s += "E_6^{{ {0} }}".format(b)
elif b == 0 and a != 0:
s += "E_4^{{ {0} }}".format(a)
else:
s += "E_4^{{ {0} }}E_6^{{ {1} }}".format(a, b)
info['explicit_formulas'] += s
info['explicit_formulas'] += " \)"
cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \
'&label=' + str(label)
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if (label_other != label):
s = 'Modular form '
if character:
s += newform_label(level, weight, character, label_other)
else:
s += newform_label(level, weight, 1, label_other)
url = url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character,
label=label_other)
friends.append((s, url))
s = 'L-Function '
if character:
s += newform_label(level, weight, character, label)
else:
s += newform_label(level, weight, 1, label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = '/L' + url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character,
label=label)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + '.' + label
s = 'Elliptic curve isogeny class ' + llabel
url = '/EllipticCurve/Q/' + llabel
friends.append((s, url))
info['properties2'] = properties2
info['friends'] = friends
info['max_cn'] = WNF.max_available_prec()
return info
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
if len(request.args) > 0:
return elliptic_curve_search(to_dict(request.args))
bread = get_bread()
# the dict data will hold additional information to be displayed on
# the main browse and search page
data = {}
# data['fields'] holds data for a sample of number fields of different
# signatures for a general browse:
ecnfstats = db_ecnfstats()
fields_by_deg = ecnfstats.find_one({'_id': 'fields_by_degree'})
fields_by_sig = ecnfstats.find_one({'_id': 'fields_by_signature'})
data['fields'] = []
# Rationals
data['fields'].append([
'the rational field',
(('1.1.1.1', [url_for('ec.rational_elliptic_curves'), '$\Q$']), )
])
# Real quadratics (sample)
rqfs = ['2.2.{}.1'.format(d) for d in [5, 89, 229, 497]]
niqfs = len(fields_by_sig['0,1'])
nrqfs = len(fields_by_sig['2,0'])
data['fields'].append([
'{} real quadratic fields, including'.format(nrqfs),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in rqfs)
])
# Imaginary quadratics (sample)
iqfs = ['2.0.{}.1'.format(d) for d in [4, 8, 3, 7, 11]]
data['fields'].append([
'{} imaginary quadratic fields, including'.format(niqfs),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in iqfs)
])
# Cubics (sample)
cubics = ['3.1.23.1'] + ['3.3.{}.1'.format(d) for d in [49, 148, 1957]]
ncubics = len(fields_by_deg['3'])
data['fields'].append([
'{} cubic fields, including'.format(ncubics),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in cubics)
])
# Quartics (sample)
quartics = ['4.4.{}.1'.format(d) for d in [725, 2777, 9909, 19821]]
nquartics = len(fields_by_deg['4'])
data['fields'].append([
'{} totally real quartic fields, including'.format(nquartics),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in quartics)
])
# Quintics (sample)
quintics = [
'5.5.{}.1'.format(d) for d in [14641, 24217, 36497, 38569, 65657]
]
nquintics = len(fields_by_deg['5'])
data['fields'].append([
'{} totally real quintic fields, including'.format(nquintics),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in quintics)
])
# Sextics (sample)
sextics = [
'6.6.{}.1'.format(d) for d in [300125, 371293, 434581, 453789, 485125]
]
nsextics = len(fields_by_deg['6'])
data['fields'].append([
'{} totally real sextic fields, including'.format(nsextics),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in sextics)
])
data['degrees'] = sorted(
[int(d) for d in fields_by_deg.keys() if d != '_id'])
# data['highlights'] holds data (URL and descriptive text) for a
# sample of elliptic curves with interesting features:
data['highlights'] = []
data['highlights'].append([
'A curve with $C_3\\times C_3$ torsion',
url_for('.show_ecnf',
nf='2.0.3.1',
class_label='a',
conductor_label='2268.36.18',
number=int(1))
])
data['highlights'].append([
'A curve with $C_4\\times C_4$ torsion',
url_for('.show_ecnf',
nf='2.0.4.1',
class_label='b',
conductor_label='5525.870.5',
number=int(9))
])
data['highlights'].append([
'A curve with CM by $\\sqrt{-267}$',
url_for('.show_ecnf',
nf='2.2.89.1',
class_label='a',
conductor_label='81.1',
number=int(1))
])
data['highlights'].append([
'An isogeny class with isogenies of degree $3$ and $89$ (and $267$)',
url_for('.show_ecnf_isoclass',
nf='2.2.89.1',
class_label='a',
conductor_label='81.1')
])
data['highlights'].append([
'A curve with everywhere good reduction, but no global minimal model',
url_for('.show_ecnf',
nf='2.2.229.1',
class_label='a',
conductor_label='1.1',
number=int(1))
])
return render_template("ecnf-index.html",
title="Elliptic Curves over Number Fields",
data=data,
bread=bread,
learnmore=learnmore_list_remove('Completeness'))
def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info['level'] = level
info['weight'] = weight
info['character'] = character
info['label'] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, 'prec', default_prec, int)
bprec = my_get(info, 'bprec', default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label)
emf_logger.critical("defined webnewform for rendering!")
# if info.has_key('download') and info.has_key('tempfile'):
# WNF._save_to_file(info['tempfile'])
# info['filename']=str(weight)+'-'+str(level)+'-'+str(character)+'-'+label+'.sobj'
# return info
except IndexError as e:
WNF = None
info['error'] = e.message
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character)
bread = [(EMF_TOP, url1)]
bread.append(("of level %s" % level, url2))
bread.append(("weight %s" % weight, url3))
if int(character) == 0:
bread.append(("trivial character", url4))
else:
bread.append(("\( %s \)" % (WNF.character.latex_name), url4))
info['bread'] = bread
properties2 = list()
friends = list()
space_url = url_for('emf.render_elliptic_modular_forms',level=level, weight=weight, character=character)
friends.append(('\( S_{%s}(%s, %s)\)'%(WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url:
friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty, WNF.base_ring.lmfdb_url))
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_label:
friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_url))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url()))
if WNF.dimension==0:
info['error'] = "This space is empty!"
