def statistics_by_signature(d, r):
if d == 1:
return redirect(url_for("ec.statistics"))
info = {}
ecnfstats = db_ecnfstats()
sigs_by_deg = ecnfstats.find_one({'_id': 'signatures_by_degree'},
projection={'_id': False})
if not str(d) in sigs_by_deg:
info[
'error'] = "The database does not contain any elliptic curves defined over fields of degree %s" % d
else:
info['degree'] = d
if not r in range(d % 2, d + 1, 2):
info['error'] = "Invalid signature %s" % info['sig']
s = (d - r) // 2
info['sig'] = sig = '%s,%s' % (r, s)
info['summary'] = ecnf_signature_summary(sig)
fields_by_sig = ecnfstats.find_one({'_id': 'fields_by_signature'},
projection={'_id': False})
counts_by_field = ecnfstats.find_one({'_id': 'conductor_norm_by_field'},
projection={'_id': False})
def field_counts(f):
ff = f.replace(".", ":")
return [f, counts_by_field[ff]]
sorted_fields = sorted(fields_by_sig[sig], key=sort_field)
info['sig_stats'] = [field_counts(f) for f in sorted_fields]
credit = 'John Cremona'
if info['sig'] == '2,0':
t = 'Elliptic curves over real quadratic number fields'
elif info['sig'] == '0,1':
t = 'Elliptic curves over imaginary quadratic number fields'
elif info['sig'] == '3,0':
t = 'Elliptic curves over totally real cubic number fields'
elif info['sig'] == '1,1':
t = 'Elliptic curves over mixed cubic number fields'
elif info['sig'] == '4,0':
t = 'Elliptic curves over totally real quartic number fields'
elif info['sig'] == '5,0':
t = 'Elliptic curves over totally real quintic number fields'
elif info['sig'] == '6,0':
t = 'Elliptic curves over totally real sextic number fields'
else:
t = 'Elliptic curves over number fields of degree %s, signature (%s)' % (
d, info['sig'])
bread = [('Elliptic Curves', url_for("ecnf.index")),
('degree %s' % d, url_for("ecnf.statistics_by_degree", d=d)),
('signature (%s)' % info['sig'], ' ')]
return render_template("ecnf-by-signature.html",
info=info,
credit=credit,
title=t,
bread=bread,
learnmore=learnmore_list())
def get_torsion_structures():
global torsion_structures
if torsion_structures==None:
#print("Getting list of torsion structures from the database")
ecnfstats = db_ecnfstats()
torsion_structures = [t[0] for t in ecnfstats.find_one({'_id':'torsion_structure'})['counts']]
torsion_structures = [[int(str(n)) for n in t.split(",")] for t in torsion_structures if t]
torsion_structures.sort()
return torsion_structures
def browse():
data = db_ecnfstats().find_one({'_id':'signatures_by_degree'}, projection={'_id': False})
# We could use the dict directly but then could not control the order
# of the keys (degrees), so we use a list
info = [[d,data[str(d)]] for d in sorted([int(d) for d in data.keys()])]
credit = 'John Cremona'
t = 'Elliptic curves over number fields'
bread = [('Elliptic Curves', url_for("ecnf.index")),
('Browse', ' ')]
return render_template("ecnf-stats.html", info=info, credit=credit, title=t, bread=bread, learnmore=learnmore_list())
def get_torsion_structures():
ecnfstats = db_ecnfstats()
torsion_structures = [
t[0]
for t in ecnfstats.find_one({'_id': 'torsion_structure'})['counts']
]
torsion_structures = [[int(str(n)) for n in t.split(",")]
for t in torsion_structures if t]
torsion_structures.sort()
return torsion_structures
def browse():
data = db_ecnfstats().find_one({'_id':'signatures_by_degree'}, projection={'_id': False})
# We could use the dict directly but then could not control the order
# of the keys (degrees), so we use a list
info = [[d,data[str(d)]] for d in sorted([int(d) for d in data.keys()])]
credit = 'John Cremona'
t = 'Elliptic Curves over Number Fields'
bread = [('Elliptic Curves', url_for("ecnf.index")),
('Browse', ' ')]
return render_template("ecnf-stats.html", info=info, credit=credit, title=t, bread=bread, learnmore=learnmore_list())
def statistics_by_degree(d):
if d == 1:
return redirect(url_for("ec.statistics"))
