def make_cm_query(cm_disc_str):
cm_list = parse_ints_to_list_flash(cm_disc_str, "CM discriminant", max_val=None)
for d in cm_list:
if not ((d < 0) and (d % 4 in [0,1])):
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
def parse_dim_args(dim_args, default_dim_args):
res={}
for v in ['k','j']:
arg = dim_args.get(v) if dim_args else None
if not arg:
arg = default_dim_args.get(v) if default_dim_args else None
if arg:
res[v] = parse_ints_to_list_flash(arg, "$"+v+"$")
args={}
if 'k' in res:
if res['k'][-1] > MAXWT:
flash(Markup("Error: <span style='color:black'>$k$</span> cannot exceed <span style='color:black'>%s</span>." % MAXWT), "error")
raise ValueError("dim_args")
if len(res['k']) > MAXWTRANGE:
flash(Markup("Error: range for <span style='color:black'>$k$</span> cannot include more than %s</span> values." % MAXWTRANGE), "error")
raise ValueError("dim_args")
args = {'k_range':res['k']}
if 'j' in res:
if res['j'][-1] > MAXJ:
flash(Markup("Error: <span style='color:black'>$j$</span> cannot exceed <span style='color:black'>%s</span>." % MAXJ), "error")
raise ValueError("dim_args")
args['j_range'] = [j for j in res['j'] if j%2 == 0]
if not args['j_range']:
flash(Markup("Error: <span style='color:black'>$j$</span> should be a nonnegative even integer."), "error")
raise ValueError("dim_args")
return args
def parse_dim_args(dim_args, default_dim_args):
res = {}
for v in ['k', 'j']:
arg = dim_args.get(v) if dim_args else None
if not arg:
arg = default_dim_args.get(v) if default_dim_args else None
if arg:
res[v] = parse_ints_to_list_flash(arg, "$" + v + "$")
args = {}
if 'k' in res:
if res['k'][-1] > MAXWT:
flash_error(
"<span style='color:black'>$k$</span> cannot exceed %s.",
MAXWT)
raise ValueError("dim_args")
if len(res['k']) > MAXWTRANGE:
flash_error(
"range for <span style='color:black'>$k$</span> cannot include more than %s values.",
MAXWTRANGE)
raise ValueError("dim_args")
args = {'k_range': res['k']}
if 'j' in res:
if res['j'][-1] > MAXJ:
flash_error(
"<span style='color:black'>$j$</span> cannot exceed %s.", MAXJ)
raise ValueError("dim_args")
args['j_range'] = [j for j in res['j'] if j % 2 == 0]
if not args['j_range']:
flash_error(
"<span style='color:black'>$j$</span> should be a nonnegative even integer."
)
raise ValueError("dim_args")
return args
def search(info):
""" query processing for Sato-Tate groups -- returns rendered results page """
if 'jump' in info:
return redirect(url_for('.by_label', label=info['jump']), 301)
if 'label' in info:
return redirect(url_for('.by_label', label=info['label']), 301)
search_type = info.get("search_type", info.get("hst", "List"))
template_kwds = {
'bread': [('Sato-Tate Groups', url_for('.index')),
('Search Results', '.')],
'credit': credit_string,
'learnmore': learnmore_list()
}
title = 'Sato-Tate Group Search Results'
err_title = 'Sato-Tate Groups Search Input Error'
count = parse_count(info, 50)
start = parse_start(info)
# if user clicked refine search always restart at 0
if 'refine' in info:
start = 0
ratonly = (info.get('include_irrational', 'no').strip().lower() == 'no')
if search_type == "Random" and not ratonly:
info['err'] = err = 'Cannot select random irrational Sato-Tate group'
flash_error(err)
return render_template('st_results.html',
info=info,
title=err_title,
**template_kwds)
query = {'rational': True} if ratonly else {}
try:
parse_ints(info, query, 'weight', 'weight')
if 'weight' in query:
weight_list = parse_ints_to_list_flash(info.get('weight'),
'weight')
parse_ints(info, query, 'degree', 'degree')
if 'degree' in query:
degree_list = parse_ints_to_list_flash(info.get('degree'),
'degree')
if info.get('identity_component'):
query['identity_component'] = info['identity_component']
parse_ints(info, query, 'components', 'components')
if 'components' in query:
components_list = parse_ints_to_list_flash(info.get('components'),
'components')
parse_rational(info, query, 'trace_zero_density', 'trace zero density')
except ValueError as err:
info['err'] = str(err)
return render_template('st_results.html',
info=info,
title=err_title,
**template_kwds)
# Check mu(n) groups first (these are not stored in the database)
results = []
