def input_box(default=None, label=None, type=lambda x: x, width=80, height=1, **kwargs):
r"""
An input box interactive control. Use this in conjunction
with the :func:`interact` command.
INPUT:
- ``default`` - an object; the default put in this input box
- ``label`` - a string; the label rendered to the left of the
box.
- ``type`` - a type; coerce inputs to this; this doesn't
have to be an actual type, since anything callable will do.
- ``height`` - an integer (default: 1); the number of rows.
If greater than 1 a value won't be returned until something
outside the textarea is clicked.
- ``width`` - an integer; width of text box in characters
- ``kwargs`` - a dictionary; additional keyword options
EXAMPLES::
sage: input_box("2+2", 'expression')
Interact input box labeled 'expression' with default value '2+2'
sage: input_box('sage', label="Enter your name", type=str)
Interact input box labeled 'Enter your name' with default value 'sage'
sage: input_box('Multiline\nInput',label='Click to change value',type=str,height=5)
Interact input box labeled 'Click to change value' with default value 'Multiline\nInput'
"""
# TODO: make input_box take a type
from sagenb.misc.misc import Color
if type is Color:
# kwargs are only used if the type is Color.
widget=kwargs.get('widget', None)
hide_box=kwargs.get('hide_box', False)
return color_selector(default=default, label=label,
widget=widget, hide_box=hide_box)
if not isinstance(default, basestring):
default=repr(default)
from sage.all import sage_eval
if type is None:
adapter = lambda x, globs: sage_eval(x, globs)
elif type is str:
adapter=lambda x, globs: x
else:
adapter = lambda x, globs: sage_eval(x, globs)
return InputBox(default=default, width=width,
label=label, adapter=adapter, height=height)
def getConstraints(m, resultAsDict=False):
"""
Input a model m, returns its set of constraints in either
1) sage dict {x:7,y:10}
1) z3 expr [x==7,y==0]
sage: S = z3.Solver()
sage: S.add(z3.Int('x') + z3.Int('y') == z3.IntVal('7'))
sage: S.check()
sat
sage: M = S.model()
sage: d = getConstraints(M, resultAsDict=True)
sage: sorted(d.items(), key=lambda(k,_): str(k))
[(x, 7), (y, 0)]
sage: getConstraints(M)
[y == 0, x == 7]
sage: S.reset()
"""
assert m is not None, m
if resultAsDict: #sage format
rs = [(var(str(v())),sage_eval(str(m[v]))) for v in m]
rs = dict(rs)
assert all(is_sage_expr(x) for x in rs.keys())
assert all(is_sage_real(x) or is_sage_int(x) for x in rs.values())
else: #z3 format
rs = [v()==m[v] for v in m]
assert all(z3.is_expr(x) for x in rs)
return rs
def compute_conjectural_ranks(level_norms, address='localhost:29000'):
"""
For each level norm in the input list, compute the
*conjectural* rank of the elliptic curve, and put
that data in a field "r?" in the database.
This uses Simon 2-descent if it works, and otherwise
uses Dokchitser. This should not be considered
super-reliable!
"""
from sage.all import sage_eval
import util
C = util.ellcurves_sqrt5(address)
for N in level_norms:
for E in C.find({
'Nlevel': int(N),
'r?': {
'$exists': False
},
'weq': {
'$exists': True
}
}):
print E
weq = sage_eval(E['weq'], {'a': F.gen()})
E['r?'] = int(rank(weq))
print E
C.update({'_id': E['_id']}, E, safe=True)
def eval_lambda(f, d, is_formula=True):
"""
Evaluates lambda function f
Examples:
sage: assert InvMPP.eval_lambda('lambda x,y: max(x - 13,-3) >= y', {'x':11,'y':100}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 6', {'x':2,'y':3,'d':7}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y', {'x':2,'y':3,'d':7}, is_formula=False) == 5
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 10 or x + y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 1 or x + y == 2', {'x':2,'y':3,'d':7}) == False