# emf_logger.debug("WNF={0}".format(WNF))
#info['name'] = name
info['title'] = 'Modular Form ' + WNF.hecke_orbit_label
if 'error' in info:
return info
# info['name']=WNF._name
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
## Example to illustrate the different cases
## base = CyclotomicField(n) -- of degree phi(n)
## coefficient_field = NumberField( p(x)) for some p in base['x'] of degree m
## we would then have cdeg = m*phi(n) and bdeg = phi(n)
## and rdeg = m
## Unfortunately, for e.g. base = coefficient_field = CyclotomicField(6)
## we get coefficient_field.relative_degree() == 2 although it should be 1
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if cdeg == 1:
rdeg = 1
else:
## just setting rdeg = WNF.coefficient_field.relative_degree() does not give correct result...
##
rdeg = QQ(cdeg)/QQ(bdeg)
cf_is_QQ = (cdeg == 1)
br_is_QQ = (bdeg == 1)
if cf_is_QQ:
info['satake'] = WNF.satake
info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ')
info['qexp_display'] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label)
info['max_cn_qexp'] = WNF.q_expansion.prec()
## All combinations should be tested...
## 13/4/4/a -> base ring = coefficient_field = QQ(zeta_6)
## 13/3/8/a -> base_ring = QQ(zeta_4), coefficient_field has poly x^2+(2\zeta_4+2x-3\zeta_$ over base_ring
## 13/4/3/a -> base_ring = coefficient_field = QQ(zeta_3)
## 13/4/1/a -> all rational
## 13/6/1/a/ -> base_ring = QQ, coefficient_field = Q(sqrt(17))
## These are variables which needs to be set properly below
info['polvars'] = {'base_ring':'x','coefficient_field':'\\alpha'}
if not cf_is_QQ:
if rdeg>1: # not WNF.coefficient_field == WNF.base_ring:
## Here WNF.base_ring should be some cyclotomic field and we have an extension over this.
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = web_latex_poly(p1, '\\alpha') # make the variable \alpha
c_pol_ltx_x = web_latex_poly(p1, 'x')
zeta = p1.base_ring().gens()[0]
# p2 = zeta.minpoly() #this is not used anymore
# b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore
z1 = zeta.multiplicative_order()
info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'),c_pol_ltx_x, z1]
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label]
if z1==4:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(c_pol_ltx)
info['polvars']['base_ring']='i'
elif z1<=2:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx)
else:
info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\) '%(c_pol_ltx, z1,z1)
info['polvars']['base_ring']='\zeta_{{ {0} }}'.format(z1)
if z1==3:
info['polynomial_st'] += 'is a primitive cube root of unity.'
else:
info['polynomial_st'] += 'is a primitive {0}-th root of unity.'.format(z1)
elif not br_is_QQ:
## Now we have base and coefficient field being equal, meaning that since the coefficient field is not QQ it is some cyclotomic field
## generated by some \zeta_n
p1 = WNF.coefficient_field.absolute_polynomial()
z1 = WNF.coefficient_field.gens()[0].multiplicative_order()
c_pol_ltx = web_latex_poly(p1, '\\zeta_{{{0}}}'.format(z1))
c_pol_ltx_x = web_latex_poly(p1, 'x')
info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x]
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label]
if z1==4:
info['polynomial_st'] = '<div class="where">where \(\zeta_4=e^{{\\frac{{\\pi i}}{{ 2 }} }}=i \).</div>'.format(c_pol_ltx)
info['polvars']['coefficient_field']='i'
elif z1<=2:
info['polynomial_st'] = ''
else:
info['polynomial_st'] = '<div class="where">where \(\zeta_{{{0}}}=e^{{\\frac{{2\\pi i}}{{ {0} }} }}\) '.format(z1)
info['polvars']['coefficient_field']='\zeta_{{{0}}}'.format(z1)
if z1==3:
info['polynomial_st'] += 'is a primitive cube root of unity.</div>'
else:
info['polynomial_st'] += 'is a primitive {0}-th root of unity.</div>'.format(z1)
else:
info['polynomial_st'] = ''
info['degree'] = int(cdeg)
if cdeg==1:
info['is_rational'] = 1
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty ]
else:
info['is_rational'] = 0
# info['q_exp_embeddings'] = WNF.print_q_expansion_embeddings()
# if(int(info['degree'])>1 and WNF.dimension()>1):
# s = 'One can embed it into \( \mathbb{C} \) as:'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# elif(int(info['degree'])>1):
# s = 'There are '+str(info['degree'])+' embeddings into \( \mathbb{C} \):'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# else:
# info['embeddings'] = ''
emf_logger.debug("PREC2: {0}".format(prec))
info['embeddings'] = WNF._embeddings['values'] #q_expansion_embeddings(prec, bprec,format='latex')
info['embeddings_len'] = len(info['embeddings'])
properties2 = []
if (ZZ(level)).is_squarefree():
info['twist_info'] = WNF.twist_info
if isinstance(info['twist_info'], list) and len(info['twist_info'])>0:
info['is_minimal'] = info['twist_info'][0]
if(info['twist_info'][0]):
s = 'Is minimal<br>'
else:
s = 'Is a twist of lower level<br>'
properties2 = [('Twist info', s)]
else:
info['twist_info'] = 'Twist info currently not available.'
