def render_isogeny_class(iso_class):
info = {}
credit = 'John Cremona'
label = iso_class
C = base.getDBConnection()
data = C.ellcurves.isogeny.find_one({'label': label})
if data is None:
return "No such isogeny class"
ainvs = [int(a) for a in data['ainvs_for_optimal_curve']]
E = EllipticCurve(ainvs)
info = {'label': label}
info['optimal_ainvs'] = ainvs
if 'imag' in data:
info['imag'] = data['imag']
if 'real' in data:
info['real'] = data['real']
info['rank'] = data['rank']
info['isogeny_matrix'] = latex(matrix(eval(data['isogeny_matrix'])))
info['modular_degree'] = data['degree']
#info['f'] = ajax_more(E.q_eigenform, 10, 20, 50, 100, 250)
info['f'] = web_latex(E.q_eigenform(10))
G = E.isogeny_graph()
n = G.num_verts()
G.relabel(range(1, n + 1)) # proper cremona labels...
info['graph_img'] = image_src(G.plot(edge_labels=True))
curves = data['label_of_curves_in_the_class']
info['curves'] = list(curves)
info['download_qexp_url'] = url_for('download_qexp',
limit=100,
ainvs=','.join([str(a)
for a in ainvs]))
info['download_all_url'] = url_for('download_all', label=str(label))
friends = [('Elliptic Curve %s' % l, "/EllipticCurve/Q/%s" % l)
for l in data['label_of_curves_in_the_class']]
friends.append(('Quadratic Twist', "/quadratic_twists/%s" % (label)))
friends.append(('Modular Form',
url_for("emf.render_classical_modular_form_from_label",
label="%s" % (label))))
info['friends'] = friends
t = "Elliptic Curve Isogeny Class %s" % info['label']
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),
('isogeny class %s' % info['label'], ' ')]
return render_template("elliptic_curve/iso_class.html",
info=info,
bread=bread,
credit=credit,
title=t)
def padic_data():
info = {}
label = request.args['label']
p = int(request.args['p'])
info['p'] = p
N, iso, number = cremona_label_regex.match(label).groups()
#print N, iso, number
if request.args['rank'] == '0':
info['reg'] = 1
elif number == '1':
C = base.getDBConnection()
data = C.ellcurves.padic_db.find_one({'label': N + iso, 'p': p})
info['data'] = data
if data is None:
info['reg'] = 'no data'
else:
reg = sage.all.Qp(p, data['prec'])(int(data['unit'])) * sage.all.Integer(p)**int(data['val'])
reg = reg.add_bigoh(min(data['prec'], data['prec'] + data['val']))
info['reg'] = web_latex(reg)
else:
info['reg'] = "no data"
return render_template("elliptic_curve/elliptic_curve_padic.html", info = info)
def render_isogeny_class(iso_class):
info = {}
credit = 'John Cremona'
label=iso_class
C = base.getDBConnection()
data = C.ellcurves.isogeny.find_one({'label': label})
if data is None:
return "No such isogeny class"
ainvs = [int(a) for a in data['ainvs_for_optimal_curve']]
E = EllipticCurve(ainvs)
info = {'label': label}
info['optimal_ainvs'] = ainvs
if 'imag' in data:
info['imag']=data['imag']
if 'real' in data:
info['real']=data['real']
info['rank'] = data['rank']
info['isogeny_matrix']=latex(matrix(eval(data['isogeny_matrix'])))
info['modular_degree']=data['degree']
#info['f'] = ajax_more(E.q_eigenform, 10, 20, 50, 100, 250)
info['f'] = web_latex(E.q_eigenform(10))
G = E.isogeny_graph(); n = G.num_verts()
G.relabel(range(1,n+1)) # proper cremona labels...
