def chartable(n, t):
if int(n) == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
ctable = str(G.CharacterTable().Display())
ctable = re.sub("^.*\n", '', ctable)
ctable = re.sub("^.*\n", '', ctable)
return ctable
def generators(n, t):
if str(n) == "1":
return "None needed"
else:
G = gap.TransitiveGroup(n, t)
gens = G.SmallGeneratingSet()
gens = str(gens)
gens = re.sub("[\[\]]", '', gens)
return gens
def chartable(n, t):
if int(n) == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
CT = G.CharacterTable()
ctable = gap.eval("Display(%s)" % CT.name())
ctable = re.sub("^.*\n", '', ctable)
ctable = re.sub("^.*\n", '', ctable)
return ctable
def cclasses(n, t):
if int(n) == 1:
G = gap.SmallGroup(1, 1)
else:
G = gap.TransitiveGroup(n, t)
cc = conjclasses(G, n)
html = """<div>
<table class="ntdata">
<thead><tr><td>Cycle Type</td><td>Size</td><td>Order</td><td>Representative</td></tr></thead>
<tbody>
"""
for c in cc:
html += '<tr><td>' + str(c[3]) + '</td>'
html += '<td>' + str(c[2]) + '</td>'
html += '<td>' + str(c[1]) + '</td>'
html += '<td>' + str(c[0]) + '</td>'
html += """</tr></tbody>
</table>
"""
return html
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Galois Group:' + label
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if ZZ(order) == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
# if len(data['resolve']) == 0: data['resolve'] = 'None'
data['otherreps'] = otherrep_display(n, t, C, data['repns'])
prop2 = [
('Order:', '\(%s\)' % order),
('n:', '\(%s\)' % data['n']),
('Cyclic:', yesno(data['cyc'])),
('Abelian:', yesno(data['ab'])),
('Solvable:', yesno(data['solv'])),
('Primitive:', yesno(data['prim'])),
('$p$-group:', yesno(pgroup)),
('Name:', group_display_short(n, t, C)),
]
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html",
credit=GG_credit,
title=title,
bread=bread,
info=info,
properties2=prop2)
import flask
from lmfdb import base
from lmfdb.base import app, getDBConnection
from flask import render_template, render_template_string, request, abort, Blueprint, url_for, make_response
from lmfdb.utils import ajax_more, image_src, web_latex, to_dict, parse_range, parse_range2, make_logger, clean_input
import os
import re
from lmfdb.galois_groups import galois_groups_page, logger
import sage.all
from sage.all import ZZ, latex, gap
# Test to see if this gap installation knows about transitive groups
# logger = make_logger("GG")
try:
G = gap.TransitiveGroup(9, 2)
except:
logger.fatal(
"It looks like the SPKGes gap_packages and database_gap are not installed on the server. Please install them via 'sage -i ...' and try again."
)
from lmfdb.transitive_group import group_display_short, group_display_long, group_display_inertia, group_knowl_guts, subfield_display, otherrep_display, resolve_display, conjclasses, generators, chartable, aliastable, WebGaloisGroup
GG_credit = 'GAP, Magma, and J. Jones'
def get_bread(breads=[]):
bc = [("Galois Groups", url_for(".index"))]
for b in breads:
bc.append(b)
return bc
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
data['label_raw'] = label.lower()
title = 'Galois Group: ' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
data['otherreps'] = wgg.otherrep_list()
if len(data['otherreps']) == 0:
data['otherreps'] = "There is no other low degree representation."