properties2 = [('Twist info', 'not available')]
args = list()
for x in range(5, 200, 10):
args.append({'digits': x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key('tau') and len(CM['tau']) != 0:
info['CM_values'] = CM
info['is_cm'] = WNF.is_cm
if WNF.is_cm == 1:
info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc)
info['cm_disc'] = WNF.cm_disc
info['cm_field_knowl'] = nf_display_knowl(info['cm_field'], getDBConnection(), field_pretty(info['cm_field']))
info['cm_field_url'] = url_for("number_fields.by_label", label=info["cm_field"])
if WNF.is_cm is None or WNF.is_cm==-1:
s = '- Unknown (insufficient data)<br>'
elif WNF.is_cm == 1:
s = 'Is a CM-form<br>'
else:
s = 'Is not a CM-form<br>'
properties2.append(('CM info', s))
alev = WNF.atkin_lehner_eigenvalues()
info['atkinlehner'] = None
if isinstance(alev,dict) and len(alev.keys())>0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if(level == 1):
poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6','')
if poly != '':
d,monom,coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info['explicit_formulas'] = '\('
for i in range(len(coeffs)):
c = QQ(coeffs[i])
s = ""
if d>1 and i >0 and c>0:
s="+"
if c<0:
s="-"
if c.denominator()>1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(abs(c.numerator()),c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]; b = monom[i][1]
if a == 0 and b != 0:
s+="E_6^{{ {0} }}".format(b)
elif b ==0 and a != 0:
s+="E_4^{{ {0} }}".format(a)
else:
s+="E_4^{{ {0} }}E_6^{{ {1} }}".format(a,b)
info['explicit_formulas'] += s
info['explicit_formulas'] += " \)"
cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \
'&label=' + str(label)
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if(label_other != label):
s = 'Modular form '
if character:
s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label_other)
else:
s = s + str(level) + '.' + str(weight) + str(label_other)
url = url_for('emf.render_elliptic_modular_forms', level=level,
weight=weight, character=character, label=label_other)
friends.append((s, url))
s = 'L-Function '
if character:
s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label)
else:
s = s + str(level) + '.' + str(weight) + str(label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = '/L' + url_for(
'emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + '.' + label
s = 'Elliptic curve isogeny class ' + llabel
url = '/EllipticCurve/Q/' + llabel
friends.append((s, url))
info['properties2'] = properties2
info['friends'] = friends
info['max_cn'] = WNF.max_cn()
return info
def render_sample_page(family, sam, args, bread):
info = { 'args': to_dict(args), 'sam': sam, 'latex': latex, 'type':sam.type(), 'name':sam.name(), 'full_name': sam.full_name(), 'weight':sam.weight(), 'fdeg':sam.degree_of_field(), 'is_eigenform':sam.is_eigenform(), 'field_poly': sam.field_poly()}
if sam.is_integral() != None:
info['is_integral'] = sam.is_integral()
if 'Sp4Z' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=0)
if 'Sp4Z_2' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=2)
info['space'] = '$'+family.latex_name.replace('k', '{' + str(sam.weight()) + '}')+'$'
if 'space_url' in info:
bread.append((info['space'], info['space_url']))
info['space_href'] = '<a href="%s">%s</d>'%(info['space_url'],info['space']) if 'space_url' in info else info['space']
if info['field_poly'].disc() < 10**10:
label = poly_to_field_label(info['field_poly'])
if label:
info['field_label'] = label
info['field_url'] = url_for('number_fields.by_label', label=label)
info['field_href'] = '<a href="%s">%s</a>'%(info['field_url'], field_pretty(label))
bread.append((info['name'], ''))
title='Siegel modular forms sample ' + info['full_name']
properties = [('Space', info['space_href']),
('Name', info['name']),
('Type', '<br>'.join(info['type'].split(','))),
('Weight', str(info['weight'])),
('Hecke eigenform', str(info['is_eigenform'])),
('Field degree', str(info['fdeg']))]
try:
evs_to_show = parse_ints_to_list_flash(args.get('ev_index'), 'list of $l$')
fcs_to_show = parse_ints_to_list_flash(args.get('fc_det'), 'list of $\\det(F)$')
except ValueError:
evs_to_show = []
fcs_to_show = []
info['evs_to_show'] = sorted([n for n in (evs_to_show if len(evs_to_show) else sam.available_eigenvalues()[:10])])
info['fcs_to_show'] = sorted([n for n in (fcs_to_show if len(fcs_to_show) else sam.available_Fourier_coefficients()[1:6])])
info['evs_avail'] = [n for n in sam.available_eigenvalues()]
info['fcs_avail'] = [n for n in sam.available_Fourier_coefficients()]
# Do not attempt to constuct a modulus ideal unless the field has a reasonably small discriminant
# otherwise sage may not even be able to factor the discriminant
info['field'] = sam.field()
if info['field_poly'].disc() < 10**80:
null_ideal = sam.field().ring_of_integers().ideal(0)
info['modulus'] = null_ideal
modulus = args.get('modulus','').strip()
m = 0
if modulus:
try:
O = sam.field().ring_of_integers()
m = O.ideal([O(str(b)) for b in modulus.split(',')])
except Exception:
info['error'] = True
flash_error("Unable to construct modulus ideal from specified generators %s.", modulus)
if m == 1:
info['error'] = True
flash_error("The ideal %s is the unit ideal, please specify a different modulus.", '('+modulus+')')
m = 0
info['modulus'] = m
# Hack to reduce polynomials and to handle non integral stuff
def redc(c):
return m.reduce(c*c.denominator())/m.reduce(c.denominator())
def redp(f):
c = f.dict()
return f.parent()(dict((e,redc(c[e])) for e in c))
def safe_reduce(f):
if not m:
return latex(f)
try:
if f in sam.field():
return latex(redc(f))
else:
return latex(redp(f))
except ZeroDivisionError:
return '\\textrm{Unable to reduce} \\bmod\\mathfrak{m}'
info['reduce'] = safe_reduce
else:
info['reduce'] = latex
# check that explicit formula is not ridiculously big
if sam.explicit_formula():
info['explicit_formula_bytes'] = len(sam.explicit_formula())
