def test_list_to_latex_matrix(self):
r"""
Checking utility: list_to_latex_matrix
"""
identity_list = [[1, 0], [0, 1]]
identity_rep = '\\left(\\begin{array}{*{2}{r}}1 & 0\\\\0 & 1\\end{array}\\right)'
self.assertEqual(list_to_latex_matrix(identity_list), identity_rep)
# malformed matrices should work
malformed = [[1, 0], [0]]
malform_rep = '\\left(\\begin{array}{*{2}{r}}1 & 0\\\\0\\end{array}\\right)'
self.assertEqual(list_to_latex_matrix(malformed), malform_rep)
def test_list_to_latex_matrix(self):
r"""
Checking utility: list_to_latex_matrix
"""
identity_list = [[1,0], [0,1]]
identity_rep = '\\left(\\begin{array}{*{2}{r}}1 & 0\\\\0 & 1\\end{array}\\right)'
self.assertEqual(list_to_latex_matrix(identity_list), identity_rep)
# malformed matrices should work
malformed = [[1,0], [0]]
malform_rep = '\\left(\\begin{array}{*{2}{r}}1 & 0\\\\0\\end{array}\\right)'
self.assertEqual(list_to_latex_matrix(malformed), malform_rep)
def galois_module_knowl_guts(n, t, index):
mymod = db.gps_gmodules.lucky({'n': int(n), 't': int(t), 'index': int(index)}, ['name','dim','gens'])
if mymod is None:
return 'Database call failed'
name = mymod['name']
out = "$\\Z[G]$ module %s with $G=$ " % str(name)
out += group_display_knowl(n, t)
out += " = %sT%s " %(n, t)
out += "<blockquote>"
out += "Dimension: %s" % str(mymod['dim'])
out += r"<br>Action: $$\begin{aligned}"
for g in mymod['gens']:
matg = list_to_latex_matrix(g[1])
out += "%s &\\mapsto %s \\\\" %(str(g[0]), matg)
out = out[:-2]
out += r"\end{aligned}$$"
out += "</blockquote>"
return out
def render_group_webpage(args):
data = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
data = db.gps_transitive.lookup(label)
if data is None:
if re.match(r'^\d+T\d+$', label):
flash_error("Group %s was not found in the database.", label)
else:
flash_error("%s is not a valid label for a Galois group.",
label)
return redirect(url_for(".index"))
title = 'Galois group: ' + label
wgg = WebGaloisGroup.from_nt(data['n'], data['t'])
data['wgg'] = wgg
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%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 wgg.num_conjclasses() < 50:
data['cclasses'] = wgg.conjclasses()
if ZZ(order) < ZZ(10000000) and wgg.num_conjclasses() < 21:
data['chartable'] = chartable(n, t)
data['gens'] = wgg.generator_string()
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['num_cc'] = comma(wgg.num_conjclasses())
data['parity'] = "$%s$" % data['parity']
data['subinfo'] = subfield_display(n, data['subfields'])
data['resolve'] = resolve_display(data['quotients'])
if data['gapid'] == 0:
data['gapid'] = "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 = data['arith_equiv']
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.'
elif ae > -1:
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.'
else:
data[
'arith_equiv'] = r'Data on whether or not a number field with this Galois group has <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields has not been computed.'
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({'galois_label': "%dT%d" % (n, 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),
('Degree', prop_int_pretty(data['n'])),
('Order', prop_int_pretty(order)),
('Cyclic', yesno(data['cyc'])),
('Abelian', yesno(data['ab'])),
('Solvable', yesno(data['solv'])),
('Primitive', yesno(data['prim'])),
('$p$-group', yesno(pgroup)),
]
pretty = group_display_short(n, t, emptyifnotpretty=True)
if len(pretty) > 0:
prop2.extend([('Group:', pretty)])
data['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)', r'_{\1}', data['name'])
data['name'] = re.sub(r'\^(\d+)', r'^{\1}', data['name'])
data['nilpotency'] = '$%s$' % data['nilpotency']
if data['nilpotency'] == '$-1$':
data['nilpotency'] += ' (not nilpotent)'
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html",
credit=GG_credit,
title=title,
bread=bread,
info=data,
properties=prop2,
friends=friends,
KNOWL_ID="gg.%s" % label,
learnmore=learnmore_list())
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
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')
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")
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
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)