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert is_dict(d), d
assert all(is_str(k) for k in d), d.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
assert set(vs) <= set(d.keys()), (vs,d.keys())
#if d has more keys than variables in f then remove those extra keys
d=dict([(k,d[k]) for k in vs])
rs = f(**d)
return bool(rs) if is_formula else rs
def scl(chain, mode='local', args=[], return_fraction=True, verbose=1):
if chain == '' or chain == ['']:
return 0
if type(chain) == str:
C = [chain]
else :
C = [x for x in chain if x != '']
cmd_list = ['scallop', '-' + mode] + args + C
if verbose > 1:
print "Running " + SCALLOPDIR+str(cmd_list)
if sys.version[:3] == '2.7':
mout = subprocess.check_output(cmd_list, \
executable=SCALLOPDIR+cmd_list[0], \
stderr=FNULL, \
cwd=SCALLOPDIR)
else:
sca = subprocess.Popen(cmd_list, \
executable=SCALLOPDIR+cmd_list[0], \
stdout=subprocess.PIPE, \
stderr=FNULL, \
cwd=SCALLOPDIR)
mout = sca.communicate()[0]
if return_fraction:
dat = mout.split(' ')[-3]
if dat == 'feasible':
raise NameError('No feasible solution found')
return sage_eval(dat)
else:
dat = mout.split(' ')[-1]
return eval(dat)
def scylla(gen_string, C_input, method=None):
if type(C_input) == str:
C = [C_input]
else :
C = [x for x in C_input if x != '']
run_string = ['scallop']
if method != None:
run_string += ['-m' + method]
run_string += [gen_string]
run_string += C
if sys.version[:3] == '2.7':
mout = subprocess.check_output(run_string, \
executable=SCALLOPDIR+run_string[0], \
stderr=FNULL, \
cwd=SCALLOPDIR)
else:
sca = subprocess.Popen(run_string, \
executable=SCALLOPDIR+run_string[0], \
stdout=subprocess.PIPE, \
stderr=FNULL, \
cwd=SCALLOPDIR)
mout = sca.communicate()[0]
dat = mout.split(' ')[-3]
return sage_eval(dat)
def eval_lambda(f, d, is_formula=True):
"""
Evaluates lambda function f
Examples:
sage: assert InvMPP.eval_lambda('lambda x,y: max(x - 13,-3) >= y', {'x':11,'y':100}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 6', {'x':2,'y':3,'d':7}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y', {'x':2,'y':3,'d':7}, is_formula=False) == 5
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 10 or x + y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 1 or x + y == 2', {'x':2,'y':3,'d':7}) == False
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert is_dict(d), d
assert all(is_str(k) for k in d), d.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
assert set(vs) <= set(d.keys()), (vs, d.keys())
#if d has more keys than variables in f then remove those extra keys
d = dict([(k, d[k]) for k in vs])
rs = f(**d)
return bool(rs) if is_formula else rs
def download_search(info):
lang = info["submit"]
filename = 'Hecke_algebras' + download_file_suffix[lang]
mydate = time.strftime("%d %B %Y")
# reissue saved query here
if 'ell' in info["query"]:
res = hecke_l_adic_db().find(ast.literal_eval(info["query"])).sort([
('level', ASC), ('weight', ASC), ('num_orbits', ASC)
])
else:
res = hecke_orbits_db().find(ast.literal_eval(info["query"])).sort([
('level', ASC), ('weight', ASC), ('num_orbits', ASC)
])
last = res.count() - 1
c = download_comment_prefix[lang]
s = '\n'
if 'ell' in info["query"]:
s += c + ' Hecke algebras downloaded from the LMFDB on %s. Found %s algebras. The data is given in the following format: it is a list of lists, each containing level, weight and the Hecke orbits for which l-adic data is available.\n\n' % (
mydate, res.count())
else:
s += c + ' Hecke algebras downloaded from the LMFDB on %s. Found %s algebras. The data is given in the following format: it is a list of lists, each containing level, weight, list of the first 10 Hecke operators (to download more operators for a given algebra, please visit its webpage).\n\n' % (
mydate, res.count())