if ((not 'weight' in query or 0 in weight_list)
and (not 'degree' in query or 1 in degree_list)
and (not 'identity_component' in query
or query['identity_component'] == 'SO(1)')
and (not 'trace_zero_density' in query
or query['trace_zero_density'] == '0')):
if not 'components' in query:
components_list = range(1, 3 if ratonly else start + count + 1)
elif ratonly:
components_list = [n for n in range(1, 3) if n in components_list]
nres = len(
components_list) if 'components' in query or ratonly else INFINITY
if search_type == "Random" and nres > 0:
# Need to return mu(1) and mu(2) sometimes
otherlen = db.gps_sato_tate.count(query)
r = ZZ.random_element(nres + otherlen)
if r < nres:
return redirect(
url_for(".by_label", label="0.1.%d" % components_list[r]),
307)
for n in itertools.islice(components_list, start, start + count):
results.append(mu_info(n))
else:
nres = 0
if 'result_count' in info:
nres += db.gps_sato_tate.count(query)
return jsonify({"nres": str(nres)})
if search_type == "Random":
label = db.gps_sato_tate.random(query, "label")
if label is not None:
return redirect(url_for(".by_label", label=label), 307)
# Now lookup other (rational) ST groups in database
if nres != INFINITY:
start2 = start - nres if start > nres else 0
proj = [
'label', 'weight', 'degree', 'real_dimension',
'identity_component', 'name', 'pretty', 'components',
'component_group', 'trace_zero_density', 'moments'
]
try:
res = db.gps_sato_tate.search(query,
proj,
limit=max(count - len(results), 0),
offset=start2,
info=info)
except QueryCanceledError as err:
ctx = ctx_proc_userdata()
flash_error(
'The search query took longer than expected! Please help us improve by reporting this error <a href="%s" target=_blank>here</a>.'
% ctx['feedbackpage'])
info['err'] = str(err)
return render_template('st_results.html',
info=info,
title=err_title,
**template_kwds)
info['number'] += nres
if start < info['number'] and len(results) < count:
for v in res:
v['identity_component'] = st0_pretty(v['identity_component'])
v['component_group'] = sg_pretty(v['component_group'])
v['trace_moments'] = trace_moments(v['moments'])
results.append(v)
else:
info['number'] = 'infinity'
info['start'] = start
info['count'] = count
info['st0_list'] = st0_list
info['st0_dict'] = st0_dict
info['results'] = results
info['stgroup_url'] = lambda dbc: url_for('.by_label', label=dbc['label'])
return render_template('st_results.html',
info=info,
title=title,
**template_kwds)
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() is not 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', prop_int_pretty(info['weight'])),
('Hecke eigenform', "yes" if info['is_eigenform'] else "no"),
('Field degree', prop_int_pretty(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,
properties=properties,
info=info)
def search(**args):
""" query processing for Sato-Tate groups -- returns rendered results page """
info = to_dict(args)
if 'jump' in info:
return redirect(url_for('.by_label', label=info['jump']), 301)
if 'label' in info:
return redirect(url_for('.by_label', label=info['label']), 301)
template_kwds = {'bread':[('Sato-Tate Groups', url_for('.index')),('Search Results', '.')],
'credit':credit_string,
'learnmore':learnmore_list()}
title = 'Sato-Tate Group Search Results'
err_title = 'Sato-Tate Groups Search Input Error'
count = parse_count(info, 25)
start = parse_start(info)
# if user clicked refine search always restart at 0
if 'refine' in info:
start = 0
ratonly = True if info.get('rational_only','no').strip().lower() == 'yes' else False
query = {'rational':True} if ratonly else {}
try:
parse_ints(info,query,'weight','weight')
if 'weight' in query:
weight_list = parse_ints_to_list_flash(info.get('weight'),'weight')
parse_ints(info,query,'degree','degree')
if 'degree' in query:
degree_list = parse_ints_to_list_flash(info.get('degree'),'degree')
if info.get('identity_component'):
query['identity_component'] = info['identity_component']
parse_ints(info,query,'components','components')
if 'components' in query:
components_list = parse_ints_to_list_flash(info.get('components'), 'components')