info['graph_img'] = image_src(G.plot(edge_labels=True))
curves = data['label_of_curves_in_the_class']
info['curves'] = list(curves)
info['download_qexp_url'] = url_for('download_qexp', limit=100, ainvs=','.join([str(a) for a in ainvs]))
info['download_all_url'] = url_for('download_all', label=str(label))
friends=[('Elliptic Curve %s' % l , "/EllipticCurve/Q/%s" % l) for l in data['label_of_curves_in_the_class']]
friends.append(('Quadratic Twist', "/quadratic_twists/%s" % (label)))
friends.append(('Modular Form', url_for("emf.render_classical_modular_form_from_label",label="%s" %(label))))
info['friends'] = friends
t= "Elliptic Curve Isogeny Class %s" % info['label']
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),('isogeny class %s' %info['label'],' ')]
return render_template("elliptic_curve/iso_class.html", info = info,bread=bread, credit=credit,title = t)
def padic_data():
info = {}
label = request.args['label']
p = int(request.args['p'])
info['p'] = p
N, iso, number = cremona_label_regex.match(label).groups()
#print N, iso, number
if request.args['rank'] == '0':
info['reg'] = 1
elif number == '1':
C = base.getDBConnection()
data = C.ellcurves.padic_db.find_one({'label': N + iso, 'p': p})
info['data'] = data
if data is None:
info['reg'] = 'no data'
else:
reg = sage.all.Qp(p, data['prec'])(int(
data['unit'])) * sage.all.Integer(p)**int(data['val'])
reg = reg.add_bigoh(min(data['prec'], data['prec'] + data['val']))
info['reg'] = web_latex(reg)
else:
info['reg'] = "no data"
return render_template("elliptic_curve/elliptic_curve_padic.html",
info=info)
def render_field_webpage(args):
data = None
if 'label' in args:
label = str(args['label'])
import base
C = base.getDBConnection()
data = C.numberfields.fields.find_one({'label': label})
if data is None:
return "No such field: " + label + " in the database"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
K = coeff_to_nf(data['coefficients'])
D = data['discriminant']
h = data['class_number']
data['galois_group'] = str(data['galois_group'][3])
data['class_group_invs'] = data['class_group']
if data['class_group_invs'] == []:
data['class_group_invs'] = 'Trivial'
data['class_group'] = str(AbelianGroup(data['class_group']))
sig = data['signature']
D = ZZ(data['discriminant'])
ram_primes = D.prime_factors()
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
Gorder, Gsign, Gab = GG_data(data['galois_group'])
if Gab:
Gab = 'abelian'
else:
Gab = 'non-abelian'
unit_rank = sig[0] + sig[1] - 1
if unit_rank == 0:
reg = 1
else:
reg = K.regulator()
UK = K.unit_group()
info.update(data)
info.update({
'label':
field_pretty(label),
'polynomial':
web_latex(K.defining_polynomial()),
'ram_primes':
ram_primes,
'integral_basis':
web_latex(K.integral_basis()),
'regulator':
web_latex(reg),
'unit_rank':
unit_rank,
'root_of_unity':
web_latex(UK.torsion_generator()),
'fund_units':
', '.join([web_latex(u) for u in UK.fundamental_units()]),
'Gorder':
Gorder,
'Gsign':
Gsign,
'Gab':
Gab
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
# info['friends'] = [('L-function', '/')]
info['friends'] = [('L-function', "/L/NumberField/%s" % label)]
info['learnmore'] = [
('Number Field labels', url_for("render_labels_page")),
('Galois group labels', url_for("render_groups_page")),
('Discriminant ranges', url_for("render_discriminants_page"))
]
bread = [('Number Fields', url_for("number_field_render_webpage")),
('%s' % info['label'], ' ')]
t = "Number Field %s" % info['label']
properties = ['<br>']
properties.extend('<table>')
properties.extend(
'<tr><td align=left><b>Degree:</b><td align=left> %s</td>' %
data['degree'])
properties.extend(
'<tr><td align=left><b>Signature:</b><td align=left>%s</td>' %
data['signature'])
properties.extend(
'<tr><td align=left><b>Discriminant:</b><td align=left>%s</td>' %
data['discriminant'])
if npr == 1:
properties.extend(
'<tr><td align=left><b>Ramified prime:</b><td align=left>%s</td>' %
ram_primes)
else:
if npr == 0:
properties.extend(
'<tr><td align=left><b>Ramified primes:</b><td align=left>%s</td>'
% "None")
else:
properties.extend(
'<tr><td align=left><b>Ramified primes:</b><td align=left>%s</td>'
% ram_primes)
properties.extend(
'<tr><td align=left><b>Class number:</b><td align=left>%s</td>' %
data['class_number'])
properties.extend(
'<tr><td align=left><b>Class group:</b><td align=left>%s</td>' %
data['class_group_invs'])
properties.extend(
'<tr><td align=left><b>Galois group:</b><td align=left>%s</td>' %
data['galois_group'])
properties.extend('</table>')
del info['_id']
return render_template("number_field/number_field.html",
properties=properties,
credit=NF_credit,
title=t,
bread=bread,
friends=info.pop('friends'),
info=info)
def render_curve_webpage_by_label(label):