# The list entries are matrices of different sizes. Sage and gp
# do not mind this but Magma requires a different sort of list.
list_start = '[*' if lang == 'magma' else '['
list_end = '*]' if lang == 'magma' else ']'
s += download_assignment_start[lang] + list_start + '\\\n'
mat_start = "Mat(" if lang == 'gp' else "Matrix("
mat_end = "~)" if lang == 'gp' else ")"
entry = lambda r: "".join([mat_start, str(r), mat_end])
# loop through all search results and grab the Hecke operators stored
for c, rr in enumerate(res):
s += list_start
s += ",".join([str(rr['level']), str(rr['weight']), ""])
if 'ell' in info["query"]:
s += '"%s"' % (str(rr['orbit_label']))
else:
s += ",".join([
entry(r) for r in [
sage_eval(rr['hecke_op'])[i]
for i in range(0, min(10, rr['num_hecke_op']))
]
])
if c != last:
s += list_end + ',\\\n'
else:
s += list_end
s += list_end
s += download_assignment_end[lang]
s += '\n'
strIO = StringIO.StringIO()
strIO.write(s)
strIO.seek(0)
return send_file(strIO,
attachment_filename=filename,
as_attachment=True,
add_etags=False)
def convert_magma_matrix_to_sage(m, K):
try:
return Matrix(K, m.sage());
except:
pass
from re import sub
text = str(m)
text = sub('\[\s+', '[', text)
text = sub('\s+', ' ', text)
text = '['+text.replace(' ',', ').replace('\n',', ')+']'
return Matrix(K, sage_eval(text, locals={'r': K.gens()}))
def curve_string_parser(self, rec):
curve_str = rec["curve"]
curve_str = curve_str.replace("^", "**")
K = self.make_base_field(rec)
nu = K.gens()[0]
S0 = PolynomialRing(K, "x")
x = S0.gens()[0]
S = PolynomialRing(S0, "y")
y = S.gens()[0]
parts = curve_str.split("=")
lhs_poly = sage_eval(parts[0], locals={"x": x, "y": y, "nu": nu})
lhs_cs = lhs_poly.coefficients()
if len(lhs_cs) == 1:
h = 0
elif len(lhs_cs) == 2: # if there is a cross-term
h = lhs_poly.coefficients()[0]
else:
raise NotImplementedError("for genus > 2")
# rhs_poly = sage_eval(parts[1], locals = {'x':x, 'y':y, 'nu':nu})
f = sage_eval(parts[1], locals={"x": x, "y": y, "nu": nu})
return f, h
def __init__(self, default="", label=None, width=0, height=1, adapter=None, evaluate=True):
if not isinstance(default, basestring):
default = repr(default)
self.default=default
self.width=int(width)
self.height=int(height)
self.raw=False
self.label=label
if evaluate:
from sage.all import sage_eval
if adapter is not None:
self.adapter = lambda x,globs: adapter(sage_eval(x,globs), globs)
else:
self.adapter = lambda x,globs: sage_eval(x,globs)
elif adapter is not None:
self.adapter = lambda x,globs: adapter(x,globs)
if self.height > 1:
self.subtype = "textarea"
self.raw = True
else:
self.subtype = "input"
def __init__(self, nrows=1, ncols=1, default='0', adapter=None, width=0, label=None,
element_adapter=None, evaluate=True):
self.nrows = int(nrows)
self.ncols = int(ncols)
self.width = int(width)
self.label = label
if evaluate:
from sage.all import sage_eval
if element_adapter is not None:
self.element_adapter = lambda x,globs: element_adapter(sage_eval(x,globs), globs)
else:
self.element_adapter = lambda x,globs: sage_eval(x,globs)
else:
if element_adapter is not None:
self.element_adapter = lambda x,globs: element_adapter(x,globs)
else:
self.element_adapter = lambda x,globs: x
if adapter is None:
self.adapter = lambda x,globs: [[self.element_adapter(i,globs) for i in xi] for xi in x]
else:
self.adapter = lambda x,globs: adapter([[self.element_adapter(i,globs) for i in xi] for xi in x],globs)
def makestring(x):
if isinstance(x, basestring):
return x
else:
return repr(x)
if not isinstance(default, list):
self.default = [[makestring(default) for _ in range(self.ncols)] for _ in range(self.nrows)]
# Allows 1-row input grids to use non-nested lists for default values
elif not all(isinstance(entry, (list,tuple)) for entry in default):