if len(sam.explicit_formula()) < 100000:
info['explicit_formula'] = sam.explicit_formula()
return render_template("ModularForm_GSp4_Q_sample.html", title=title, bread=bread, properties2=properties, info=info)
def render_sample_page(family, sam, args, bread):
info = {
'args': to_dict(args),
'sam': sam,
'latex': latex,
'type': sam.type(),
'name': sam.name(),
'full_name': sam.full_name(),
'weight': sam.weight(),
'fdeg': sam.degree_of_field(),
'is_eigenform': sam.is_eigenform(),
'field_poly': sam.field_poly()
}
if sam.is_integral() != None:
info['is_integral'] = sam.is_integral()
if 'Sp4Z' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=0)
if 'Sp4Z_2' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=2)
info['space'] = '$' + family.latex_name.replace(
'k', '{' + str(sam.weight()) + '}') + '$'
if 'space_url' in info:
bread.append((info['space'], info['space_url']))
info['space_href'] = '<a href="%s">%s</d>' % (
info['space_url'],
info['space']) if 'space_url' in info else info['space']
if info['field_poly'].disc() < 10**10:
label = poly_to_field_label(info['field_poly'])
if label:
info['field_label'] = label
info['field_url'] = url_for('number_fields.by_label', label=label)
info['field_href'] = '<a href="%s">%s</a>' % (info['field_url'],
field_pretty(label))
bread.append((info['name'], ''))
title = 'Siegel modular forms sample ' + info['full_name']
properties = [('Space', info['space_href']), ('Name', info['name']),
('Type', '<br>'.join(info['type'].split(','))),
('Weight', str(info['weight'])),
('Hecke eigenform', str(info['is_eigenform'])),
('Field degree', str(info['fdeg']))]
try:
evs_to_show = parse_ints_to_list_flash(args.get('ev_index'),
'list of $l$')
fcs_to_show = parse_ints_to_list_flash(args.get('fc_det'),
'list of $\\det(F)$')
except ValueError:
evs_to_show = []
fcs_to_show = []
info['evs_to_show'] = sorted([
n for n in
(evs_to_show if len(evs_to_show) else sam.available_eigenvalues()[:10])
])
info['fcs_to_show'] = sorted([
n for n in (fcs_to_show if len(fcs_to_show) else sam.
available_Fourier_coefficients()[1:6])
])
info['evs_avail'] = [n for n in sam.available_eigenvalues()]
info['fcs_avail'] = [n for n in sam.available_Fourier_coefficients()]
# Do not attempt to constuct a modulus ideal unless the field has a reasonably small discriminant
# otherwise sage may not even be able to factor the discriminant
info['field'] = sam.field()
if info['field_poly'].disc() < 10**80:
null_ideal = sam.field().ring_of_integers().ideal(0)
info['modulus'] = null_ideal
modulus = args.get('modulus', '').strip()
m = 0
if modulus:
try:
O = sam.field().ring_of_integers()
m = O.ideal([O(str(b)) for b in modulus.split(',')])
except Exception:
info['error'] = True
flash_error(
"Unable to construct modulus ideal from specified generators %s.",
modulus)
if m == 1:
info['error'] = True
flash_error(
"The ideal %s is the unit ideal, please specify a different modulus.",
'(' + modulus + ')')
m = 0
info['modulus'] = m
# Hack to reduce polynomials and to handle non integral stuff
def redc(c):
return m.reduce(c * c.denominator()) / m.reduce(c.denominator())
def redp(f):
c = f.dict()
return f.parent()(dict((e, redc(c[e])) for e in c))
def safe_reduce(f):
if not m:
return latex(f)
try:
if f in sam.field():
return latex(redc(f))
else:
return latex(redp(f))
except ZeroDivisionError:
return '\\textrm{Unable to reduce} \\bmod\\mathfrak{m}'
info['reduce'] = safe_reduce
else:
info['reduce'] = latex
# check that explicit formula is not ridiculously big
if sam.explicit_formula():
info['explicit_formula_bytes'] = len(sam.explicit_formula())
if len(sam.explicit_formula()) < 100000:
info['explicit_formula'] = sam.explicit_formula()
return render_template("ModularForm_GSp4_Q_sample.html",
title=title,
bread=bread,
properties2=properties,
info=info)
try:
nf, cond_label, iso_label, number = split_full_label(label.strip())
except ValueError:
info['err'] = ''
return redirect(url_for("ecnf.index"))
return redirect(url_for(".show_ecnf", nf=nf, conductor_label=cond_label, class_label=iso_label, number=number), 301)
@search_wrap(template="ecnf-search-results.html",
table=db.ec_nfcurves,
title='Elliptic Curve Search Results',
err_title='Elliptic Curve Search Input Error',
shortcuts={'jump':elliptic_curve_jump,
'download':download_search},
cleaners={'numb':lambda e: str(e['number']),
'field_knowl':lambda e: nf_display_knowl(e['field_label'], field_pretty(e['field_label']))},
bread=lambda:[('Elliptic Curves', url_for(".index")), ('Search Results', '.')],
credit=lambda:ecnf_credit)
def elliptic_curve_search(info, query):
parse_nf_string(info,query,'field',name="base number field",qfield='field_label')
if query.get('field_label') == '1.1.1.1':
return redirect(url_for("ec.rational_elliptic_curves", **request.args), 301)
parse_ints(info,query,'conductor_norm')
parse_noop(info,query,'conductor_label')
parse_ints(info,query,'torsion',name='Torsion order',qfield='torsion_order')
parse_bracketed_posints(info,query,'torsion_structure',maxlength=2)
if 'torsion_structure' in query and not 'torsion_order' in query:
query['torsion_order'] = reduce(mul,[int(n) for n in query['torsion_structure']],1)
parse_ints(info,query,field='isodeg',qfield='isogeny_degrees')
def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = str(curve['Lhash'])
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(
data['cond']))).replace(r"-1 \cdot", "-")
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['mw_rank'] = ZZ(0) if curve.get('mw_rank') is None else ZZ(
curve['mw_rank']) # 0 will be marked as a lower bound
data['mw_rank_proved'] = curve['mw_rank_proved']
data['analytic_rank_proved'] = curve['analytic_rank_proved']
data['hasse_weil_proved'] = curve['hasse_weil_proved']
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1, 4, data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page",
cond=data['slabel'][0],
