def number_field_render_webpage():
args = request.args
sig_list = sum([[[d - 2 * r2, r2] for r2 in range(
1 + (d // 2))] for d in range(1, 7)], []) + sum([[[d, 0]] for d in range(7, 11)], [])
sig_list = sig_list[:10]
if len(args) == 0:
init_nf_count()
discriminant_list_endpoints = [-10000, -1000, -100, 0, 100, 1000, 10000]
discriminant_list = ["%s..%s" % (start, end - 1) for start, end in zip(
discriminant_list_endpoints[:-1], discriminant_list_endpoints[1:])]
info = {
'degree_list': range(1, max_deg + 1),
'signature_list': sig_list,
'class_number_list': range(1, 6) + ['6..10'],
'count': '20',
'nfields': comma(nfields),
'maxdeg': max_deg,
'discriminant_list': discriminant_list
}
t = 'Global Number Fields'
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
info['learnmore'] = [(Completename, url_for(".render_discriminants_page")), ('How data was computed', url_for(".how_computed_page")), ('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), ('Quadratic imaginary class groups', url_for(".render_class_group_data"))]
return render_template("number_field_all.html", info=info, credit=NF_credit, title=t, bread=bread, learnmore=info.pop('learnmore'))
else:
return number_field_search(**args)
def render_groups_page():
info = {}
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page")) ]
t = 'Galois group labels'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Galois group labels', ' ')]
C = base.getDBConnection()
return render_template("galois_groups.html", al=aliastable(C), info=info, credit=NF_credit, title=t, bread=bread, learnmore=info.pop('learnmore'))
def render_groups_page():
info = {}
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page"))]
t = 'Galois group labels'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Galois group labels', ' ')]
C = base.getDBConnection()
return render_template("galois_groups.html", al=aliastable(C), info=info, credit=NF_credit, title=t, bread=bread, learnmore=info.pop('learnmore'))
def number_field_render_webpage():
args = request.args
sig_list = sum([[[d - 2 * r2, r2] for r2 in range(1 + (d // 2))] for d in range(1, 7)], []) + sum(
[[[d, 0]] for d in range(7, 11)], []
)
sig_list = sig_list[:10]
if len(args) == 0:
init_nf_count()
discriminant_list_endpoints = [-10000, -1000, -100, 0, 100, 1000, 10000]
discriminant_list = [
"%s..%s" % (start, end - 1)
for start, end in zip(discriminant_list_endpoints[:-1], discriminant_list_endpoints[1:])
]
info = {
"degree_list": range(1, max_deg + 1),
"signature_list": sig_list,
"class_number_list": range(1, 6) + ["6..10"],
"count": "20",
"nfields": comma(nfields),
"maxdeg": max_deg,
"discriminant_list": discriminant_list,
}
t = "Global Number Fields"
bread = [("Global Number Fields", url_for(".number_field_render_webpage"))]
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
]
return render_template(
"number_field_all.html", info=info, credit=NF_credit, title=t, bread=bread
) # , learnmore=info.pop('learnmore'))
else:
return number_field_search(**args)
def number_field_render_webpage():
args = request.args
sig_list = sum([[[d - 2 * r2, r2] for r2 in range(
1 + (d // 2))] for d in range(1, 7)], []) + sum([[[d, 0]] for d in range(7, 11)], [])
sig_list = sig_list[:10]
if len(args) == 0:
init_nf_count()
discriminant_list_endpoints = [-10000, -1000, -100, 0, 100, 1000, 10000]
discriminant_list = ["%s..%s" % (start, end - 1) for start, end in zip(
discriminant_list_endpoints[:-1], discriminant_list_endpoints[1:])]
info = {
'degree_list': range(1, max_deg + 1),
'signature_list': sig_list,
'class_number_list': range(1, 6) + ['6..10'],
'count': '20',
'nfields': comma(nfields),
'maxdeg': max_deg,
'discriminant_list': discriminant_list
}
t = 'Global Number Fields'
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page"))]
return render_template("number_field_all.html", info=info, credit=NF_credit, title=t, bread=bread) # , learnmore=info.pop('learnmore'))
else:
return number_field_search(**args)
def how_computed_page():
info = {}
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")) ]
t = 'How Number Field Data was Computed'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Galois group labels', ' ')]
return render_template("single.html", kid='dq.nf.howcomputed',
credit=NF_credit, title=t, bread=bread, learnmore=info.pop('learnmore'))
def how_computed_page():
info = {}
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")) ]
t = 'How Number Field Data was Computed'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Galois group labels', ' ')]
return render_template("single.html", kid='dq.nf.howcomputed',
credit=NF_credit, title=t, bread=bread, learnmore=info.pop('learnmore'))
def how_computed_page():
info = {}
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
]
t = "How Number Field Data was Computed"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), ("Galois group labels", " ")]
return render_template(
"single.html", kid="dq.nf.howcomputed", credit=NF_credit, title=t, bread=bread, learnmore=info.pop("learnmore")