# the above test will exhaust an iterator...
self.default = [[makestring(default[i * self.ncols + j]) for j in range(self.ncols)] for i in range(self.nrows)]
else:
self.default = [[makestring(default[r][c]) for c in range(self.ncols)] for r in range(self.nrows)]
def build_poly(vts, ts, is_max_plus):
"""
Build a MPP convex polyhedron over vts and
return a set of constraints
Examples:
sage: var('y')
y
sage: rs = IeqMPPGen.build_poly([[0,0,0],[3,3,0]], [x,y,SR(0)], is_max_plus=True)
dig_polynomials:Debug:Build (gen max-plus) poly from 2 vts in 3 dims: [x, y, 0]
sage: print '\n'.join(map(str,sorted(rs)))
('lambda x,y: -x + y >= 0', 'y >= x')
('lambda x,y: x - y >= 0', 'x >= y')
('lambda x: -x + 3 >= 0', '0 >= x - 3')
('lambda x: x >= 0', 'x >= 0')
('lambda y: -y + 3 >= 0', '0 >= y - 3')
('lambda y: y >= 0', 'y >= 0')
"""
opt_arg = ''
if is_max_plus:
mpp_str = 'max-plus'
else:
mpp_str = 'min-plus'
opt_arg = "{} {}".format(opt_arg, '-{}'.format(mpp_str))
# if any vertex is float
if any(any(not SR(v).is_integer() for v in vt) for vt in vts):
vts = [[RR(v).n() for v in vt] for vt in vts]
opt_arg = "{} -numerical-data ocaml_float".format(opt_arg)
# exec external program
# important, has to end with newline \n !!!
vts_s = '\n'.join(str(vt).replace(' ', '') for vt in vts) + '\n'
logger.debug('Build (gen {}) poly from {} vts in {} dims: {}'.format(
mpp_str, len(vts), len(vts[0]), ts))
cmd = 'compute_ext_rays_polar {} {}'.format(opt_arg, len(vts[0]))
rs, _ = vcmd(cmd, vts_s)
rs = [sage_eval(s.replace('oo', 'Infinity')) for s in rs.split()]
rs = IeqMPPGen.group_rs(rs)
rs = map(lambda ls: [x + y for x, y in zip(ls, ts + ts)], rs)
rs = map(lambda ls: IeqMPP.sparse(ls, is_max_plus), rs)
rs = filter(None, rs)
return rs
def build_poly(vts, ts, is_max_plus):
"""
Build a MPP convex polyhedron over vts and
return a set of constraints
Examples:
sage: logger.set_level(VLog.DEBUG)
sage: IeqMPP.build_poly([[0,0,0],[3,3,0]], is_max_plus=True)
dig_polynomials:Debug:Ieq: Build (MPP max-plus) polyhedra from 2 vertices in 3 dim
['[-oo,0,-oo,-oo,-oo,0]', '[0,-oo,-oo,-oo,-oo,0]', '[0,-oo,-oo,-oo,-oo,-oo]',
'[-oo,0,-oo,-oo,-oo,-oo]', '[-oo,-oo,0,-oo,-oo,-oo]', '[-oo,-oo,0,-oo,-oo,0]',
'[-oo,-oo,0,-oo,-3,-oo]', '[-oo,-oo,0,-3,-oo,-oo]', '[-oo,0,-oo,-oo,0,-oo]',
'[-oo,0,-oo,0,-oo,-oo]', '[0,-oo,-oo,-oo,0,-oo]', '[0,-oo,-oo,0,-oo,-oo]']
"""
opt_arg = ''
if is_max_plus:
mpp_str = 'max-plus'
else:
mpp_str = 'min-plus'
opt_arg = "{} {}".format(opt_arg, '-{}'.format(mpp_str))
#if any vertex is float
if any(any(not SR(v).is_integer() for v in vt) for vt in vts):
vts = [[RR(v).n() for v in vt] for vt in vts]
opt_arg = "{} -numerical-data ocaml_float".format(opt_arg)
#exec external program
#important, has to end with newline \n !!!