x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [(c[0],
list_to_factored_poly_otherorder(c[1]))
for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['abs_disc'])
data['disc'] = curve['disc_sign'] * data['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = min_eqns_pretty(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(
data['disc'])).replace(r"-1 \cdot", "-")
data['igusa_clebsch'] = [
ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])
]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [
web_latex(zfactor(i)).replace(r"-1 \cdot", "-")
for i in data['igusa_clebsch']
]
data['igusa_factor_latex'] = [
web_latex(zfactor(j)).replace(r"-1 \cdot", "-")
for j in data['igusa']
]
data['aut_grp'] = small_group_label_display_knowl(
'%d.%d' % tuple(literal_eval(curve['aut_grp_id'])))
data['geom_aut_grp'] = small_group_label_display_knowl(
'%d.%d' % tuple(literal_eval(curve['geom_aut_grp_id'])))
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['has_square_sha'] = "square" if curve[
'has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",", 1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['torsion_order'] = curve['torsion_order']
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['real_period'] = decimal_pretty(str(curve['real_period']))
data['regulator'] = decimal_pretty(
str(curve['regulator']
)) if curve['regulator'] > -0.5 else 'unknown'
if data['mw_rank'] == 0 and data['mw_rank_proved']:
data['regulator'] = '1' # display an exact 1 when we know this
data['tamagawa_product'] = ZZ(
curve['tamagawa_product']) if curve.get(
'tamagawa_product') else 0
data['analytic_sha'] = ZZ(
curve['analytic_sha']) if curve.get('analytic_sha') else 0
data['leading_coeff'] = decimal_pretty(
str(curve['leading_coeff']
)) if curve['leading_coeff'] else 'unknown'
data['rat_pts'] = ratpts['rat_pts']
data['rat_pts_v'] = ratpts['rat_pts_v']
data['rat_pts_table'] = ratpts_table(ratpts['rat_pts'],
ratpts['rat_pts_v'])
data['mw_gens_v'] = ratpts['mw_gens_v']
lower = len([n for n in ratpts['mw_invs'] if n == 0])
upper = data['analytic_rank']
invs = ratpts[
'mw_invs'] if data['mw_gens_v'] or lower >= upper else [
0 for n in range(upper - lower)
] + ratpts['mw_invs']
if len(invs) == 0:
data['mw_group'] = 'trivial'
else:
data['mw_group'] = r'\(' + r' \times '.join([
(r'\Z' if n == 0 else r'\Z/{%s}\Z' % n) for n in invs
]) + r'\)'
if lower >= upper:
data['mw_gens_table'] = mw_gens_table(ratpts['mw_invs'],
ratpts['mw_gens'],
ratpts['mw_heights'],
ratpts['rat_pts'])
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl(
curve['two_torsion_field'][0],
field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data[
'two_torsion_field_knowl'] = r"splitting field of \(%s\) with Galois group %s" % (
intlist_to_poly(
t[1]), group_display_knowl(t[2][0], t[2][1]))
tamalist = [[item['p'], item['tamagawa_number']] for item in tama]
data['local_table'] = local_table(data['abs_disc'], data['cond'],
tamalist,
data['bad_lfactors_pretty'])
else:
# invariants specific to isogeny class
curves_data = list(
db.g2c_curves.search({"class": curve['class']},
['label', 'eqn']))
if not curves_data:
raise KeyError(
"No curves found in database for isogeny class %s of genus 2 curve %s."
% (curve['class'], curve['label']))
data['curves'] = [{
"label":
c['label'],
"equation_formatted":
min_eqn_pretty(literal_eval(c['eqn'])),
"url":
url_for_curve_label(c['label'])
} for c in curves_data]
lfunc_data = db.lfunc_lfunctions.lucky(
{'Lhash': str(curve['Lhash'])})
if not lfunc_data:
raise KeyError(
"No Lfunction found in database for isogeny class of genus 2 curve %s."
% curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [
[nth_prime(n + 1), lfunc_data['euler_factors'][n]]
for n in range(len(lfunc_data['euler_factors']))
if nth_prime(n + 1) < 30 and (data['cond'] %
nth_prime(n + 1))
]
data['good_lfactors_pretty'] = [
(c[0], list_to_factored_poly_otherorder(c[1]))
for c in data['good_lfactors']
]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(
endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = (
r"Endomorphism %s over \(\Q\):<br>" %
("ring" if is_curve else "algebra") +
end_statement(data['factorsQQ_base'],
endo['factorsRR_base'],
ring=data['end_ring_base'] if is_curve else None))
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(
data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(
data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = (
r"Endomorphism %s over \(\overline{\Q}\):" %
("ring" if is_curve else "algebra") + end_statement(
data['factorsQQ_geom'],
data['factorsRR_geom'],
field=r'\overline{\Q}',
ring=data['end_ring_geom'] if is_curve else None))
data['real_geom_end_alg_name'] = real_geom_end_alg_name(
curve['real_geom_end_alg'])
data['geom_end_alg_name'] = geom_end_alg_name(curve['geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(
data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(
data['is_simple_geom'], data['split_field_label'],
data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(
endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'],
data.get('split_labels'),
data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
plot_from_db = db.g2c_plots.lucky({"label": curve['label']})
if (plot_from_db is None):
self.plot = encode_plot(
eqn_list_to_curve_plot(
data['min_eqn'], ratpts['rat_pts'] if ratpts else []))
else:
self.plot = plot_from_db['plot']
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(
self.plot)
properties += [
(None, plot_link),
('Conductor', str(data['cond'])),
('Discriminant', str(data['disc'])),
]
if data['mw_rank_proved']:
properties += [('Mordell-Weil group', data['mw_group'])]
properties += [
('Sato-Tate group', data['st_group_link']),
(r'\(\End(J_{\overline{\Q}}) \otimes \R\)',
r'\(%s\)' % data['real_geom_end_alg_name']),
(r'\(\End(J_{\overline{\Q}}) \otimes \Q\)',
r'\(%s\)' % data['geom_end_alg_name']),
(r'\(\overline{\Q}\)-simple', bool_pretty(data['is_simple_geom'])),