)
def render_discriminants_page():
info = {}
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
]
t = "Completeness of Global Number Field Data"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), (Completename, " ")]
return render_template(
"discriminant_ranges.html", info=info, credit=NF_credit, title=t, bread=bread, learnmore=info.pop("learnmore")
)
def render_class_group_data():
info = to_dict(request.args)
#nf_logger.info('******************* ')
#for k in info.keys():
# nf_logger.info(str(k) + ' ---> ' + str(info[k]))
#nf_logger.info('******************* ')
info['learnmore'] = [
('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page"))
]
t = 'Class Groups of Quadratic Imaginary Fields'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")),
(t, ' ')]
info['message'] = ''
info['filename'] = 'none'
if 'Fetch' in info:
if 'k' in info:
# remove non-digits
k = re.sub(r'\D', '', info['k'])
if k == "":
info['message'] = 'The value of k is either invalid or empty'
return class_group_request_error(info, bread)
k = int(k)
if k > 4095:
info['message'] = 'The value of k is too large'
return class_group_request_error(info, bread)
else:
info['message'] = 'The value of k is missing'
return class_group_request_error(info, bread)
info['filenamebase'] = str(info['filenamebase'])
if info['filenamebase'] in [
'cl3mod8', 'cl7mod8', 'cl4mod16', 'cl8mod16'
]:
filepath = "%s/%s/%s.%d.gz" % (class_group_data_directory,
info['filenamebase'],
info['filenamebase'], k)
if os.path.isfile(filepath) and os.access(filepath, os.R_OK):
return send_file(filepath, as_attachment=True)
else:
info['message'] = 'File not found'
return class_group_request_error(info, bread)
else:
info['message'] = 'Invalid congruence requested'
return class_group_request_error(info, bread)
return render_template("class_group_data.html",
info=info,
credit="A. Mosunov and M. J. Jacobson, Jr.",
title=t,
bread=bread,
learnmore=info.pop('learnmore'))
def render_labels_page():
info = {}
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
("Quadratic imaginary class groups", url_for(".render_class_group_data")),
]
t = "Number field labels"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), ("Number field labels", "")]
return render_template(
"number_field_labels.html", info=info, credit=NF_credit, title=t, bread=bread, learnmore=info.pop("learnmore")
)
def render_class_group_data():
info = to_dict(request.args)
# nf_logger.info('******************* ')
# for k in info.keys():
# nf_logger.info(str(k) + ' ---> ' + str(info[k]))
# nf_logger.info('******************* ')
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
]
t = "Class Groups of Quadratic Imaginary Fields"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), (t, " ")]
info["message"] = ""
info["filename"] = "none"
if "Fetch" in info:
if "k" in info:
# remove non-digits
k = re.sub(r"\D", "", info["k"])
if k == "":
info["message"] = "The value of k is either invalid or empty"
return class_group_request_error(info, bread)
k = int(k)
if k > 4095:
info["message"] = "The value of k is too large"
return class_group_request_error(info, bread)
else:
info["message"] = "The value of k is missing"
return class_group_request_error(info, bread)
info["filenamebase"] = str(info["filenamebase"])
if info["filenamebase"] in ["cl3mod8", "cl7mod8", "cl4mod16", "cl8mod16"]:
filepath = "%s/%s/%s.%d.gz" % (class_group_data_directory, info["filenamebase"], info["filenamebase"], k)
if os.path.isfile(filepath) and os.access(filepath, os.R_OK):
return send_file(filepath, as_attachment=True)
else:
info["message"] = "File not found"
return class_group_request_error(info, bread)
else:
info["message"] = "Invalid congruence requested"
return class_group_request_error(info, bread)
return render_template(
"class_group_data.html",
info=info,
credit="A. Mosunov and M. J. Jacobson, Jr.",
title=t,
bread=bread,
learnmore=info.pop("learnmore"),
)
def random_nfglobal():
label = random_object_from_collection( getDBConnection().numberfields.fields )['label']
#This version leaves the word 'random' in the URL:
#return render_field_webpage({'label': label})
#This version uses the number field's own URL:
#url =
return redirect(url_for(".by_label", label= label))
def random_nfglobal():
label = random_object_from_collection( getDBConnection().numberfields.fields )['label']
#This version leaves the word 'random' in the URL:
#return render_field_webpage({'label': label})
#This version uses the number field's own URL:
#url =
return redirect(url_for(".by_label", label= label))
def render_discriminants_page():
info = {}
info['learnmore'] = [
('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page"))
]
t = 'Completeness of Global Number Field Data'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")),
(Completename, ' ')]
return render_template("discriminant_ranges.html",
info=info,
credit=NF_credit,
title=t,
bread=bread,
learnmore=info.pop('learnmore'))
def by_label(label):
try:
return render_field_webpage({'label': nf_string_to_label(label)})