parse_rational(info,query,'trace_zero_density','trace zero density')
except ValueError as err:
info['err'] = str(err)
return render_template('st_results.html', info=info, title=err_title, **template_kwds)
# Check mu(n) groups first (these are not stored in the database)
results = []
if (not 'weight' in query or 0 in weight_list) and \
(not 'degree' in query or 1 in degree_list) and \
(not 'identity_component' in query or query['identity_component'] == 'SO(1)') and \
(not 'trace_zero_density' in query or query['trace_zero_density'] == '0'):
if not 'components' in query:
components_list = xrange(1,3 if ratonly else start+count+1)
elif ratonly:
components_list = [n for n in range(1,3) if n in components_list]
nres = len(components_list) if 'components' in query or ratonly else INFINITY
for n in itertools.islice(components_list,start,start+count):
results.append(mu_info(n))
else:
nres = 0
if 'result_count' in info:
nres += db.gps_sato_tate.count(query)
return jsonify({"nres":str(nres)})
# Now lookup other (rational) ST groups in database
if nres != INFINITY:
start2 = start - nres if start > nres else 0
proj = ['label','weight','degree','real_dimension','identity_component','name','pretty','components','component_group','trace_zero_density','moments']
try:
res = db.gps_sato_tate.search(query, proj, limit=max(count - len(results), 0), offset=start2, info=info)
except QueryCanceledError as err:
ctx = ctx_proc_userdata()
flash_error('The search query took longer than expected! Please help us improve by reporting this error <a href="%s" target=_blank>here</a>.' % ctx['feedbackpage'])
info['err'] = str(err)
return render_template('st_results.html', info=info, title=err_title, **template_kwds)
info['number'] += nres
if start < info['number'] and len(results) < count:
for v in res:
v['identity_component'] = st0_pretty(v['identity_component'])
v['component_group'] = sg_pretty(v['component_group'])
v['trace_moments'] = trace_moments(v['moments'])
results.append(v)
else:
info['number'] = 'infinity'
info['start'] = start
info['count'] = count
info['st0_list'] = st0_list
info['st0_dict'] = st0_dict
info['results'] = results
info['stgroup_url'] = lambda dbc: url_for('.by_label', label=dbc['label'])
return render_template('st_results.html', info=info, title=title, **template_kwds)
def search(info):
""" query processing for Sato-Tate groups -- returns rendered results page """
if 'jump' in info:
return redirect(url_for('.by_label', label=info['jump']), 301)
if 'label' in info:
return redirect(url_for('.by_label', label=info['label']), 301)
search_type = info.get("search_type", info.get("hst", "List"))
template_kwds = {
'bread': get_bread("Search results"),
'credit': credit_string,
'learnmore': learnmore_list()
}
title = 'Sato-Tate group search results'
err_title = 'Sato-Tate group search input error'
count = parse_count(info, 50)
start = parse_start(info)
# if user clicked refine search always restart at 0
if 'refine' in info:
start = 0
ratonly = (info.get('include_irrational', 'no').strip().lower() == 'no')
if search_type == "Random" and not ratonly:
info['err'] = err = 'Cannot select random irrational Sato-Tate group'
flash_error(err)
return render_template('st_results.html',
info=info,
title=err_title,
**template_kwds)
query = {'rational': True} if ratonly else {}
try:
parse_ints(info, query, 'weight', 'weight')
if 'weight' in query:
weight_list = parse_ints_to_list_flash(info.get('weight'),
'weight')
parse_ints(info, query, 'degree', 'degree')
if 'degree' in query:
degree_list = parse_ints_to_list_flash(info.get('degree'),
'degree')
if info.get('identity_component'):
query['identity_component'] = info['identity_component']
parse_ints(info, query, 'components', 'components')
def refine_components(current, condition):
if current is None:
return condition
return [x for x in condition if x in current]
# The following are used to constraint which mu(n) will show up in search results
components_list = None
ommitted = set()
if 'components' in query:
components_list = parse_ints_to_list_flash(info.get('components'),