vts_s = '\n'.join(str(vt).replace(' ', '') for vt in vts) + '\n'
logger.debug('Build (gen {}) poly from {} vts in {} dims: {}'.format(
mpp_str, len(vts), len(vts[0]), ts))
cmd = 'compute_ext_rays_polar {} {}'.format(opt_arg, len(vts[0]))
rs, _ = vcmd(cmd, vts_s)
rs = [sage_eval(s.replace('oo', 'Infinity')) for s in rs.split()]
rs = IeqMPPGen.group_rs(rs)
rs = map(lambda ls: [x + y for x, y in zip(ls, ts + ts)], rs)
rs = map(lambda ls: IeqMPP.sparse(ls, is_max_plus), rs)
rs = filter(None, rs)
return rs
def build_poly(vts, ts, is_max_plus):
"""
Build a MPP convex polyhedron over vts and
return a set of constraints
Examples:
sage: logger.set_level(VLog.DEBUG)
sage: IeqMPP.build_poly([[0,0,0],[3,3,0]], is_max_plus=True)
dig_polynomials:Debug:Ieq: Build (MPP max-plus) polyhedra from 2 vertices in 3 dim
['[-oo,0,-oo,-oo,-oo,0]', '[0,-oo,-oo,-oo,-oo,0]', '[0,-oo,-oo,-oo,-oo,-oo]',
'[-oo,0,-oo,-oo,-oo,-oo]', '[-oo,-oo,0,-oo,-oo,-oo]', '[-oo,-oo,0,-oo,-oo,0]',
'[-oo,-oo,0,-oo,-3,-oo]', '[-oo,-oo,0,-3,-oo,-oo]', '[-oo,0,-oo,-oo,0,-oo]',
'[-oo,0,-oo,0,-oo,-oo]', '[0,-oo,-oo,-oo,0,-oo]', '[0,-oo,-oo,0,-oo,-oo]']
"""
opt_arg = ''
if is_max_plus:
mpp_str = 'max-plus'
else:
mpp_str = 'min-plus'
opt_arg = "{} {}".format(opt_arg, '-{}'.format(mpp_str))
#if any vertex is float
if any(any(not SR(v).is_integer() for v in vt) for vt in vts):
vts = [[RR(v).n() for v in vt] for vt in vts]
opt_arg = "{} -numerical-data ocaml_float".format(opt_arg)
#exec external program
#important, has to end with newline \n !!!
vts_s = '\n'.join(str(vt).replace(' ','') for vt in vts) + '\n'
logger.debug('Build (gen {}) poly from {} vts in {} dims: {}'
.format(mpp_str,len(vts),len(vts[0]),ts))
cmd = 'compute_ext_rays_polar {} {}'.format(opt_arg, len(vts[0]))
rs,_ = vcmd(cmd, vts_s)
rs = [sage_eval(s.replace('oo','Infinity')) for s in rs.split()]
rs = IeqMPPGen.group_rs(rs)
rs = map(lambda ls:[x+y for x,y in zip(ls,ts+ts)], rs)
rs = map(lambda ls: IeqMPP.sparse(ls,is_max_plus), rs)
rs = filter(None, rs)
return rs
def safe_sage_eval(code, globs):
"""
Evaluate an expression using sage_eval,
returning an ``Exception`` object if an exception occurs
:arg str code: the expression to evaluate
:arg dict globs: the global dictionary in which to evaluate the expression
:returns: the value of the expression. If an exception occurs, the
``Exception`` object is returned instead.
"""
try:
try:
from sage.all import sage_eval
return sage_eval(code, globs)
except ImportError:
return eval(code, globs)
except Exception as e:
return e
def download_search(info):
lang = info["submit"]
filename = 'Hecke_algebras' + download_file_suffix[lang]
mydate = time.strftime("%d %B %Y")
# reissue saved query here
if 'ell' in info["query"]:
res = list(db.hecke_ladic.search(ast.literal_eval(info["query"])))
else:
res = list(db.hecke_orbits.search(ast.literal_eval(info["query"])))
last = len(res)
c = download_comment_prefix[lang]
s = '\n'
if 'ell' in info["query"]:
s += c + ' Hecke algebras downloaded from the LMFDB on %s. Found %s algebras. The data is given in the following format: it is a list of lists, each containing level, weight and the Hecke orbits for which l-adic data is available.\n\n'%(mydate, len(res))
else:
s += c + ' Hecke algebras downloaded from the LMFDB on %s. Found %s algebras. The data is given in the following format: it is a list of lists, each containing level, weight, list of the first 10 Hecke operators (to download more operators for a given algebra, please visit its webpage).\n\n'%(mydate, len(res))