except ValueError:
bread = [('Global Number Fields',
url_for(".number_field_render_webpage")),
('Search results', ' ')]
return search_input_error({'err': ''}, bread)
def render_labels_page():
info = {}
info['learnmore'] = [
('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
('Quadratic imaginary class groups',
url_for(".render_class_group_data"))
]
t = 'Number field labels'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")),
('Number field labels', '')]
return render_template("number_field_labels.html",
info=info,
credit=NF_credit,
title=t,
bread=bread,
learnmore=info.pop('learnmore'))
def search():
if request.method == "GET":
val = request.args.get("val", "no value")
bread = get_bread([("Search for '%s'" % val, url_for('.search'))])
return render_template("nfgg-search.html", title="Search for Galois groups of number fields", bread=bread, val=val)
elif request.method == "POST":
return "ERROR: we always do http get to explicitly display the search parameters"
else:
return flask.redirect(404)
def random_nfglobal():
from sage.misc.prandom import randint
C = getDBConnection()
init_nf_count()
n = randint(0, nfields - 1)
label = C.numberfields.fields.find()[n]['label']
#This version leaves the word 'random' in the URL:
#return render_field_webpage({'label': label})
#This version uses the number field's own URL:
#url =
return redirect(url_for(".by_label", label=label))
def random_nfglobal():
from sage.misc.prandom import randint
C = getDBConnection()
init_nf_count()
n = randint(0,nfields-1)
label = C.numberfields.fields.find()[n]['label']
#This version leaves the word 'random' in the URL:
#return render_field_webpage({'label': label})
#This version uses the number field's own URL:
#url =
return redirect(url_for(".by_label", label= label))
def search():
if request.method == "GET":
val = request.args.get("val", "no value")
bread = get_bread([("Search for '%s'" % val, url_for('.search'))])
return render_template(
"nfgg-search.html",
title="Search for Galois groups of number fields",
bread=bread,
val=val)
elif request.method == "POST":
return "ERROR: we always do http get to explicitly display the search parameters"
else:
return flask.redirect(404)
def render_class_group_data():
info = to_dict(request.args)
#nf_logger.info('******************* ')
#for k in info.keys():
# nf_logger.info(str(k) + ' ---> ' + str(info[k]))
#nf_logger.info('******************* ')
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page"))]
t = 'Class Groups of Quadratic Imaginary Fields'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), (t, ' ')]
info['message'] = ''
info['filename']='none'
if 'Fetch' in info:
if 'k' in info:
# remove non-digits
k = re.sub(r'\D', '', info['k'])
if k == "":
info['message'] = 'The value of k is either invalid or empty'
return class_group_request_error(info, bread)
k = int(k)
if k>4095:
info['message'] = 'The value of k is too large'
return class_group_request_error(info, bread)
else:
info['message'] = 'The value of k is missing'
return class_group_request_error(info, bread)
info['filenamebase'] = str(info['filenamebase'])
if info['filenamebase'] in ['cl3mod8', 'cl7mod8', 'cl4mod16', 'cl8mod16']:
filepath = "%s/%s/%s.%d.gz" % (class_group_data_directory,info['filenamebase'],info['filenamebase'],k)
if os.path.isfile(filepath) and os.access(filepath, os.R_OK):
return send_file(filepath, as_attachment=True)
else:
info['message'] = 'File not found'
return class_group_request_error(info, bread)
else:
info['message'] = 'Invalid congruence requested'
return class_group_request_error(info, bread)
return render_template("class_group_data.html", info=info, credit="A. Mosunov and M. J. Jacobson, Jr.", title=t, bread=bread, learnmore=info.pop('learnmore'))
def global_numberfield_summary():
init_nf_count()
return r'This database contains %s <a title="global number fields" knowl="nf">global number fields</a> of <a title="degree" knowl="nf.degree">degree</a> $n\leq %d$. In addition, extensive data on <a href="%s">class groups of quadratic imaginary fields</a> is available for download.' %(comma(nfields),max_deg,url_for('number_fields.render_class_group_data'))
def number_field_search(**args):
info = to_dict(args)
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page")), ('Quadratic imaginary class groups', url_for(".render_class_group_data"))]
t = 'Global Number Field search results'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search results', ' ')]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if 'natural' in info:
query = {'label_orig': info['natural']}
try:
parse_nf_string(info,query,'natural',name="Label",qfield='label')
return redirect(url_for(".by_label", label= clean_input(query['label'])))
except ValueError:
query['err'] = info['err']
return search_input_error(query, bread)
query = {}
try:
parse_galgrp(info,query, qfield='galois')
parse_ints(info,query,'degree')
parse_bracketed_posints(info,query,'signature',split=False,exactlength=2)