'components')
parse_component_group(info, query)
gps = query.get("component_group")
if gps:
if isinstance(gps, dict):
gps = gps["$in"]
else:
gps = [gps]
if not ratonly:
cyclic_labels = set(
db.gps_small.search({"cyclic": True}, "label"))
irrat = []
for code in gps:
if code in cyclics:
irrat.append(cyclics[code])
elif re.match(cyclicre, code):
irrat.append(int(code[1:]))
elif not ratonly and code in cyclic_labels:
irrat.append(int(code.split(".")[0]))
components_list = refine_components(components_list, irrat)
parse_rational(info, query, 'trace_zero_density', 'trace zero density')
parse_ints(info, query, 'second_trace_moment')
parse_ints(info, query, 'fourth_trace_moment')
for name, ones in [('second_trace_moment', [1, 2]),
('fourth_trace_moment', [1, 2, 4])]:
if name in query:
# E(x^2) for mu(1) and mu(2) are 1; others are 0
# E(x^4) for mu(1), mu(2) and mu(4) are 1; others are 0
E = parse_ints_to_list_flash(info.get(name),
name.replace("_", " "))
if 0 not in E:
components_list = refine_components(components_list, ones)
if 1 not in E:
ommitted.update(ones)
parse_ints(info, query, 'first_a2_moment')
parse_bool(info, query, 'maximal')
if "first_a2_moment" in query or query.get("maximal"):
# mu(n) do not have a2 moments and none are maximal
components_list = []
except ValueError as err:
info['err'] = str(err)
return render_template('st_results.html',
info=info,
title=err_title,
**template_kwds)
# Check mu(n) groups first (these are not stored in the database)
results = []
if ((not 'weight' in query or 0 in weight_list)
and (not 'degree' in query or 1 in degree_list)
and (not 'identity_component' in query
or query['identity_component'] == 'SO(1)')
and (not 'trace_zero_density' in query
or query['trace_zero_density'] == '0')):
nres = None
if components_list is None:
components_list = range(
1, 3 if ratonly else (start + count + 1 + len(ommitted)))
nres = None if ratonly else INFINITY
elif ratonly:
components_list = [n for n in range(1, 3) if n in components_list]
components_list = [n for n in components_list if n not in ommitted]
if nres is None:
nres = len(components_list)
if search_type == "Random" and nres > 0:
# Need to return mu(1), mu(2) or mu(4) sometimes
otherlen = db.gps_sato_tate.count(query)
r = ZZ.random_element(nres + otherlen)
if r < nres:
return redirect(
url_for(".by_label", label="0.1.%d" % components_list[r]),
307)
for n in itertools.islice(components_list, start, start + count):
results.append(mu_info(n))
else:
nres = 0
if 'result_count' in info:
nres += db.gps_sato_tate.count(query)
return jsonify({"nres": str(nres)})
if search_type == "Random":
label = db.gps_sato_tate.random(query, "label")
if label is not None:
return redirect(url_for(".by_label", label=label), 307)
# Now lookup other (rational) ST groups in database
if nres != INFINITY:
start2 = start - nres if start > nres else 0
proj = [
'label', 'weight', 'degree', 'real_dimension',
'identity_component', 'name', 'pretty', 'components',
'component_group', 'trace_zero_density', 'moments'
]
try:
res = db.gps_sato_tate.search(query,
proj,
limit=max(count - len(results), 0),
offset=start2,
info=info)
except QueryCanceledError as err:
ctx = ctx_proc_userdata()
flash_error(
'The search query took longer than expected! Please help us improve by reporting this error <a href="%s" target=_blank>here</a>.'
% ctx['feedbackpage'])
info['err'] = str(err)
return render_template('st_results.html',
info=info,
title=err_title,
**template_kwds)
info['number'] += nres
if start < info['number'] and len(results) < count:
for v in res:
v['identity_component'] = st0_pretty(v['identity_component'])
v['component_group'] = sg_pretty(v['component_group'])
v['trace_moments'] = trace_moments(v['moments'])
results.append(v)
else:
info['number'] = 'infinity'
info['start'] = start
info['count'] = count
info['st0_list'] = st0_list
info['st0_dict'] = st0_dict
info['results'] = results
info['stgroup_url'] = lambda dbc: url_for('.by_label', label=dbc['label'])
return render_template('st_results.html',
info=info,
title=title,
**template_kwds)
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)