(r'\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = []
if is_curve:
friends.append(('Isogeny class %s.%s' %
(data['slabel'][0], data['slabel'][1]),
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1])))
# first deal with EC
ecs = []
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
ecs.append(("Elliptic curve " + friend_label,
url_for_ec(friend_label)))
else:
ecs.append(
("Isogeny class " + ec_label_class(friend_label),
url_for_ec_class(friend_label)))
ecs.sort(key=lambda x: key_for_numerically_sort(x[0]))
# then again EC from lfun
instances = []
for elt in db.lfunc_instances.search(
{
'Lhash': data['Lhash'],
'type': 'ECQP'
}, 'url'):
instances.extend(elt.split('|'))
# and then the other isogeny friends
instances.extend([
elt['url'] for elt in get_instances_by_Lhash_and_trace_hash(
data["Lhash"], 4, int(data["Lhash"]))
])
exclude = {
elt[1].rstrip('/').lstrip('/')
for elt in self.friends if elt[1]
}
exclude.add(data['lfunc_url'].lstrip('/L/').rstrip('/'))
for elt in ecs + names_and_urls(instances, exclude=exclude):
# because of the splitting we must use G2C specific code
add_friend(friends, elt)
if is_curve:
friends.append(('Twists',
url_for(".index_Q",
g20=str(data['g2'][0]),
g21=str(data['g2'][1]),
g22=str(data['g2'][2]))))
friends.append(('L-function', data['lfunc_url']))
# Breadcrumbs
self.bread = bread = [('Genus 2 Curves', url_for(".index")),
(r'$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0],
url_for(".by_conductor",
cond=data['slabel'][0])),
('%s' % data['slabel'][1],
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1]))]
if is_curve:
bread += [('%s' % data['slabel'][2],
url_for(".by_url_isogeny_class_discriminant",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2])),
('%s' % data['slabel'][3],
url_for(".by_url_curve_label",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2],
num=data['slabel'][3]))]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else
"Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage': '', 'magma': ''} # use default show names
f, h = fh = data['min_eqn']
g = simplify_hyperelliptic(fh)
code['curve'] = {
'sage':
'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s));'
% (f, h),
'magma':
'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'
% (f, h)
}
code['simple_curve'] = {
'sage': 'X = HyperellipticCurve(R(%s))' % (g),
'magma': 'X,pi:= SimplifiedModel(C);'
}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation(' + str(
data['cond']) + ',2),R!' + str(bad2) + '>*]'
else:
magma_cond_option = ''
code['cond'] = {
'magma':
'Conductor(LSeries(C%s)); Factorization($1);' % magma_cond_option
}
code['disc'] = {
'magma': 'Discriminant(C); Factorization(Integers()!$1);'
}
code['geom_inv'] = {
'sage':
'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':
'IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);'
}
code['aut'] = {'magma': 'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {
'magma':
'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'
}
code['num_rat_wpts'] = {
'magma': '#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'
}
if ratpts:
code['rat_pts'] = {
'magma':
'[' + ','.join([
"C![%s,%s,%s]" % (p[0], p[1], p[2])
for p in ratpts['rat_pts']
]) + '];'
}
code['mw_group'] = {'magma': 'MordellWeilGroupGenus2(Jacobian(C));'}
code['two_selmer'] = {
'magma': 'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'
}
code['has_square_sha'] = {'magma': 'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {
'magma':
'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'
}
code['torsion_subgroup'] = {
'magma':
'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'
}
def make_torsion_growth(self):
try:
tor_gro = self.tor_gro
except AttributeError: # for curves with norsion growth data
tor_gro = None
if tor_gro is None:
self.torsion_growth_data_exists = False
return
self.torsion_growth_data_exists = True
self.tg = tg = {}
tg['data'] = tgextra = []
# find all base-changes of this curve in the database, if any
bcs = list(
db.ec_nfcurves.search(
{'base_change': {
'$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'] = '\(' + ' \\times '.join(
['\Z/{}\Z'.format(n) for n in T.split(",")]) + '\)'
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 end_statement(factorsQQ, factorsRR, field='', ring=None):
# field is a latex string describing the basechange field (default is empty)
# ring is optional, if unspecified only endomorphism algebra is described
statement = """<table class="g2">"""
factorsQQ_number = len(factorsQQ)
factorsQQ_pretty = [ field_pretty(fac[0]) for fac in factorsQQ if fac[0] ]
# endomorphism ring is an invariant of the curve but not the isogeny class, so we make it optional
if ring:
# First row: description of the endomorphism ring as an order in the endomorphism algebra
statement += """<tr><td>\(\End (J_{%s})\)</td><td>\(\simeq\)</td><td>""" % field
# First the case of a maximal order:
if ring[0] == 1:
# Single factor:
if factorsQQ_number == 1:
# Number field or not:
if factorsQQ[0][2] == -1:
# Prettify in quadratic case:
if len(factorsQQ[0][1]) in [2, 3]:
statement += """\(%s\)""" % ring_pretty(factorsQQ[0][1], 1)
else:
statement += """the maximal order of \(\End (J_{%s}) \otimes \Q\)""" % field
else:
# Use M_2 over integers if this applies:
if factorsQQ[0][2] == 1 and factorsQQ[0][0] == '1.1.1.1':
statement += """\(\mathrm{M}_2 (\Z)\)"""
# TODO: Add flag that indicates whether we are over a PID, in
# which case we can use the following lines:
#if factorsQQ[0][2] == 1:
# statement += """\(\mathrm{M}_2 (%s)\)"""\
# % ring_pretty(factorsQQ[0][1], 1)
else:
statement += """a maximal order of \(\End (J_{%s}) \otimes \Q\)""" % field
# If there are two factors, then they are both at most quadratic
# and we can prettify them
else:
statement += r'\(' + ' \\times '.join([ ring_pretty(factorQQ[1], 1) for factorQQ in factorsQQ ]) + r'\)'
