def browse():
data = ECNF_stats().sigs_by_deg
# We could use the dict directly but then could not control the order
# of the keys (degrees), so we use a list
info = [[d,['%s,%s'%sig for sig in data[d]]] for d in sorted(data.keys())]
t = 'Elliptic curves over number fields'
bread = [('Elliptic curves', url_for("ecnf.index")),
('Browse', ' ')]
return render_template("ecnf-stats.html", info=info, title=t, bread=bread, learnmore=learnmore_list())
def browse():
data = ECNF_stats().sigs_by_deg
# We could use the dict directly but then could not control the order
# of the keys (degrees), so we use a list
info = [[d,['%s,%s'%sig for sig in data[d]]] for d in sorted(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_signature(d, r):
if d == 1:
return redirect(url_for("ec.statistics"))
info = {}
sigs_by_deg = ECNF_stats().sigs_by_deg
if d not 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
sig = (r, s)
info['sig'] = '%s,%s' % sig
info['summary'] = ECNF_stats().signature_summary(sig)
fields_by_sig = ECNF_stats().fields_by_sig
counts_by_field = ECNF_stats().field_normstats
def field_counts(f):
return [f, counts_by_field[f]]
info['sig_stats'] = [field_counts(f) for f in fields_by_sig[sig]]
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 statistics_by_degree(d):
if d == 1:
return redirect(url_for("ec.statistics"))
info = {}
sigs_by_deg = ECNF_stats().sigs_by_deg
if d not 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 = ECNF_stats().fields_by_sig
counts_by_sig = ECNF_stats().sig_normstats
counts_by_field = ECNF_stats().field_normstats
def field_counts(f):
return [f, counts_by_field[f]]
def sig_counts(sig):
return [
'%s,%s' % sig, counts_by_sig[sig],
[field_counts(f) for f in fields_by_sig[sig]]
]
info['summary'] = ECNF_stats().degree_summary(d)
info['sig_stats'] = [sig_counts(sig) for sig in sigs_by_deg[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 tor_struct_search_nf(prefill="any"):
def fix(t):
return t + ' selected = "yes"' if prefill == t else t
def cyc(n):
return [fix("[" + str(n) + "]"), "C{}".format(n)]
def cyc2(m, n):
return [fix("[{},{}]".format(m, n)), "C{}×C{}".format(m, n)]
gps = [[fix(""), "any"], [fix("[]"), "trivial"]]
tors = ECNF_stats().torsion_counts
# The following was the set as of 24/4/2017:
# assert tors == [[2], [2, 2], [2, 4], [2, 6], [2, 8], [2, 10], [2, 12], [2, 14], [2, 16], [2, 18], [3], [3, 3], [3, 6], [4], [4, 4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [25], [27], [37]]
for t in tors:
if len(t) == 1:
gps.append(cyc(t[0]))
elif len(t) == 2:
gps.append(cyc2(*t))
return "\n".join(
["<select name='torsion_structure'>"] +
["<option value={}>{}</option>".format(a, b)
for a, b in gps] + ["</select>"])
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 statistics():
title = "Elliptic curves: statistics"
bread = get_bread("Statistics")
return render_template("display_stats.html",
info=ECNF_stats(),
title=title,
bread=bread,
learnmore=learnmore_list())
def __init__(self):
field = TextBox(name="field",
label="Base field",
knowl="ag.base_field",
example="2.2.5.1",
example_span="2.2.5.1 or Qsqrt5")
include_base_change = ExcludeOnlyBox(name="include_base_change",
label="Base change curves",
knowl="ec.base_change")
include_Q_curves = ExcludeOnlyBox(name="include_Q_curves",
label=r"\(\Q\)-curves",
knowl="ec.q_curve")
conductor_norm = TextBox(name="conductor_norm",
label="Conductor norm",
knowl="ec.conductor",
example="31",
example_span="31 or 1-100")
one = SelectBox(name="one",
label="Curves per isogeny class",
knowl="ec.isogeny_class",
example="all, one",
options=[("", "all"), ("yes", "one")])
include_cm = SelectBox(name="include_cm",
label="CM",