info = {}
ecnfstats = db_ecnfstats()
sigs_by_deg = ecnfstats.find_one({'_id': 'signatures_by_degree'},
projection={'_id': False})
if not str(d) in sigs_by_deg:
info[
'error'] = "The database does not contain any elliptic curves defined over fields of degree %s" % d
else:
info['degree'] = d
fields_by_sig = ecnfstats.find_one({'_id': 'fields_by_signature'},
projection={'_id': False})
counts_by_sig = ecnfstats.find_one({'_id': 'conductor_norm_by_signature'},
projection={'_id': False})
counts_by_field = ecnfstats.find_one({'_id': 'conductor_norm_by_field'},
projection={'_id': False})
def field_counts(f):
ff = f.replace(".", ":")
return [f, counts_by_field[ff]]
def sig_counts(sig):
sorted_fields = sorted(fields_by_sig[sig], key=sort_field)
return [
sig, counts_by_sig[sig], [field_counts(f) for f in sorted_fields]
]
info['summary'] = ecnf_degree_summary(d)
info['sig_stats'] = [sig_counts(sig) for sig in sigs_by_deg[str(d)]]
credit = 'John Cremona'
if d == 2:
t = 'Elliptic curves over quadratic number fields'
elif d == 3:
t = 'Elliptic curves over cubic number fields'
elif d == 4:
t = 'Elliptic curves over quartic number fields'
elif d == 5:
t = 'Elliptic curves over quintic number fields'
elif d == 6:
t = 'Elliptic curves over sextic number fields'
else:
t = 'Elliptic curves over number fields of degree {}'.format(d)
bread = [('Elliptic Curves', url_for("ecnf.index")),
('degree %s' % d, ' ')]
return render_template("ecnf-by-degree.html",
info=info,
credit=credit,
title=t,
bread=bread,
learnmore=learnmore_list())
def statistics_by_signature(d,r):
if d==1:
return redirect(url_for("ec.statistics"))
info = {}
ecnfstats = db_ecnfstats()
sigs_by_deg = ecnfstats.find_one({'_id':'signatures_by_degree'}, projection={'_id': False})
if not str(d) in sigs_by_deg:
info['error'] = "The database does not contain any elliptic curves defined over fields of degree %s" % d
else:
info['degree'] = d
if not r in range(d%2,d+1,2):
info['error'] = "Invalid signature %s" % info['sig']
s = (d-r)//2
info['sig'] = sig = '%s,%s' % (r,s)
info['summary'] = ecnf_signature_summary(sig)
fields_by_sig = ecnfstats.find_one({'_id':'fields_by_signature'}, projection={'_id': False})
counts_by_field = ecnfstats.find_one({'_id':'conductor_norm_by_field'}, projection={'_id': False})
def field_counts(f):
ff = f.replace(".",":")
return [f,counts_by_field[ff]]
sorted_fields = sorted(fields_by_sig[sig], key=sort_field)
info['sig_stats'] = [field_counts(f) for f in sorted_fields]
credit = 'John Cremona'
if info['sig'] == '2,0':
t = 'Elliptic curves over real quadratic number fields'
elif info['sig'] == '0,1':
t = 'Elliptic curves over imaginary quadratic number fields'
elif info['sig'] == '3,0':
t = 'Elliptic curves over totally real cubic number fields'
elif info['sig'] == '1,1':
t = 'Elliptic curves over mixed cubic number fields'
elif info['sig'] == '4,0':
t = 'Elliptic curves over totally real quartic number fields'
elif info['sig'] == '5,0':
t = 'Elliptic curves over totally real quintic number fields'
elif info['sig'] == '6,0':
t = 'Elliptic curves over totally real sextic number fields'
else:
t = 'Elliptic curves over number fields of degree %s, signature (%s)' % (d,info['sig'])
bread = [('Elliptic Curves', url_for("ecnf.index")),
('Degree %s' % d,url_for("ecnf.statistics_by_degree", d=d)),
('Signature (%s)' % info['sig'],' ')]
return render_template("ecnf-by-signature.html", info=info, credit=credit, title=t, bread=bread, learnmore=learnmore_list())
def ecnf_degree_summary(d):
ec_knowl = '<a knowl="ec">elliptic curves</a>'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny classes</a>'
nf_knowl = '<a knowl="nf">number fields</a>'
cond_knowl = '<a knowl="ec.conductor">conductors</a>'
data = db_ecnfstats().find_one({'_id':'conductor_norm_by_degree'})[str(d)]
ncurves = data['ncurves']