# Then the case where there is still a single factor:
elif factorsQQ_number == 1:
# Number field case:
if factorsQQ[0][2] == -1:
# Prettify in quadratic case:
if len(factorsQQ[0][1]) in [2, 3]:
statement += """\(%s\)""" % ring_pretty(factorsQQ[0][1], ring[0])
else:
statement += """an order of conductor of norm \(%s\) in \(\End (J_{%s}) \otimes \Q\)""" % (ring[0], field)
# Otherwise mention whether the order is Eichler:
elif ring[1] == 1:
statement += """an Eichler order of index \(%s\) in a maximal order of \(\End (J_{%s}) \otimes \Q\)""" % (ring[0], field)
else:
statement += """a non-Eichler order of index \(%s\) in a maximal order of \(\End (J_{%s}) \otimes \Q\)""" % (ring[0], field)
# Finally the case of two factors. We can prettify to some extent, since we
# can describe the maximal order here
else:
statement += """an order of index \(%s\) in \(%s\)""" % (ring[0], ' \\times '.join([ ring_pretty(factorQQ[1], 1) for factorQQ in factorsQQ ]))
# End of first row:
statement += """</td></tr>"""
# Second row: description of endomorphism algebra factors (this is the first row if ring=None)
statement += """<tr><td>\(\End (J_{%s}) \otimes \Q \)</td><td>\(\simeq\)</td><td>""" % field
# In the case of only one factor we either get a number field or a
# quaternion algebra:
if factorsQQ_number == 1:
# First we deal with the number field case,
# in which we have set the discriminant to be -1
if factorsQQ[0][2] == -1:
# Prettify if labels available, otherwise return defining polynomial:
if factorsQQ_pretty:
statement += """<a href=%s>%s</a>""" % (url_for("number_fields.by_label", label=factorsQQ[0][0]), factorsQQ_pretty[0])
else:
statement += """the number field with defining polynomial \(%s\)""" % intlist_to_poly(factorsQQ[0][1])
# Detect CM by presence of a quartic polynomial:
if len(factorsQQ[0][1]) == 5:
statement += """ (CM)"""
# TODO: Get the following line to work
#statement += """ ({{ KNOWL('ag.complex_multiplication', title='CM') }})"""
# Up next is the case of a matrix ring (trivial disciminant), with
# labels and full prettification always available:
elif factorsQQ[0][2] == 1:
statement += """\(\mathrm{M}_2(\)<a href=%s>%s</a>\()\)""" % (url_for("number_fields.by_label", label=factorsQQ[0][0]), factorsQQ_pretty[0])
# And finally we deal with quaternion algebras over the rationals:
else:
statement += """the quaternion algebra over <a href=%s>%s</a> of discriminant %s"""\
% (url_for("number_fields.by_label", label=factorsQQ[0][0]), factorsQQ_pretty[0], factorsQQ[0][2])
# If there are two factors, then we get two at most quadratic fields:
else:
statement += """<a href=%s>%s</a> \(\\times\) <a href=%s>%s</a>"""\
% (url_for("number_fields.by_label", label=factorsQQ[0][0]),
factorsQQ_pretty[0], url_for("number_fields.by_label",
label=factorsQQ[1][0]), factorsQQ_pretty[1])
# End of second row:
statement += """</td></tr>"""
# Third row: description of algebra tensored with RR (this is the second row if ring=None)
statement += """<tr><td>\(\End (J_{%s}) \otimes \R\)</td><td>\(\simeq\)</td> <td>\(%s\)</td></tr>""" % (field, factorsRR_raw_to_pretty(factorsRR))
# End of statement:
statement += """</table>"""
return statement
def make_object(self, curve, endo, tama, ratpts, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = str(curve['Lhash'])
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex(factor(int(data['cond'])))
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1, 4, data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page",
cond=data['slabel'][0],
x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [(c[0],
list_to_factored_poly_otherorder(c[1]))
for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['abs_disc'])
data['disc'] = curve['disc_sign'] * data['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = list_to_min_eqn(data['min_eqn'])
data['disc_factor_latex'] = web_latex(factor(data['disc']))
data['igusa_clebsch'] = [
ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])
]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [
web_latex(zfactor(i)) for i in data['igusa_clebsch']
]
data['igusa_factor_latex'] = [
web_latex(zfactor(j)) for j in data['igusa']
]
data['aut_grp_id'] = curve['aut_grp_id']
data['geom_aut_grp_id'] = curve['geom_aut_grp_id']
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['has_square_sha'] = "square" if curve[
'has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",", 1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['torsion_order'] = curve['torsion_order']
data['torsion_factors'] = [
ZZ(a) for a in literal_eval(curve['torsion_subgroup'])
]
if len(data['torsion_factors']) == 0:
data['torsion_subgroup'] = '\mathrm{trivial}'
else:
data['torsion_subgroup'] = ' \\times '.join(
['\Z/{%s}\Z' % n for n in data['torsion_factors']])
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['tama'] = ''
for item in tama:
if item['tamagawa_number'] > 0:
tamgwnr = str(item['tamagawa_number'])
else:
tamgwnr = 'N/A'
data['tama'] += tamgwnr + ' (p = ' + str(item['p']) + '), '
data['tama'] = data['tama'][:-2] # trim last ", "
if ratpts:
if len(ratpts['rat_pts']):
data['rat_pts'] = ', '.join(
web_latex('(' + ' : '.join(map(str, P)) + ')')
for P in ratpts['rat_pts'])
data['rat_pts_v'] = 2 if ratpts['rat_pts_v'] else 1
# data['mw_rank'] = ratpts['mw_rank']
# data['mw_rank_v'] = ratpts['mw_rank_v']
else:
data['rat_pts_v'] = 0
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl(
curve['two_torsion_field'][0],
field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data[
'two_torsion_field_knowl'] = """splitting field of \(%s\) with Galois group %s""" % (
intlist_to_poly(
t[1]), group_display_knowl(t[2][0], t[2][1]))
else:
# invariants specific to isogeny class
curves_data = list(
db.g2c_curves.search({"class": curve['class']},
['label', 'eqn']))
if not curves_data:
raise KeyError(
"No curves found in database for isogeny class %s of genus 2 curve %s."