knowl="ec.complex_multiplication",
options=[('', ''), ('PCM', 'potential CM'),
('PCMnoCM', 'potential CM but no CM'),
('CM', 'CM'),
('noPCM', 'no potential CM')])
cm_disc = TextBox(name="cm_disc",
label="CM discriminant",
example="-4",
example_span="-4 or -3,-8",
knowl="ec.complex_multiplication")
jinv = TextBox(name="jinv",
label="j-invariant",
knowl="ec.j_invariant",
width=685,
short_width=160,
colspan=(1, 4, 1),
example_span_colspan=2,
example="105474/49 + a*34213/49",
example_span="")
rank = TextBox(name="rank",
label="Rank*",
knowl="ec.rank",
example="2")
torsion = TextBox(name="torsion",
label="Torsion order",
knowl="ec.torsion_order",
example="2")
bf_deg = SelectBox(name="bf_deg",
label="Base field degree",
knowl="nf.degree",
options=[("", ""), ("2", "2"), ("3", "3"),
("4", "4"), ("5", "5"), ("6", "6")])
tor_opts = (
[("", ""), ("[]", "trivial")] +
[disp_tor(tuple(t)) for t in ECNF_stats().torsion_counts if t])
torsion_structure = SelectBox(name="torsion_structure",
label="Torsion structure",
knowl="ec.torsion_subgroup",
options=tor_opts)
sha = TextBox(name="sha",
label="Analytic order* of Ш",
knowl="ec.analytic_sha_order",
example="4")
regulator = TextBox(name="regulator",
label="Regulator*",
knowl="ec.regulator",
example="8.4-9.1")
bad_quant = SubsetBox(name="bad_quantifier")
bad_primes = TextBoxWithSelect(name="conductor_norm_factors",
label="Bad primes",
knowl="ec.reduction_type",
example="5,13",
select_box=bad_quant)
isodeg = TextBox(name="isodeg",
label="Cyclic isogeny degree",
knowl="ec.isogeny",
example="16")
semistable = YesNoBox(name="semistable",
label="Semistable",
example="Yes",
knowl="ec.semistable")
potential_good_reduction = YesNoBox(
name="potential_good_reduction",
label="Potential good reduction",
example="Yes",
knowl="ec.potential_good_reduction")
class_size = TextBox(name="class_size",
label="Isogeny class size",
knowl="ec.isogeny",
example="4")
class_deg = TextBox(name="class_deg",
label="Isogeny class degree",
knowl="ec.isogeny",
example="16")
count = CountBox()
self.browse_array = [[field, bf_deg],
[conductor_norm, include_base_change],
[rank, include_Q_curves],
[torsion, torsion_structure],
[cm_disc, include_cm], [sha, regulator],
[isodeg, one], [class_size, class_deg],
[semistable, potential_good_reduction], [jinv],
[count, bad_primes]]
self.refine_array = [
[field, conductor_norm, rank, torsion, cm_disc],
[
bf_deg, include_base_change, include_Q_curves,
torsion_structure, include_cm
],
[sha, isodeg, class_size, semistable, jinv],
[regulator, one, class_deg, potential_good_reduction, bad_primes],
]
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 'torsion_order' not in query:
t_o = 1
for n in query['torsion_structure']:
t_o *= int(n)
query['torsion_order'] = t_o
parse_element_of(info,
query,
'isodeg',
split_interval=1000,
contained_in=ECNF_stats().isogeny_degrees)
parse_bool(info, query, 'semistable', 'semistable')
parse_bool(info, query, 'potential_good_reduction',
'potential_good_reduction')
parse_ints(info, query, 'class_size', 'class_size')
parse_ints(info, query, 'class_deg', 'class_deg')
parse_ints(info, query, 'sha', 'analytic order of Ш')
parse_floats(info, query, 'reg', 'regulator')
parse_nf_jinv(info,
query,
'jinv',
'j-invariant',
field_label=query.get('field_label'))
if info.get('one') == "yes":
info['number'] = 1
query['number'] = 1
if 'include_base_change' in info:
if info['include_base_change'] in [
'exclude', 'off'
]: # off for backward compat with urls
query['base_change'] = []
if info['include_base_change'] == 'only':
query['base_change'] = {'$ne': []}
else:
info['include_base_change'] = "on"
if 'include_Q_curves' in info:
if info['include_Q_curves'] == 'exclude':