query = {'galois': bson.SON([('n', n), ('t', t)])}
C = base.getDBConnection()
intreps = C.transitivegroups.Gmodules.find({
'n': n,
't': t
}).sort('index', pymongo.ASCENDING)
# turn cursor into a list
intreps = [z for z in intreps]
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens'] = [
list_to_latex_matrix(z[1]) for z in onerep['gens']
]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]),
mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[
modules2string(n, t, z[0]),
modules2string(n, t, z[1])
] for z in data['isoms']]
print dcq[1]
print data['isoms']
friends = []
one = C.numberfields.fields.find_one(query)
if one:
friends.append(
('Number fields with this Galois group',
url_for('number_fields.number_field_render_webpage') +
"?galois_group=%dT%d" % (n, t)))
prop2 = [
('Order:', '\(%s\)' % order),
('n:', '\(%s\)' % data['n']),
('Cyclic:', yesno(data['cyc'])),
('Abelian:', yesno(data['ab'])),
('Solvable:', yesno(data['solv'])),
('Primitive:', yesno(data['prim'])),
('$p$-group:', yesno(pgroup)),
]
pretty = group_display_pretty(n, t, C)
if len(pretty) > 0:
prop2.extend([('Group:', pretty)])
info['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)', r'_{\1}', data['name'])
data['name'] = re.sub(r'\^(\d+)', r'^{\1}', data['name'])
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html",
credit=GG_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends)
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
data = db.gps_transitive.lookup(label)
if data is None:
bread = get_bread([("Search Error", ' ')])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
data['label_raw'] = label.lower()
title = 'Galois Group: ' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(n, data['subs'])
data['resolve'] = resolve_display(data['resolve'])
if data['gapid'] == 0:
data['gapid'] = "Data not available"
else:
data['gapid'] = small_group_display_knowl(int(data['order']),
int(data['gapid']),
str([int(data['order']), int(data['gapid'])]))
data['otherreps'] = wgg.otherrep_list()
ae = wgg.arith_equivalent()
if ae>0:
if ae>1:
data['arith_equiv'] = r'A number field with this Galois group has %d <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'% ae
else:
data['arith_equiv'] = r'A number field with this Galois group has exactly one <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> field.'
else:
data['arith_equiv'] = r'A number field with this Galois group has no <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'
if len(data['otherreps']) == 0:
data['otherreps']="There is no other low degree representation."
intreps = list(db.gps_gmodules.search({'n': n, 't': t}))
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens']=[list_to_latex_matrix(z[1]) for z in onerep['gens']]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]), mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[modules2string(n,t,z[0]), modules2string(n,t,z[1])] for z in data['isoms']]
#print dcq[1]
#print data['isoms']
friends = []
if db.nf_fields.exists({'degree': n, 'galt': t}):
friends.append(('Number fields with this Galois group', url_for('number_fields.number_field_render_webpage')+"?galois_group=%dT%d" % (n, t) ))
prop2 = [('Label', label),
('Order', '\(%s\)' % order),
('n', '\(%s\)' % data['n']),
('Cyclic', yesno(data['cyc'])),
('Abelian', yesno(data['ab'])),
('Solvable', yesno(data['solv'])),
('Primitive', yesno(data['prim'])),
('$p$-group', yesno(pgroup)),
]
pretty = group_display_pretty(n,t)
if len(pretty)>0:
prop2.extend([('Group:', pretty)])
info['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)',r'_{\1}',data['name'])
data['name'] = re.sub(r'\^(\d+)',r'^{\1}',data['name'])
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html", credit=GG_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
return 1
return 0
from jjgals import gals # this reads in the list called quads
print "finished importing gals, number = %s" % len(gals)
# for deg in [3,4,5]:
for deg in range(1, len(gals)):
# for deg in None:
d = gals[deg]
for myg in range(len(d)):
n = deg + 1
t = myg + 1
g = gap.TransitiveGroup(n, t)
gal = d[myg]
solv = tf(g.IsSolvable())
prim = tf(g.IsPrimitive())
cyc = tf(g.IsCyclic())
ab = tf(g.IsAbelian())
pretty = None
if len(gal) > 700:
pretty = gal[700]
data = {
'n': n,
't': t,
'cyc': cyc,
'ab': ab,
'solv': solv,
'prim': prim,