# The list entries are matrices of different sizes. Sage and gp
# do not mind this but Magma requires a different sort of list.
list_start = '[*' if lang=='magma' else '['
list_end = '*]' if lang=='magma' else ']'
s += download_assignment_start[lang] + list_start + '\\\n'
mat_start = "Mat(" if lang == 'gp' else "Matrix("
mat_end = "~)" if lang == 'gp' else ")"
entry = lambda r: "".join([mat_start,str(r),mat_end])
# loop through all search results and grab the Hecke operators stored
for c, rr in enumerate(res):
s += list_start
s += ",".join([str(rr['level']), str(rr['weight']),""])
if 'ell' in info["query"]:
s += '"%s"' % (str(rr['orbit_label']))
else:
s += ",".join([entry(r) for r in [sage_eval(rr['hecke_op'])[i] for i in range(0, min(10, rr['num_hecke_op']))]])
if c != last:
s += list_end + ',\\\n'
else:
s += list_end
s += list_end
s += download_assignment_end[lang]
s += '\n'
strIO = StringIO.StringIO()
strIO.write(s)
strIO.seek(0)
return send_file(strIO, attachment_filename=filename, as_attachment=True, add_etags=False)
def download_hecke_algebras_full_lists_op(**args):
label = str(args['orbit_label'])
res = hecke_orbits_db().find_one({'orbit_label': label})
mydate = time.strftime("%d %B %Y")
if res is None:
return "No operators available"
lang = args['lang']
c = download_comment_prefix[lang]
mat_start = "Mat(" if lang == 'gp' else "Matrix("
mat_end = "~)" if lang == 'gp' else ")"
entry = lambda r: "".join([mat_start,str(r),mat_end])
outstr = c + 'Hecke algebra for Gamma0(%s) and weight %s, orbit label %s. List of Hecke operators T_1, ..., T_%s. Downloaded from the LMFDB on %s. \n\n'%(res['level'], res['weight'], res['orbit_label'],res['num_hecke_op'], mydate)
outstr += download_assignment_start[lang] + '[\\\n'
outstr += ",\\\n".join([entry(r) for r in [sage_eval(res['hecke_op'])[i] for i in range(0,res['num_hecke_op'])]])
outstr += ']'
outstr += download_assignment_end[lang]
outstr += '\n'
return outstr
def download_hecke_algebras_full_lists_op(**args):
label = str(args['orbit_label'])
res = db.hecke_orbits.lucky({'orbit_label': label})
mydate = time.strftime("%d %B %Y")
if res is None:
return "No operators available"
lang = args['lang']
c = download_comment_prefix[lang]
mat_start = "Mat(" if lang == 'gp' else "Matrix("
mat_end = "~)" if lang == 'gp' else ")"
entry = lambda r: "".join([mat_start,str(r),mat_end])
outstr = c + 'Hecke algebra for Gamma0(%s) and weight %s, orbit label %s. List of Hecke operators T_1, ..., T_%s. Downloaded from the LMFDB on %s. \n\n'%(res['level'], res['weight'], res['orbit_label'],res['num_hecke_op'], mydate)
outstr += download_assignment_start[lang] + '[\\\n'
outstr += ",\\\n".join([entry(r) for r in [sage_eval(res['hecke_op'])[i] for i in range(0,res['num_hecke_op'])]])
outstr += ']'
outstr += download_assignment_end[lang]
outstr += '\n'
return outstr
def compute_conjectural_ranks(level_norms, address='localhost:29000'):
"""
For each level norm in the input list, compute the
*conjectural* rank of the elliptic curve, and put
that data in a field "r?" in the database.
This uses Simon 2-descent if it works, and otherwise
uses Dokchitser. This should not be considered
super-reliable!
"""
from sage.all import sage_eval
import util
C = util.ellcurves_sqrt5(address)
for N in level_norms:
for E in C.find({'Nlevel':int(N), 'r?':{'$exists':False}, 'weq':{'$exists':True}}):
print E
weq = sage_eval(E['weq'], {'a':F.gen()})
E['r?'] = int(rank(weq))
print E
C.update({'_id':E['_id']}, E, safe=True)
def get_constraints(m, result_as_dict=False):
"""
Input a model m, returns its set of constraints in either
1) sage dict {x:7,y:10}
1) z3 expr [x==7,y==0]
sage: S = z3.Solver()
sage: S.add(z3.Int('x') + z3.Int('y') == z3.IntVal('7'))
sage: S.check()
sat
sage: M = S.model()
sage: d = SMT_Z3.get_constraints(M, result_as_dict=True)
sage: sorted(d.items(), key=lambda(k,_): str(k))
[(x, 7), (y, 0)]
sage: SMT_Z3.get_constraints(M)
[y == 0, x == 7]
sage: S.reset()
"""
assert m is not None, m
if result_as_dict: #sage format