query['q_curve'] = False
elif info['include_Q_curves'] == 'only':
query['q_curve'] = True
parse_cm_list(info,
query,
field='cm_disc',
qfield='cm',
name="CM discriminant")
if 'include_cm' in info:
if info['include_cm'] == 'PCM':
tmp = {'$ne': 0}
if 'cm' in query:
query['cm'] = {'$and': [tmp, query['cm']]}
else:
query['cm'] = tmp
elif info['include_cm'] == 'PCMnoCM':
tmp = {'$lt': 0}
if 'cm' in query:
query['cm'] = {'$and': [tmp, query['cm']]}
else:
query['cm'] = tmp
elif info['include_cm'] == 'CM':
tmp = {'$gt': 0}
if 'cm' in query:
query['cm'] = {'$and': [tmp, query['cm']]}
else:
query['cm'] = tmp
elif info['include_cm'] == 'noPCM':
tmp = 0
if 'cm' in query:
query['cm'] = {'$and': [tmp, query['cm']]}
else:
query['cm'] = tmp
parse_primes(info,
query,
'conductor_norm_factors',
name='bad primes',
qfield='conductor_norm_factors',
mode=info.get('bad_quantifier'))
info['field_pretty'] = field_pretty
parse_ints(info,
query,
'bf_deg',
name='Base field degree',
qfield='degree')
def __init__(self):
field = TextBox(name="field",
label="Base field",
knowl="nf",
example="2.2.5.1",
example_span="2.2.5.1 or Qsqrt5")
include_base_change = ExcludeOnlyBox(name="include_base_change",
label="Base change curves",
knowl="ec.base_change")
include_Q_curves = ExcludeOnlyBox(name="include_Q_curves",
label=r"\(\Q\)-curves",
knowl="ec.q_curve")
conductor_norm = TextBox(name="conductor_norm",
label="Conductor norm",
knowl="ec.conductor",
example="31",
example_span="31 or 1-100")
one = SelectBox(name="one",
label="Curves per isogeny class",
knowl="ec.isogeny_class",
options=[("", ""), ("yes", "one")])
include_cm = ExcludeOnlyBox(name="include_cm",
label="CM",
knowl="ec.complex_multiplication")
cm_disc = TextBox(name="cm_disc",
label="CM discriminant",
example="-4",
example_span="-4 or -3,-8",
knowl="ec.complex_multiplication")
jinv = TextBox(name="jinv",
label="j-invariant",
knowl="ec.j_invariant",
width=675,
short_width=160,
colspan=(1, 4, 1),
example_span_colspan=2,
example="105474/49 + a*34213/49",
example_span="")
torsion = TextBox(name="torsion",
label="Torsion order",
knowl="ec.torsion_order",
example="2")
def disp_tor(t):
if len(t) == 1:
return "[%s]" % t, "C%s" % t
else:
return "[%s,%s]" % t, "C%s×C%s" % t
tor_opts = (
[("", ""), ("[]", "trivial")] +
[disp_tor(tuple(t)) for t in ECNF_stats().torsion_counts if t])
torsion_structure = SelectBox(name="torsion_structure",
label="Torsion structure",
knowl="ec.torsion_subgroup",
options=tor_opts)
isodeg = TextBox(name="isodeg",
label="Cyclic isogeny degree",
knowl="ec.isogeny",
example="16")
count = CountBox()
self.browse_array = [[jinv], [field, include_base_change],
[conductor_norm, include_Q_curves],
[torsion, torsion_structure],
[cm_disc, include_cm], [isodeg, one], [count]]
self.refine_array = [[
field, conductor_norm, jinv, include_base_change, include_Q_curves
], [isodeg, torsion, torsion_structure, include_cm, cm_disc], [one]]
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
info = to_dict(request.args, search_array=ECNFSearchArray())
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:
fields_by_deg = ECNF_stats().fields_by_deg
fields_by_sig = ECNF_stats().fields_by_sig
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 [5, 89, 229, 497]]
niqfs = len(fields_by_sig[0, 1])
nrqfs = len(fields_by_sig[2, 0])
info['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]]
info['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])
info['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])
info['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])
info['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])
info['fields'].append([
'{} totally real sextic fields, including'.format(nsextics),