% (curve['class'], curve['label']))
data['curves'] = [{
"label":
c['label'],
"equation_formatted":
list_to_min_eqn(literal_eval(c['eqn'])),
"url":
url_for_curve_label(c['label'])
} for c in curves_data]
lfunc_data = db.lfunc_lfunctions.lucky(
{'Lhash': str(curve['Lhash'])})
if not lfunc_data:
raise KeyError(
"No Lfunction found in database for isogeny class of genus 2 curve %s."
% curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [
[nth_prime(n + 1), lfunc_data['euler_factors'][n]]
for n in range(len(lfunc_data['euler_factors']))
if nth_prime(n + 1) < 30 and (data['cond'] %
nth_prime(n + 1))
]
data['good_lfactors_pretty'] = [
(c[0], list_to_factored_poly_otherorder(c[1]))
for c in data['good_lfactors']
]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(
endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = """Endomorphism %s over \(\Q\):<br>""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_base'], endo['factorsRR_base'], ring=data['end_ring_base'] if is_curve else None)
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(
data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(
data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = """Endomorphism %s over \(\overline{\Q}\):""" %("ring" if is_curve else "algebra") + \
end_statement(data['factorsQQ_geom'], data['factorsRR_geom'], field=r'\overline{\Q}', ring=data['end_ring_geom'] if is_curve else None)
data['real_geom_end_alg_name'] = end_alg_name(
curve['real_geom_end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(
data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(
data['is_simple_geom'], data['split_field_label'],
data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(
endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'],
data.get('split_labels'),
data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
self.plot = encode_plot(
eqn_list_to_curve_plot(
data['min_eqn'],
data['rat_pts'].split(',') if 'rat_pts' in data else []))
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(
self.plot)
properties += [
(None, plot_link),
('Conductor', str(data['cond'])),
('Discriminant', str(data['disc'])),
]
properties += [
('Sato-Tate group', data['st_group_link']),
('\(\\End(J_{\\overline{\\Q}}) \\otimes \\R\)',
'\(%s\)' % data['real_geom_end_alg_name']),
('\(\\overline{\\Q}\)-simple',
bool_pretty(data['is_simple_geom'])),
('\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = [('L-function', data['lfunc_url'])]
if is_curve:
friends.append(('Isogeny class %s.%s' %
(data['slabel'][0], data['slabel'][1]),
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1])))
for friend_url in db.lfunc_instances.search({'Lhash': data['Lhash']},
'url'):
if '|' in friend_url:
for url in friend_url.split('|'):
add_friend(friends, lfunction_friend_from_url(url))
else:
add_friend(friends, lfunction_friend_from_url(friend_url))
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
add_friend(friends, ("Elliptic curve " + friend_label,
url_for_ec(friend_label)))
else:
add_friend(
friends,
("EC isogeny class " + ec_label_class(friend_label),
url_for_ec_class(friend_label)))
if is_curve:
friends.append(('Twists',
url_for(".index_Q",
g20=str(data['g2'][0]),
g21=str(data['g2'][1]),
g22=str(data['g2'][2]))))
# Breadcrumbs
self.bread = bread = [('Genus 2 Curves', url_for(".index")),
('$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0],
url_for(".by_conductor",
cond=data['slabel'][0])),
('%s' % data['slabel'][1],
url_for(".by_url_isogeny_class_label",
cond=data['slabel'][0],
alpha=data['slabel'][1]))]
if is_curve:
bread += [('%s' % data['slabel'][2],
url_for(".by_url_isogeny_class_discriminant",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2])),
('%s' % data['slabel'][3],
url_for(".by_url_curve_label",
cond=data['slabel'][0],
alpha=data['slabel'][1],
disc=data['slabel'][2],
num=data['slabel'][3]))]
# Title
self.title = "Genus 2 " + ("Curve " if is_curve else
"Isogeny Class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage': '', 'magma': ''} # use default show names
code['curve'] = {
'sage':
'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s))'
% (data['min_eqn'][0], data['min_eqn'][1]),
'magma':
'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'
% (data['min_eqn'][0], data['min_eqn'][1])
}
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation(' + str(
data['cond']) + ',2),R!' + str(bad2) + '>*]'
else:
magma_cond_option = ''
code['cond'] = {
'magma':
'Conductor(LSeries(C%s)); Factorization($1);' % magma_cond_option
}
code['disc'] = {
'magma': 'Discriminant(C); Factorization(Integers()!$1);'
}
code['igusa_clebsch'] = {
'sage':
'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':
'IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];'
}
code['igusa'] = {
'magma':
'IgusaInvariants(C); [Factorization(Integers()!a): a in $1];'
}
code['g2'] = {'magma': 'G2Invariants(C);'}
code['aut'] = {'magma': 'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {
'magma':
'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'
}
code['num_rat_wpts'] = {
'magma': '#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'
}
if ratpts:
code['rat_pts'] = {
'magma':
'[' + ','.join([
"C![%s,%s,%s]" % (p[0], p[1], p[2])
for p in ratpts['rat_pts']
]) + '];'
}
code['two_selmer'] = {
'magma': 'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'
}
code['has_square_sha'] = {'magma': 'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {
'magma':
'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'
}
code['torsion_subgroup'] = {
'magma':
'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'
}