rs = [(var(str(v())),sage_eval(str(m[v]))) for v in m]
rs = dict(rs)
if __debug__:
assert all(is_sage_expr(x) for x in rs.keys())
assert all(is_sage_real(x) or is_sage_int(x) for x in rs.values())
else: #z3 format
rs = [v()==m[v] for v in m]
if __debug__:
assert all(z3.is_expr(x) for x in rs)
return rs
def verify_curve_lmfdb(label, Lhash = None):
from lmfdb import getDBconnection
C = getDBconnection()
curve = C.genus2_curves.curves.find_one({"label" : label})
endo_query = C.genus2_curves.endomorphisms.find_one({'label': label})
rank_endo = rank_from_endo(endo_query['factorsRR_geom'])
if not endo_query['is_simple_geom']:
assert rank_endo > 1
if Lhash is None:
Lhash = curve['Lhash'];
Lfunction_query = C. Lfunctions.Lfunctions.find_one({'Lhash': Lhash})
ef = sage_eval(str(Lfunction_query['euler_factors']))
rank_euler, _ = upperrank(ef)
if endo_query['factorsRR_geom'] == [u'RR', u'RR'] and curve['is_simple_geom']:
RM_coeff = endo_query['factorsQQ_geom'][0][1]
checkRM = RM_and_notCM(ef, RM_coeff)
return rank_endo == rank_euler and checkRM, rank_endo, rank_euler
return rank_endo == rank_euler, rank_endo, rank_euler
def render_hecke_algebras_webpage_l_adic(**args):
data = None
if 'orbit_label' in args and 'prime' in args:
lab = clean_input(args.get('orbit_label'))
if lab != args.get('orbit_label'):
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
try:
ell = int(args.get('prime'))
except ValueError:
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
data = hecke_l_adic_db().find_one({'orbit_label': lab , 'ell': ell})
if data is None:
t = "Hecke Agebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for the ℓ-adic information for an Hecke Algebra orbit in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
res = hecke_l_adic_db().find({'level': data['level'],'weight': data['weight'],'orbit_label': data['orbit_label'], 'ell': data['ell']})
info['num_l_adic_orbits']=res.count()
res_clean = []
for f in res:
f_clean = {}
f_clean['index']=int(f['index'])
f_clean['orbit_label']=str(f['orbit_label'])
f_clean['ell']=int(f['ell'])
if f['idempotent'] != "":
f['dim']=len(sage_eval(f['idempotent']))
l_max= sage_eval(f['idempotent'])[0][0].ndigits()
if f['dim']>4 or l_max>5:
f_clean['idempotent_display']=[]
elif f['dim']==1:
f_clean['idempotent_display']=sage_eval(f['idempotent'])[0][0]
else:
f_clean['idempotent_display']=latex(matrix(sage_eval(f['idempotent'])))
else:
f_clean['idempotent_display']=latex(matrix([[1]]))
f_clean['download_id'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='idempotents')) for i in ['magma','sage']] # for 'gp' the code does not work, since p-adics are not implemented
try:
f_clean['deg']=int(f['field'][1])
f_clean['field_poly']=str(f['field'][2])
f_clean['dim']=int(f['structure'][0])
f_clean['num_gen']=int(f['structure'][1])
f_clean['gens']=[[int(sage_eval(f['structure'][2]).index(i)+1), str(i)] for i in sage_eval(f['structure'][2])]
f_clean['rel']=sage_eval(f['structure'][3])
f_clean['grading']=[int(i) for i in f['properties'][0]]
f_clean['gorenstein_def']=int(f['properties'][1])
if f_clean['gorenstein_def']==0:
f_clean['gorenstein']="yes"
else:
f_clean['gorenstein']="no"
f_clean['operators_mod_l']=[[int(i) for i in j] for j in f['operators']]
f_clean['num_hecke_op']=len(f_clean['operators_mod_l'])
f_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='operators')) for i in ['magma','sage']] # for 'gp' the code does not work
f_clean['size_op']=sqrt(len(f_clean['operators_mod_l'][0]))
if f_clean['size_op']>4:
f_clean['operators_mod_l_display']=[]
elif f_clean['size_op']==1:
f_clean['operators_mod_l_display']=[[i+1, f_clean['operators_mod_l'][i][0]] for i in range(0,10)]
else:
f_clean['operators_mod_l_display']=[[i+1,latex(matrix(f_clean['size_op'],f_clean['size_op'], f_clean['operators_mod_l'][i]))] for i in range(0,5)]
res_clean.append(f_clean)
except KeyError:
pass
info['l_adic_orbits']=res_clean
info['level']=int(data['level'])
info['weight']= int(data['weight'])