((nf, [url_for('.show_ecnf1', nf=nf),
field_pretty(nf)]) for nf in sextics)
])
info['degrees'] = sorted(
[int(d) for d in fields_by_deg.keys() if d != '_id'])
# info['highlights'] holds data (URL and descriptive text) for a
# sample of elliptic curves with interesting features:
info['highlights'] = []
info['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))
])
info['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))
])
info['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))
])
info['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')
])
info['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",
info=info,
bread=bread,
learnmore=learnmore_list())
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
for n in query['torsion_structure']:
t_o *= int(n)
query['torsion_order'] = t_o
parse_element_of(info,query,'isodeg',split_interval=1000,contained_in=ECNF_stats().isogeny_degrees)
parse_bool(info,query,'semistable','semistable')
parse_bool(info,query,'potential_good_reduction','potential_good_reduction')
parse_ints(info,query,'class_size','class_size')
parse_ints(info,query,'class_deg','class_deg')
parse_ints(info,query,'sha','analytic order of Ш')
parse_floats(info,query,'reg','regulator')
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')
if not 'a' in info['jinv'] and not info.get('field'): # rational j-invariant allowed for any field
parse_rational(info, query, 'jinv', name='j-invariant')
if query.get('jinv'):
query['jinv'] = {'$regex': '^' + query['jinv'] + '(,0)*$'} # nf elements like j,0,0,0
else: # j-invariant is a number field element
parse_nf_elt(info, query, 'jinv', name='j-invariant')
if query.get('jinv'):
query['jinv'] = ','.join(query['jinv'])
if info.get('one') == "yes":
info['number'] = 1
query['number'] = 1
if 'include_base_change' in info:
if info['include_base_change'] in ['exclude', 'off']: # off for backward compat with urls
query['base_change'] = []
if info['include_base_change'] == 'only':
query['base_change'] = {'$ne':[]}
else:
info['include_base_change'] = "on"
if 'include_Q_curves' in info:
if info['include_Q_curves'] == 'exclude':
query['q_curve'] = False
elif info['include_Q_curves'] == 'only':
query['q_curve'] = True
if 'include_cm' in info:
if info['include_cm'] == 'PCM':
query['cm'] = {'$ne' : 0}
elif info['include_cm'] == 'PCMnoCM':
query['cm'] = {'$lt' : 0}
elif info['include_cm'] == 'CM':
query['cm'] = {'$gt' : 0}
elif info['include_cm'] == 'noPCM':
query['cm'] = 0
parse_ints(info,query,field='cm_disc',qfield='cm')
parse_primes(info, query, 'conductor_norm_factors', name='bad primes',
qfield='conductor_norm_factors',mode=info.get('bad_quantifier'))
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
parse_ints(info,query,'bf_deg',name='Base field degree',qfield='degree')
def index():
# if 'jump' in request.args:
# return show_ecnf1(request.args['label'])
if len(request.args) > 0:
return elliptic_curve_search(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:
fields_by_deg = ECNF_stats().fields_by_deg
fields_by_sig = ECNF_stats().fields_by_sig
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())
def __init__(self):
field = TextBox(name="field",
label="Base field",
knowl="nf",
example="2.2.5.1",
example_span="2.2.5.1 or Qsqrt5")
include_base_change = IncludeOnlyBox(name="include_base_change",
label="Base change curves",
knowl="ec.base_change",
width=50,
short_width=170)
include_Q_curves = IncludeOnlyBox(name="include_Q_curves",
label=r"\(\Q\)-curves",
knowl="ec.q_curve",
width=50,
short_width=170)
conductor_norm = TextBox(name="conductor_norm",
label="Conductor norm",
knowl="ec.conductor",
example="31",