C = base.getDBConnection()
data = C.ellcurves.curves.find_one({'label': label})
if data is None:
return "No such curve"
info = {}
ainvs = [int(a) for a in data['ainvs']]
E = EllipticCurve(ainvs)
label = data['label']
N = ZZ(data['conductor'])
iso_class = data['iso']
rank = data['rank']
j_invariant = E.j_invariant()
#plot=E.plot()
discriminant = E.discriminant()
xintpoints_projective = [
E.lift_x(x)
for x in xintegral_point(data['x-coordinates_of_integral_points'])
]
xintpoints = proj_to_aff(xintpoints_projective)
G = E.torsion_subgroup().gens()
if 'gens' in data:
generator = parse_gens(data['gens'])
if len(G) == 0:
tor_struct = 'Trivial'
tor_group = 'Trivial'
else:
tor_group = ' \\times '.join(
['\mathbb{Z}/{%s}\mathbb{Z}' % a.order() for a in G])
if 'torsion_structure' in data:
info['tor_structure'] = ' \\times '.join([
'\mathbb{Z}/{%s}\mathbb{Z}' % int(a)
for a in data['torsion_structure']
])
else:
info['tor_structure'] = tor_group
info.update(data)
if rank >= 2:
lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
elif rank == 1:
lder_tex = "L%s(E,1)" % ("'" * rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info.update({
'conductor':
N,
'disc_factor':
latex(discriminant.factor()),
'j_invar_factor':
latex(j_invariant.factor()),
'label':
label,
'isogeny':
iso_class,
'equation':
web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
'f':
web_latex(E.q_eigenform(10)),
'generators':
','.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
'lder':
lder_tex,
'p_adic_primes': [
p for p in sage.all.prime_range(5, 100)
if E.is_ordinary(p) and not p.divides(N)
],
'ainvs':
format_ainvs(data['ainvs']),
'tamagawa_numbers':
r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()),
'cond_factor':
latex(N.factor()),
'xintegral_points':
','.join(web_latex(i_p) for i_p in xintpoints),
'tor_gens':
','.join(web_latex(eval(g)) for g in data['torsion_generators'])
if 'torsion_generators' in data else list(G)
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', url_for("not_yet_implemented"))]
info['friends'] = [('Isogeny class', "/EllipticCurve/Q/%s" % iso_class),
('Modular Form',
url_for("emf.render_elliptic_modular_form_from_label",
label="%s" % (iso_class))),
('L-function', "/L/EllipticCurve/Q/%s" % label)]
info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
#info['plot'] = image_src(plot)
info['plot'] = url_for('plot_ec', label=label)
info['iso_class'] = data['iso']
info['download_qexp_url'] = url_for('download_qexp',
limit=100,
ainvs=','.join([str(a)
for a in ainvs]))
properties2 = [('Label', '%s' % label),
(None, '<img src="%s" width="200" height="150"/>' %
url_for('plot_ec', label=label)),
('Conductor', '\(%s\)' % N),
('Discriminant', '\(%s\)' % discriminant),
('j-invariant', '\(%s\)' % j_invariant),
('Rank', '\(%s\)' % rank),
('Torsion Structure', '\(%s\)' % tor_group)]
#properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = 'John Cremona'
t = "Elliptic Curve %s" % info['label']
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),
('Elliptic curves %s' % info['label'], ' ')]
return render_template("elliptic_curve/elliptic_curve.html",
info=info,
properties2=properties2,
credit=credit,
bread=bread,
title=t)
def render_field_webpage(args):
data = None
if 'label' in args:
label = str(args['label'])
import base
C = base.getDBConnection()
data = C.numberfields.fields.find_one({'label': label})
if data is None:
return "No such field: " + label + " in the database"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
K = coeff_to_nf(data['coefficients'])
D = data['discriminant']
h = data['class_number']
data['galois_group'] = str(data['galois_group'][3])
data['class_group_invs'] = data['class_group']
if data['class_group_invs']==[]:
data['class_group_invs']='Trivial'
data['class_group'] = str(AbelianGroup(data['class_group']))
sig = data['signature']
D = ZZ(data['discriminant'])
ram_primes = D.prime_factors()
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
Gorder,Gsign,Gab = GG_data(data['galois_group'])
if Gab:
Gab='abelian'
else:
Gab='non-abelian'
unit_rank = sig[0]+sig[1]-1
if unit_rank==0:
reg = 1
else:
reg = K.regulator()
UK = K.unit_group()
info.update(data)
info.update({
'label': field_pretty(label),
'polynomial': web_latex(K.defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': web_latex(K.integral_basis()),
'regulator': web_latex(reg),
'unit_rank': unit_rank,
'root_of_unity': web_latex(UK.torsion_generator()),
'fund_units': ', '.join([web_latex(u) for u in UK.fundamental_units()]),
'Gorder': Gorder, 'Gsign': Gsign, 'Gab': Gab
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
# info['friends'] = [('L-function', '/')]
info['friends'] = [('L-function', "/L/NumberField/%s" % label)]