parse_signed_ints(info,query,'discriminant',qfield=('disc_sign','disc_abs_key'),parse_one=make_disc_key)
parse_ints(info,query,'class_number')
parse_bracketed_posints(info,query,'class_group',split=False,check_divisibility='increasing')
parse_primes(info,query,'ur_primes',name='Unramified primes',qfield='ramps',mode='complement',to_string=True)
if 'ram_quantifier' in info and str(info['ram_quantifier']) == 'some':
mode = 'append'
else:
mode = 'exact'
parse_primes(info,query,'ram_primes','ramified primes','ramps',mode,to_string=True)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one['label']
return redirect(url_for(".by_label", clean_input(label)))
fields = C.numberfields.fields
res = fields.find(query)
if 'download' in info and info['download'] != '0':
return download_search(info, res)
res = res.sort([('degree', ASC), ('disc_abs_key', ASC),('disc_sign', ASC)])
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['wnf'] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def by_label(label):
try:
return render_field_webpage({'label': nf_string_to_label(label)})
except ValueError:
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search results', ' ')]
return search_input_error({'err':''}, bread)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
info['err'] = 'There is no field with label %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
from lmfdb.WebNumberField import nf_display_knowl
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'Not' in str(data['class_number']):
grh_lab=''
grh_label=''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
('How data was computed', url_for(".how_computed_page"))]
if info['signature'] == [0,1]:
info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m==0: return ''
if m==1: return '*'
return '*<sup>%d</sup>'% m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def number_field_search(**args):
info = to_dict(args)
info['learnmore'] = [
('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
('Quadratic imaginary class groups',
url_for(".render_class_group_data"))
]
t = 'Global Number Field search results'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")),
('Search results', ' ')]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if 'natural' in info:
field_id = info['natural']
field_id_parsed = parse_field_string(info['natural'])
if FIELD_LABEL_RE.match(field_id_parsed):
field_id_parsed = split_label(
field_id_parsed) # allows factored labels 11.11.11e20.1
return render_field_webpage({
'label': field_id_parsed,
'label_orig': field_id
})
query = {}
dlist = []
for field in [
'galois_group', 'degree', 'signature', 'discriminant',
'class_number', 'class_group'
]:
if info.get(field):
info[field] = clean_input(info[field])
if field in ['class_group', 'signature']:
# different regex for the two types
if (field == 'signature' and PAIR_RE.match(info[field])) or (
field == 'class_group' and IF_RE.match(info[field])):
query[field] = info[field][1:-1]
else:
name = 'class group' if field == 'class_group' else 'signature'
info[
'err'] = 'Error parsing input for %s. It needs to be a pair of integers in square brackets, such as [2,3] or [3,3]' % name
return search_input_error(info, bread)
else:
if field == 'galois_group':
try:
gcs = complete_group_codes(info[field])
if len(gcs) == 1:
query['galois'] = make_galois_pair(
gcs[0][0], gcs[0][1])
# list(gcs[0])
if len(gcs) > 1:
query['galois'] = {
'$in':
[make_galois_pair(x[0], x[1]) for x in gcs]
}
except NameError as code:
info[
'err'] = 'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.' % code
return search_input_error(info, bread)
else: # not signature, class group, or galois group
ran = info[field]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
if field == 'discriminant':
# dlist will contain the disc conditions
# as sage ints
dlist = parse_discs(ran)
# now convert to a query
tmp = list_to_query(dlist)
# Two cases, could be a list of sign/inequalties
# or an '$or'
if len(tmp) == 1:
tmp = tmp[0]
else:
query[tmp[0][0]] = tmp[0][1]
tmp = tmp[1]
else:
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
else:
name = re.sub('_', ' ', field)
info[
'err'] = 'Error parsing input for %s. It needs to be an integer (such as 5), a range of integers (such as 2-100 or 2..100), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-100).' % name
return search_input_error(info, bread)
if info.get('ur_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ur_primes'] = clean_input(info['ur_primes'])
if LIST_SIMPLE_RE.match(info['ur_primes']):
ur_primes = info['ur_primes'].split(',')
# Assuming this will be the only nor in the query
query['$nor'] = [{'ramps': x} for x in ur_primes]
else:
info[
'err'] = 'Error parsing input for unramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('ram_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ram_primes'] = clean_input(info['ram_primes'])
if LIST_SIMPLE_RE.match(info['ram_primes']):
ram_primes = info['ram_primes'].split(',')
if str(info['ram_quantifier']) == 'some':
query['ramps'] = {'$all': ram_primes}
else:
query['ramps'] = ram_primes
else:
info[
'err'] = 'Error parsing input for ramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
info['count'] = count
else:
info['count'] = count_default
count = count_default
info['count'] = int(info['count'])
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if (start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one['label']
return render_field_webpage({'label': label})
fields = C.numberfields.fields
res = fields.find(query).sort([('degree', ASC), ('disc_abs_key', ASC),
('disc_sign', ASC), ('label', ASC)])
if 'download' in info and info['download'] != '0':
return download_search(info, res)
nres = res.count()
res = res.skip(start).limit(count)
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['wnf'] = WebNumberField.from_data
return render_template("number_field_search.html",
info=info,
title=t,
bread=bread)
def render_labels_page():
info = {}
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page")), ('Quadratic imaginary class groups', url_for(".render_class_group_data"))]
t = 'Number field labels'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Number field labels', '')]
return render_template("single.html", info=info, credit=NF_credit, kid='nf.label', title=t, bread=bread, learnmore=info.pop('learnmore'))
def number_field_search(**args):
info = to_dict(args)
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page")), ('Quadratic imaginary class groups', url_for(".render_class_group_data"))]
t = 'Global Number Field search results'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search results', ' ')]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if 'natural' in info:
query = {'label_orig': info['natural']}
try:
parse_nf_string(info,query,'natural',name="Label",qfield='label')
return redirect(url_for(".by_label", label= clean_input(query['label'])))
except ValueError:
query['err'] = info['err']
return search_input_error(query, bread)
query = {}
try:
parse_galgrp(info,query, qfield='galois')
parse_ints(info,query,'degree')
parse_bracketed_posints(info,query,'signature',split=False,exactlength=2)
parse_signed_ints(info,query,'discriminant',qfield=('disc_sign','disc_abs_key'),parse_one=make_disc_key)
parse_ints(info,query,'class_number')
parse_bracketed_posints(info,query,'class_group',split=False,check_divisibility='increasing')
parse_primes(info,query,'ur_primes',name='Unramified primes',qfield='ramps',mode='complement',to_string=True)
if 'ram_quantifier' in info and str(info['ram_quantifier']) == 'some':
mode = 'append'
else:
mode = 'exact'
parse_primes(info,query,'ram_primes','ramified primes','ramps',mode,to_string=True)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one['label']
return redirect(url_for(".by_label", clean_input(label)))
fields = C.numberfields.fields
res = fields.find(query)
if 'download' in info and info['download'] != '0':
return download_search(info, res)
res = res.sort([('degree', ASC), ('disc_abs_key', ASC),('disc_sign', ASC)])
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['wnf'] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for("render_Character", arg1=data['conductor'], arg2=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst('a')
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")), (Completename, url_for(".render_discriminants_page"))]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x>0 else '' for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def render_discriminants_page():
info = {}
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page"))]
t = 'Completeness of Global Number Field Data'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), (Completename, ' ')]
return render_template("discriminant_ranges.html", info=info, credit=NF_credit, title=t, bread=bread, learnmore=info.pop('learnmore'))
def number_field_search(**args):
info = to_dict(args)
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
("Quadratic imaginary class groups", url_for(".render_class_group_data")),
]
t = "Global Number Field search results"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), ("Search results", " ")]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if "natural" in info:
query = {"label_orig": info["natural"]}
try:
parse_nf_string(info, query, "natural", name="Label", qfield="label")
return redirect(url_for(".by_label", label=clean_input(query["label"])))
except ValueError:
query["err"] = info["err"]
return search_input_error(query, bread)
query = {}
try:
parse_galgrp(info, query, qfield="galois")
parse_ints(info, query, "degree")
parse_bracketed_posints(info, query, "signature", split=False, exactlength=2)
parse_signed_ints(info, query, "discriminant", qfield=("disc_sign", "disc_abs_key"), parse_one=make_disc_key)
parse_ints(info, query, "class_number")
parse_bracketed_posints(info, query, "class_group", split=False, check_divisibility="increasing")
parse_primes(
info, query, "ur_primes", name="Unramified primes", qfield="ramps", mode="complement", to_string=True
)