example_span="31 or 1-100")
include_isogenous = IncludeBox(name="include_isogenous",
label=r"Isogenous curves",
knowl="ec.isogeny",
width=50,
short_width=170)
include_cm = IncludeOnlyBox(name="include_cm",
label="CM curves",
knowl="ec.complex_multiplication",
width=50,
short_width=170)
jinv = TextBox(name="jinv",
label="j-invariant",
width=300,
short_width=160,
colspan=(1, 2, 1),
example_span_colspan=2,
example="105474/49 + a*34213/49",
example_span="105474/49 + a*34213/49 over 2.0.7.1")
torsion = TextBox(name="torsion",
label="Torsion order",
knowl="ec.torsion_order",
example="2")
def disp_tor(t):
if len(t) == 1:
return "[%s]" % t, "C%s" % t
else:
return "[%s,%s]" % t, "C%s×C%s" % t
tor_opts = (
[("", "any"), ("[]", "trivial")] +
[disp_tor(tuple(t)) for t in ECNF_stats().torsion_counts if t])
torsion_structure = SelectBox(name="torsion_structure",
label="Torsion structure",
knowl="ec.torsion_subgroup",
options=tor_opts,
width=50,
short_width=170)
isodeg = TextBox(name="isodeg",
label="Cyclic isogeny degree",
knowl="ec.isogeny",
example="16")
count = TextBox(name="count", label="Results to display", example="50")
self.browse_array = [[field, include_base_change, include_Q_curves],
[conductor_norm, include_isogenous, include_cm],
[jinv], [torsion, torsion_structure], [isodeg],
[count]]
self.refine_array = [[
field, conductor_norm, include_isogenous, include_base_change,
include_Q_curves
], [jinv, isodeg, torsion, torsion_structure, include_cm]]
def __init__(self):
field = TextBox(name="field",
label="Base field",
knowl="ag.base_field",
example="2.2.5.1",
example_span="2.2.5.1 or Qsqrt5")
include_base_change = ExcludeOnlyBox(name="include_base_change",
label="Base change curves",
knowl="ec.base_change")
include_Q_curves = ExcludeOnlyBox(name="include_Q_curves",
label=r"\(\Q\)-curves",
knowl="ec.q_curve")
conductor_norm = TextBox(name="conductor_norm",
label="Conductor norm",
knowl="ec.conductor",
example="31",
example_span="31 or 1-100")
one = SelectBox(name="one",
label="Curves per isogeny class",
knowl="ec.isogeny_class",
options=[("", ""), ("yes", "one")])
include_cm = SelectBox(name="include_cm",
label="CM",
knowl="ec.complex_multiplication",
options=[('', ''), ('PCM', 'potential CM'),
('PCMnoCM', 'potential CM but no CM'),
('CM', 'CM'),
('noPCM', 'no potential CM')])
cm_disc = TextBox(name="cm_disc",
label="CM discriminant",
example="-4",
example_span="-4 or -3,-8",
knowl="ec.complex_multiplication")
jinv = TextBox(name="jinv",
label="j-invariant",
knowl="ec.j_invariant",
width=675,
short_width=160,
colspan=(1, 4, 1),
example_span_colspan=2,
example="105474/49 + a*34213/49",
example_span="")
rank = TextBox(name="rank",
label="Rank*",
knowl="ec.rank",
example="2")
torsion = TextBox(name="torsion",
label="Torsion order",
knowl="ec.torsion_order",
example="2")
bf_deg = SelectBox(name="bf_deg",
label="Base field degree",
knowl="nf.degree",
options=[("", ""), ("2", "2"), ("3", "3"),
("4", "4"), ("5", "5"), ("6", "6")])
tor_opts = (
[("", ""), ("[]", "trivial")] +
[disp_tor(tuple(t)) for t in ECNF_stats().torsion_counts if t])
torsion_structure = SelectBox(name="torsion_structure",
label="Torsion structure",
knowl="ec.torsion_subgroup",
options=tor_opts)
isodeg = TextBox(name="isodeg",
label="Cyclic isogeny degree",
knowl="ec.isogeny",
example="16")
count = CountBox()
self.browse_array = [[jinv], [field, bf_deg],
[conductor_norm, include_base_change],
[rank, include_Q_curves],
[torsion, torsion_structure],
[cm_disc, include_cm], [isodeg, one], [count]]
self.refine_array = [[
field, bf_deg, conductor_norm, jinv, include_base_change
], [include_Q_curves, isodeg, rank, torsion, torsion_structure],
[include_cm, cm_disc, one]]