nclasses = data['nclasses']
max_norm = data['max_norm']
return ''.join([r'The database currently contains {} '.format(ncurves),
ec_knowl,
r' defined over ',
nf_knowl,
r' of degree {}, in {} '.format(d, nclasses),
iso_knowl,
r', with ',
cond_knowl,
r' of norm up to {}.'.format(max_norm)])
def statistics_by_degree(d):
if d==1:
return redirect(url_for("ec.statistics"))
info = {}
ecnfstats = db_ecnfstats()
sigs_by_deg = ecnfstats.find_one({'_id':'signatures_by_degree'}, projection={'_id': False})
if not str(d) in sigs_by_deg:
info['error'] = "The database does not contain any elliptic curves defined over fields of degree %s" % d
else:
info['degree'] = d
fields_by_sig = ecnfstats.find_one({'_id':'fields_by_signature'}, projection={'_id': False})
counts_by_sig = ecnfstats.find_one({'_id':'conductor_norm_by_signature'}, projection={'_id': False})
counts_by_field = ecnfstats.find_one({'_id':'conductor_norm_by_field'}, projection={'_id': False})
def field_counts(f):
ff = f.replace(".",":")
return [f,counts_by_field[ff]]
def sig_counts(sig):
sorted_fields = sorted(fields_by_sig[sig], key=sort_field)
return [sig, counts_by_sig[sig], [field_counts(f) for f in sorted_fields]]
info['summary'] = ecnf_degree_summary(d)
info['sig_stats'] = [sig_counts(sig) for sig in sigs_by_deg[str(d)]]
credit = 'John Cremona'
if d==2:
t = 'Elliptic curves over quadratic number fields'
elif d==3:
t = 'Elliptic curves over cubic number fields'
elif d==4:
t = 'Elliptic curves over quartic number fields'
elif d==5:
t = 'Elliptic curves over quintic number fields'
elif d==6:
t = 'Elliptic curves over sextic number fields'
else:
t = 'Elliptic curves over number fields of degree {}'.format(d)
bread = [('Elliptic Curves', url_for("ecnf.index")),
('Degree %s' % d,' ')]
return render_template("ecnf-by-degree.html", info=info, credit=credit, title=t, bread=bread, learnmore=learnmore_list())
def ecnf_signature_summary(sig):
ec_knowl = '<a knowl="ec">elliptic curves</a>'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny classes</a>'
nf_knowl = '<a knowl="nf">number fields</a>'
cond_knowl = '<a knowl="ec.conductor">conductors</a>'
r, s = [int(x) for x in sig.split(",")]
d = r+2*s
data = db_ecnfstats().find_one({'_id':'conductor_norm_by_signature'})[sig]
ncurves = data['ncurves']
nclasses = data['nclasses']
max_norm = data['max_norm']
return ''.join([r'The database currently contains {} '.format(ncurves),
ec_knowl,
r' defined over ',
nf_knowl,
r' of signature ({}) (degree {}), in {} '.format(sig, d, nclasses),
iso_knowl,
r', with ',
cond_knowl,
r' of norm up to {}.'.format(max_norm)])
def ecnf_summary():
ecnfstats = db_ecnfstats()
ec_knowl = '<a knowl="ec">elliptic curves</a>'
iso_knowl = '<a knowl="ec.isogeny_class">isogeny classes</a>'
nf_knowl = '<a knowl="nf">number fields</a>'
deg_knowl = '<a knowl="nf.degree">degree</a>'
data = ecnfstats.find_one({'_id':'conductor_norm'})
ncurves = comma(data['ncurves'])
nclasses = comma(data['nclasses'])
data = ecnfstats.find_one({'_id':'field_label'})
nfields = len(data['counts'])
data = ecnfstats.find_one({'_id':'signatures_by_degree'})
maxdeg = max(int(d) for d in data if d!='_id')
return ''.join([r'The database currently contains {} '.format(ncurves),
ec_knowl,
r' in {} '.format(nclasses),
iso_knowl,
r', over {} '.format(nfields),
nf_knowl, ' (not including $\mathbb{Q}$) of ',
deg_knowl,
r' up to {}.'.format(maxdeg)])
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 get_torsion_structures():
ecnfstats = db_ecnfstats()
torsion_structures = [t[0] for t in ecnfstats.find_one({'_id':'torsion_structure'})['counts']]
torsion_structures = [[int(str(n)) for n in t.split(",")] for t in torsion_structures if t]
torsion_structures.sort()
return torsion_structures
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 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'))