if "ram_quantifier" in info and str(info["ram_quantifier"]) == "some":
mode = "append"
else:
mode = "exact"
parse_primes(info, query, "ram_primes", "ramified primes", "ramps", mode, to_string=True)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
if info.get("paging"):
try:
paging = int(info["paging"])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info["query"] = dict(query)
if "lucky" in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one["label"]
return redirect(url_for(".by_label", clean_input(label)))
fields = C.numberfields.fields
res = fields.find(query)
if "download" in info and info["download"] != "0":
return download_search(info, res)
res = res.sort([("degree", ASC), ("disc_abs_key", ASC), ("disc_sign", ASC)])
nres = res.count()
res = res.skip(start).limit(count)
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
info["fields"] = res
info["number"] = nres
info["start"] = start
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
info["wnf"] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for("render_Character", arg1=data['conductor'], arg2=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst('a')
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")), (Completename, url_for(".render_discriminants_page"))]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x>0 else '' for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def get_bread(breads=[]):
bc = [("Number Field Galois Group", url_for(".index"))]
for b in breads:
bc.append(b)
return bc
def global_numberfield_summary():
init_nf_count()
return r'This database contains %s <a title="global number fields" knowl="nf">global number fields</a> of <a title="degree" knowl="nf.degree">degree</a> $n\leq %d$. In addition, extensive data on <a href="%s">class groups of quadratic imaginary fields</a> is available for download.' %(comma(nfields),max_deg,url_for('number_fields.render_class_group_data'))
def NF_redirect():
return redirect(url_for(".number_field_render_webpage", **request.args))
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [("Global Number Fields", url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args["label"])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(("Search results", " "))
info["err"] = "There is no field with label %s in the database" % label2
info["label"] = args["label_orig"] if "label_orig" in args else args["label"]
return search_input_error(info, bread)
info["wnf"] = nf
from lmfdb.WebNumberField import nf_display_knowl
data["degree"] = nf.degree()
data["class_number"] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data["is_galois"] = nf.is_galois()
data["is_abelian"] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data["conductor"] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars) > 0:
data["dirichlet_group"] = [
'<a href = "%s">$\chi_{%s}(%s,·)$</a>'
% (
url_for("characters.render_Dirichletwebpage", modulus=data["conductor"], number=j),
data["conductor"],
j,
)
for j in dirichlet_chars
]
data["dirichlet_group"] = r"$\lbrace$" + ", ".join(data["dirichlet_group"]) + r"$\rbrace$"
if data["conductor"].is_prime() or data["conductor"] == 1:
data["conductor"] = "\(%s\)" % str(data["conductor"])
else:
data["conductor"] = "\(%s=%s\)" % (str(data["conductor"]), latex(data["conductor"].factor()))
data["galois_group"] = group_display_knowl(n, t, C)
data["cclasses"] = cclasses_display_knowl(n, t, C)
data["character_table"] = character_table_display_knowl(n, t, C)
data["class_group"] = nf.class_group()
data["class_group_invs"] = nf.class_group_invariants()
data["signature"] = nf.signature()
data["coefficients"] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
ram_primes = D.prime_factors()
data["disc_factor"] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data["discriminant"] = "\(%s\)" % str(D)
else:
data["discriminant"] = "\(%s=%s\)" % (str(D), data["disc_factor"])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == "":
ram_primes = r"\textrm{None}"
data["frob_data"], data["seeram"] = frobs(nf.K())
data["phrase"] = group_phrase(n, t, C)
zk = nf.zk()
Ra = PolynomialRing(QQ, "a")
zk = [latex(Ra(x)) for x in zk]
zk = ["$%s$" % x for x in zk]
zk = ", ".join(zk)
grh_label = (
'<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ""
)
# Short version for properties
grh_lab = nf.short_grh_string()
if "Not" in str(data["class_number"]):
grh_lab = ""
grh_label = ""
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update(
{
"label": pretty_label,
"label_raw": label,
"polynomial": web_latex_split_on_pm(nf.K().defining_polynomial()),
"ram_primes": ram_primes,
"integral_basis": zk,
"regulator": web_latex(nf.regulator()),
"unit_rank": nf.unit_rank(),
"root_of_unity": web_latex(nf.K().primitive_root_of_unity()),
"fund_units": nf.units(),
"grh_label": grh_label,
}
)
bread.append(("%s" % info["label_raw"], " "))
info["downloads_visible"] = True
info["downloads"] = [("worksheet", "/")]
info["friends"] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D ** n) < 50000000:
info["friends"].append(("L-function", "/L/NumberField/%s" % label))
info["friends"].append(("Galois group", "/GaloisGroup/%dT%d" % (n, t)))
if "dirichlet_group" in info:
info["friends"].append(
(
"Dirichlet group",
url_for(
"characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=",".join([str(a) for a in dirichlet_chars]),
poly=info["polynomial"],
),
)
)
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
("How data was computed", url_for(".how_computed_page")),
]
if info["signature"] == [0, 1]:
info["learnmore"].append(("Quadratic imaginary class groups", url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info["label"]
if npr == 1:
primes = "prime"
else:
primes = "primes"
properties2 = [
("Degree:", "%s" % data["degree"]),
("Signature:", "$%s$" % data["signature"]),
("Discriminant:", "$%s$" % data["disc_factor"]),
("Ramified " + primes + ":", "$%s$" % ram_primes),
("Class number:", "%s %s" % (data["class_number"], grh_lab)),
("Class group:", "%s %s" % (data["class_group_invs"], grh_lab)),
("Galois Group:", group_display_short(data["degree"], t, C)),
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data["coeffs"])
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m == 0:
return ""
if m == 1:
return "*"
return "*<sup>%d</sup>" % m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
# del info['_id']
return render_template(
"number_field.html",
properties2=properties2,
credit=NF_credit,
title=title,
bread=bread,
friends=info.pop("friends"),
learnmore=info.pop("learnmore"),
info=info,
)
def number_field_search(**args):
info = to_dict(args)
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
]
t = "Global Number Field search results"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), ("Search results", " ")]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if "natural" in info:
field_id = info["natural"]
field_id_parsed = parse_field_string(info["natural"])
if FIELD_LABEL_RE.match(field_id_parsed):
field_id_parsed = split_label(field_id_parsed) # allows factored labels 11.11.11e20.1
return render_field_webpage({"label": field_id_parsed, "label_orig": field_id})
query = {}
dlist = []
for field in ["galois_group", "degree", "signature", "discriminant", "class_number", "class_group"]:
if info.get(field):
info[field] = clean_input(info[field])
if field in ["class_group", "signature"]:
# different regex for the two types
if (field == "signature" and PAIR_RE.match(info[field])) or (
field == "class_group" and IF_RE.match(info[field])
):
query[field] = info[field][1:-1]
else:
name = "class group" if field == "class_group" else "signature"
info["err"] = (
"Error parsing input for %s. It needs to be a pair of integers in square brackets, such as [2,3] or [3,3]"
% name
)
return search_input_error(info, bread)
else:
if field == "galois_group":
try:
gcs = complete_group_codes(info[field])
if len(gcs) == 1:
query["galois"] = make_galois_pair(gcs[0][0], gcs[0][1])
# list(gcs[0])
if len(gcs) > 1:
query["galois"] = {"$in": [make_galois_pair(x[0], x[1]) for x in gcs]}
except NameError as code:
info["err"] = (
'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.'
% code
)
return search_input_error(info, bread)
else: # not signature, class group, or galois group
ran = info[field]
ran = ran.replace("..", "-")
if LIST_RE.match(ran):
if field == "discriminant":
# dlist will contain the disc conditions
# as sage ints
dlist = parse_discs(ran)
# now convert to a query
tmp = list_to_query(dlist)
# Two cases, could be a list of sign/inequalties
# or an '$or'
if len(tmp) == 1:
tmp = tmp[0]
else:
query[tmp[0][0]] = tmp[0][1]
tmp = tmp[1]
else:
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == "$or" and "$or" in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query["$or"]]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
else:
name = re.sub("_", " ", field)
info["err"] = (
"Error parsing input for %s. It needs to be an integer (such as 5), a range of integers (such as 2-100 or 2..100), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-100)."
% name
)
return search_input_error(info, bread)
if info.get("ur_primes"):
# now we want a list of strings, no spaces, which might be big ints
info["ur_primes"] = clean_input(info["ur_primes"])
if LIST_SIMPLE_RE.match(info["ur_primes"]):
ur_primes = info["ur_primes"].split(",")
# Assuming this will be the only nor in the query
query["$nor"] = [{"ramps": x} for x in ur_primes]
else:
info[
"err"
] = "Error parsing input for unramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11)."
return search_input_error(info, bread)
if info.get("ram_primes"):
# now we want a list of strings, no spaces, which might be big ints
info["ram_primes"] = clean_input(info["ram_primes"])
if LIST_SIMPLE_RE.match(info["ram_primes"]):
ram_primes = info["ram_primes"].split(",")
if str(info["ram_quantifier"]) == "some":
query["ramps"] = {"$all": ram_primes}
else:
query["ramps"] = ram_primes
else:
info[
"err"
] = "Error parsing input for ramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11)."