info['learnmore'] = [('Number Field labels', url_for("render_labels_page")), ('Galois group labels',url_for("render_groups_page")), ('Discriminant ranges',url_for("render_discriminants_page"))]
bread = [('Number Fields', url_for("number_field_render_webpage")),('%s'%info['label'],' ')]
t = "Number Field %s" % info['label']
properties = ['<br>']
properties.extend('<table>')
properties.extend('<tr><td align=left><b>Degree:</b><td align=left> %s</td>'%data['degree'])
properties.extend('<tr><td align=left><b>Signature:</b><td align=left>%s</td>'%data['signature'])
properties.extend('<tr><td align=left><b>Discriminant:</b><td align=left>%s</td>'%data['discriminant'])
if npr==1:
properties.extend('<tr><td align=left><b>Ramified prime:</b><td align=left>%s</td>'%ram_primes)
else:
if npr==0:
properties.extend('<tr><td align=left><b>Ramified primes:</b><td align=left>%s</td>'%"None")
else:
properties.extend('<tr><td align=left><b>Ramified primes:</b><td align=left>%s</td>'%ram_primes)
properties.extend('<tr><td align=left><b>Class number:</b><td align=left>%s</td>'%data['class_number'])
properties.extend('<tr><td align=left><b>Class group:</b><td align=left>%s</td>'%data['class_group_invs'])
properties.extend('<tr><td align=left><b>Galois group:</b><td align=left>%s</td>'%data['galois_group'])
properties.extend('</table>')
del info['_id']
return render_template("number_field/number_field.html", properties=properties, credit=NF_credit, title = t, bread=bread, friends=info.pop('friends'), info=info )
def render_curve_webpage_by_label(label):
C = base.getDBConnection()
data = C.ellcurves.curves.find_one({'label': label})
if data is None:
return "No such curve"
info = {}
ainvs = [int(a) for a in data['ainvs']]
E = EllipticCurve(ainvs)
label=data['label']
N = ZZ(data['conductor'])
iso_class = data['iso']
rank = data['rank']
j_invariant=E.j_invariant()
#plot=E.plot()
discriminant=E.discriminant()
xintpoints_projective=[E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points'])]
xintpoints=proj_to_aff(xintpoints_projective)
G = E.torsion_subgroup().gens()
if 'gens' in data:
generator=parse_gens(data['gens'])
if len(G) == 0:
tor_struct = 'Trivial'
tor_group='Trivial'
else:
tor_group=' \\times '.join(['\mathbb{Z}/{%s}\mathbb{Z}'%a.order() for a in G])
if 'torsion_structure' in data:
info['tor_structure']= ' \\times '.join(['\mathbb{Z}/{%s}\mathbb{Z}'% int(a) for a in data['torsion_structure']])
else:
info['tor_structure'] = tor_group
info.update(data)
if rank >=2:
lder_tex = "L%s(E,1)" % ("^{("+str(rank)+")}")
elif rank ==1:
lder_tex = "L%s(E,1)" % ("'"*rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info.update({
'conductor': N,
'disc_factor': latex(discriminant.factor()),
'j_invar_factor':latex(j_invariant.factor()),
'label': label,
'isogeny':iso_class,
'equation': web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
'f' : web_latex(E.q_eigenform(10)),
'generators':','.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
'lder' : lder_tex,
'p_adic_primes': [p for p in sage.all.prime_range(5,100) if E.is_ordinary(p) and not p.divides(N)],
'ainvs': format_ainvs(data['ainvs']),
'tamagawa_numbers': r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()),
'cond_factor':latex(N.factor()),
'xintegral_points':','.join(web_latex(i_p) for i_p in xintpoints),
'tor_gens':','.join(web_latex(eval(g)) for g in data['torsion_generators']) if 'torsion_generators' in data else list(G)
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', url_for("not_yet_implemented"))]
info['friends'] = [('Isogeny class', "/EllipticCurve/Q/%s" % iso_class),
('Modular Form', url_for("emf.render_elliptic_modular_form_from_label",label="%s" %(iso_class))),
('L-function', "/L/EllipticCurve/Q/%s" % label)]
info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
#info['plot'] = image_src(plot)
info['plot'] = url_for('plot_ec', label=label)
info['iso_class'] = data['iso']
info['download_qexp_url'] = url_for('download_qexp', limit=100, ainvs=','.join([str(a) for a in ainvs]))
properties2 = [('Label', '%s' % label),
(None, '<img src="%s" width="200" height="150"/>' % url_for('plot_ec', label=label) ),
('Conductor', '\(%s\)' % N),
('Discriminant', '\(%s\)' % discriminant),
('j-invariant', '\(%s\)' % j_invariant),
('Rank', '\(%s\)' % rank),
('Torsion Structure', '\(%s\)' % tor_group)
]
#properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = 'John Cremona'
t = "Elliptic Curve %s" % info['label']
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),('Elliptic curves %s' %info['label'],' ')]
return render_template("elliptic_curve/elliptic_curve.html",
info=info, properties2=properties2, credit=credit,bread=bread, title = t)
