def lattice_search_isometric(res, info, query):
"""
We check for isometric lattices if the user enters a valid gram matrix
but not one stored in the database
This may become slow in the future: at the moment we compare against
a list of stored matrices with same dimension and determinant
(just compare with respect to dimension is slow)
"""
if info['number'] == 0 and info.get('gram'):
A = query['gram']
n = len(A[0])
d = matrix(A).determinant()
for gram in db.lat_lattices.search({'dim': n, 'det': int(d)}, 'gram'):
if isom(A, gram):
query['gram'] = gram
proj = lattice_search_projection
count = parse_count(info)
start = parse_start(info)
res = db.lat_lattices.search(query, proj, limit=count, offset=start, info=info)
break
for v in res:
v['min'] = v.pop('minimum')
return res
def lattice_search_isometric(res, info, query):
"""
We check for isometric lattices if the user enters a valid gram matrix
but not one stored in the database
This may become slow in the future: at the moment we compare against
a list of stored matrices with same dimension and determinant
(just compare with respect to dimension is slow)
"""
if info['number'] == 0 and info.get('gram'):
A = query['gram']
n = len(A[0])
d = matrix(A).determinant()
for gram in db.lat_lattices.search({'dim': n, 'det': int(d)}, 'gram'):
if isom(A, gram):
query['gram'] = gram
proj = lattice_search_projection
count = parse_count(info)
start = parse_start(info)
res = db.lat_lattices.search(query, proj, limit=count, offset=start, info=info)
break
for v in res:
v['min'] = v.pop('minimum')
return res
def bmf_field_dim_table(**args):
argsdict = to_dict(args)
argsdict.update(to_dict(request.args))
gl_or_sl = argsdict['gl_or_sl']
field_label=argsdict['field_label']
field_label = nf_string_to_label(field_label)
start = parse_start(argsdict)
info={}
info['gl_or_sl'] = gl_or_sl
# level_flag controls whether to list all levels ('all'), only
# those with positive cuspidal dimension ('cusp'), or only those
# with positive new dimension ('new'). Default is 'cusp'.
level_flag = argsdict.get('level_flag', 'cusp')
info['level_flag'] = level_flag
count = parse_count(argsdict, 50)
pretty_field_label = field_pretty(field_label)
bread = [('Bianchi Modular Forms', url_for(".index")), (
pretty_field_label, ' ')]
properties = []
query = {}
query['field_label'] = field_label
if gl_or_sl=='gl2_dims':
info['group'] = 'GL(2)'
info['bgroup'] = '\GL(2,\mathcal{O}_K)'
else:
info['group'] = 'SL(2)'
info['bgroup'] = '\SL(2,\mathcal{O}_K)'
if level_flag == 'all':
query[gl_or_sl] = {'$exists': True}
else:
# Only get records where the cuspdial/new dimension is positive for some weight
totaldim = gl_or_sl.replace('dims', level_flag) + '_totaldim'
query[totaldim] = {'$gt': 0}
t = ' '.join(['Dimensions of Spaces of {} Bianchi Modular Forms over'.format(info['group']), pretty_field_label])
data = list(db.bmf_dims.search(query, limit=count, offset=start, info=info))
nres = info['number']
if not info['exact_count']:
info['number'] = nres = db.bmf_dims.count(query)
info['exact_count'] = True
if nres > count or start != 0:
info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'Displaying all %s levels,' % nres
info['field'] = field_label
info['field_pretty'] = pretty_field_label
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
weights = set()
for dat in data:
weights = weights.union(set(dat[gl_or_sl].keys()))
weights = list([int(w) for w in weights])
weights.sort()
info['weights'] = weights
info['nweights'] = len(weights)
data.sort(key = lambda x: [int(y) for y in x['level_label'].split(".")])
dims = {}
for dat in data:
dims[dat['level_label']] = d = {}
for w in weights:
sw = str(w)
if sw in dat[gl_or_sl]:
d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
'n': dat[gl_or_sl][sw]['new_dim']}
else:
d[w] = {'d': '?', 'n': '?'}
info['nlevels'] = len(data)
dimtable = [{'level_label': dat['level_label'],
'level_norm': dat['level_norm'],
'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
'dims': dims[dat['level_label']]} for dat in data]
info['dimtable'] = dimtable
return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
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 bmf_field_dim_table(**args):
argsdict = to_dict(args)
argsdict.update(to_dict(request.args))
gl_or_sl = argsdict['gl_or_sl']
field_label=argsdict['field_label']
field_label = nf_string_to_label(field_label)
start = parse_start(argsdict)
info={}
info['gl_or_sl'] = gl_or_sl
# level_flag controls whether to list all levels ('all'), only
# those with positive cuspidal dimension ('cusp'), or only those
# with positive new dimension ('new'). Default is 'cusp'.
level_flag = argsdict.get('level_flag', 'cusp')
info['level_flag'] = level_flag
count = parse_count(argsdict, 50)
pretty_field_label = field_pretty(field_label)
bread = [('Bianchi Modular Forms', url_for(".index")), (
pretty_field_label, ' ')]
properties = []
if gl_or_sl=='gl2_dims':
info['group'] = 'GL(2)'
info['bgroup'] = '\GL(2,\mathcal{O}_K)'
else:
info['group'] = 'SL(2)'
info['bgroup'] = '\SL(2,\mathcal{O}_K)'
t = ' '.join(['Dimensions of Spaces of {} Bianchi Modular Forms over'.format(info['group']), pretty_field_label])
query = {}
query['field_label'] = field_label
query[gl_or_sl] = {'$exists': True}
data = db.bmf_dims.search(query, limit=count, offset=start, info=info)
nres = info['number']
if nres > count or start != 0:
info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'Displaying all %s levels,' % nres
# convert data to a list and eliminate levels where all
# new/cuspidal dimensions are 0. (This could be done at the
# search stage, but that requires adding new fields to each
# record.)
def filter(dat, flag):
dat1 = dat[gl_or_sl]
return any([int(dat1[w][flag])>0 for w in dat1])
flag = 'cuspidal_dim' if level_flag=='cusp' else 'new_dim'
data = [dat for dat in data if level_flag == 'all' or filter(dat, flag)]
info['field'] = field_label
info['field_pretty'] = pretty_field_label
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
weights = set()
for dat in data:
weights = weights.union(set(dat[gl_or_sl].keys()))
weights = list([int(w) for w in weights])
weights.sort()
info['weights'] = weights
info['nweights'] = len(weights)
data.sort(key = lambda x: [int(y) for y in x['level_label'].split(".")])
dims = {}
for dat in data:
dims[dat['level_label']] = d = {}
for w in weights:
sw = str(w)
if sw in dat[gl_or_sl]:
d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
'n': dat[gl_or_sl][sw]['new_dim']}
else:
d[w] = {'d': '?', 'n': '?'}
info['nlevels'] = len(data)
dimtable = [{'level_label': dat['level_label'],
'level_norm': dat['level_norm'],
'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
'dims': dims[dat['level_label']]} for dat in data]
info['dimtable'] = dimtable
return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
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)