return search_input_error(info, bread)
count_default = 20
if info.get("count"):
try:
count = int(info["count"])
except:
count = count_default
info["count"] = count
else:
info["count"] = count_default
count = count_default
info["count"] = int(info["count"])
start_default = 0
if info.get("start"):
try:
start = int(info["start"])
if start < 0:
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get("paging"):
try:
paging = int(info["paging"])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info["query"] = dict(query)
if "lucky" in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one["label"]
return render_field_webpage({"label": label})
fields = C.numberfields.fields
res = fields.find(query).sort([("degree", ASC), ("disc_abs_key", ASC), ("disc_sign", ASC), ("label", ASC)])
if "download" in info and info["download"] != "0":
return download_search(info, res)
nres = res.count()
res = res.skip(start).limit(count)
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
info["fields"] = res
info["number"] = nres
info["start"] = start
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
info["wnf"] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def by_label(label):
try:
return render_field_webpage({"label": nf_string_to_label(label)})
except ValueError:
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), ("Search results", " ")]
return search_input_error({"err": ""}, bread)
def get_bread(breads=[]):
bc = [("Number Field Galois Group", url_for(".index"))]
for b in breads:
bc.append(b)
return bc
def NF_redirect():
return redirect(url_for(".number_field_render_webpage", **request.args))
def number_field_search(**args):
info = to_dict(args)
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page"))]
t = 'Global Number Field search results'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search results', ' ')]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if 'natural' in info:
field_id = info['natural']
field_id_parsed = parse_field_string(info['natural'])
if FIELD_LABEL_RE.match(field_id_parsed):
field_id_parsed = split_label(field_id_parsed) # allows factored labels 11.11.11e20.1
return render_field_webpage({'label': field_id_parsed, 'label_orig': field_id})
query = {}
dlist = []
for field in ['galois_group', 'degree', 'signature', 'discriminant', 'class_number', 'class_group']:
if info.get(field):
info[field] = clean_input(info[field])
if field in ['class_group', 'signature']:
# different regex for the two types
if (field == 'signature' and PAIR_RE.match(info[field])) or (field == 'class_group' and IF_RE.match(info[field])):
query[field] = info[field][1:-1]
else:
name = 'class group' if field == 'class_group' else 'signature'
info['err'] = 'Error parsing input for %s. It needs to be a pair of integers in square brackets, such as [2,3] or [3,3]' % name
return search_input_error(info, bread)
else:
if field == 'galois_group':
try:
gcs = complete_group_codes(info[field])
if len(gcs) == 1:
query['galois'] = make_galois_pair(gcs[0][0], gcs[0][1])
# list(gcs[0])
if len(gcs) > 1:
query['galois'] = {'$in': [make_galois_pair(x[0], x[1]) for x in gcs]}
except NameError as code:
info['err'] = 'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.' % code
return search_input_error(info, bread)
else: # not signature, class group, or galois group
ran = info[field]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
if field == 'discriminant':
# dlist will contain the disc conditions
# as sage ints
dlist = parse_discs(ran)
# now convert to a query
tmp = list_to_query(dlist)
# Two cases, could be a list of sign/inequalties
# or an '$or'
if len(tmp) == 1:
tmp = tmp[0]
else:
query[tmp[0][0]] = tmp[0][1]
tmp = tmp[1]
else:
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
else:
name = re.sub('_', ' ', field)
info['err'] = 'Error parsing input for %s. It needs to be an integer (such as 5), a range of integers (such as 2-100 or 2..100), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-100).' % name
return search_input_error(info, bread)
if info.get('ur_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ur_primes'] = clean_input(info['ur_primes'])
if LIST_SIMPLE_RE.match(info['ur_primes']):
ur_primes = info['ur_primes'].split(',')
# Assuming this will be the only nor in the query
query['$nor'] = [{'ramps': x} for x in ur_primes]
else:
info['err'] = 'Error parsing input for unramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('ram_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ram_primes'] = clean_input(info['ram_primes'])
if LIST_SIMPLE_RE.match(info['ram_primes']):
ram_primes = info['ram_primes'].split(',')
if str(info['ram_quantifier']) == 'some':
query['ramps'] = {'$all': ram_primes}
else:
query['ramps'] = ram_primes
else:
info['err'] = 'Error parsing input for ramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
info['count'] = count
else:
info['count'] = count_default
count = count_default
info['count'] = int(info['count'])
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one['label']
return render_field_webpage({'label': label})
fields = C.numberfields.fields
res = fields.find(
query).sort([('degree', ASC), ('disc_abs_key', ASC), ('disc_sign', ASC), ('label', ASC)])
if 'download' in info and info['download'] != '0':
return download_search(info, res)
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['wnf'] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
info['err'] = 'There is no field with label %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
from lmfdb.WebNumberField import nf_display_knowl
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'Not' in str(data['class_number']):
grh_lab=''
grh_label=''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
('How data was computed', url_for(".how_computed_page"))]
if info['signature'] == [0,1]:
info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m==0: return ''
if m==1: return '*'
return '*<sup>%d</sup>'% m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)