info['base_lab']=".".join([split(data['orbit_label'])[i] for i in [0,1,2]])
info['orbit_label']= str(data['orbit_label'])
info['ell']=int(data['ell'])
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % info['base_lab'], url_for('.render_hecke_algebras_webpage', label=info['base_lab'])), ('%s' % info['ell'], ' ')]
credit = hecke_algebras_credit
info['properties'] = [
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight']),
('Characteristic', '%s' %info['ell']),
('Orbit label', '%s' %info['orbit_label'])]
info['friends'] = [('Modular form ' + info['base_lab'], url_for("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
t = "%s-adic and mod %s data for the Hecke Algebra orbit %s" % (info['ell'], info['ell'], info['orbit_label'])
return render_template("hecke_algebras_l_adic-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
def _element_constructor_(self, x):
"""
Convert ``x`` into ``self``.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: R(3) # indirect doctest
3
TESTS::
sage: F.<x,y,z> = FreeAlgebra(GF(5),3)
sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace')
sage: F(x) # indirect doctest
x
sage: F.1*L.2
y*z
sage: (F.1*L.2).parent() is F
True
::
sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K,3)
sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace')
sage: F.1+(z+1)*L.2
b + (z+1)*c
Check that :trac:`15169` is fixed::
sage: A.<x> = FreeAlgebra(CC)
sage: A(2)
2.00000000000000
We check that the string coercions work correctly over
inexact fields::
sage: F.<x,y> = FreeAlgebra(CC)
sage: F('2')
2.00000000000000
sage: F('x')
1.00000000000000*x
Check that it also converts factorizations::
sage: f = Factorization([(x,2),(y,3)]); f
1.00000000000000*x^2 * 1.00000000000000*y^3
sage: F(f)
1.00000000000000*x^2*y^3
"""
if isinstance(x, FreeAlgebraElement):
P = x.parent()
if P is self:
return x
if P is not self.base_ring():
return self.element_class(self, x)
elif hasattr(x, 'letterplace_polynomial'):
P = x.parent()
if self.has_coerce_map_from(P): # letterplace versus generic
ngens = P.ngens()
M = self._indices
def exp_to_monomial(T):
out = []
for i in range(len(T)):
if T[i]:
out.append((i % ngens, T[i]))
return M(out)
return self.element_class(
self, {
exp_to_monomial(T): c
for T, c in x.letterplace_polynomial().dict().items()
})
# ok, not a free algebra element (or should not be viewed as one).
if isinstance(x, str):
from sage.all import sage_eval
G = self.gens()
d = {str(v): G[i] for i, v in enumerate(self.variable_names())}
return self(sage_eval(x, locals=d))
R = self.base_ring()
# coercion from free monoid
if isinstance(x, FreeMonoidElement) and x.parent() is self._indices:
return self.element_class(self, {x: R.one()})
# coercion from the PBW basis
if isinstance(x, PBWBasisOfFreeAlgebra.Element) \
and self.has_coerce_map_from(x.parent()._alg):
return self(x.parent().expansion(x))
# Check if it's a factorization
from sage.structure.factorization import Factorization
if isinstance(x, Factorization):
return self.prod(f**i for f, i in x)
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self, {})
return self.element_class(self, {self.one_basis(): x})
def eval_lambda(f, idx_info, tc):
"""
Evaluate array expression p, e.g. p: A[i,j,k]=B[2i+3j+k]
if idx_info is specified then only test p on the idxs from idx_info
Assumes:
the first array in lambda is the pivot
lambda A,B,i,j,k: ... => i,j,k belong to A
inv = 'lambda B,C,D,i,j: B[i][j]=C[D[2i+3]]'
returns true if inv is true on tc
Examples:
sage: var('a,b,c,i,j')
(a, b, c, i, j)
sage: InvArray.eval_lambda('lambda a,b,c,i,j: a[i][j]==2*b[i]+c[j]', None, {'a':[[4,-5],[20,11]],'b':[1,9],'c':[2,-7]})
True
sage: InvArray.eval_lambda('lambda c,a,b,xor,i: c[i] == xor(a[i],b[i])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 334, 79, 281]})
False
sage: InvArray.eval_lambda('lambda c,a,b,xor,i1: c[i1] == xor(a[i1],b[i1])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 134, 79, 241]})
True
sage: InvArray.eval_lambda('lambda rvu, t, rvu1, rvu0: (rvu[rvu0][rvu1]) + (-t[4*rvu0 + rvu1]) == 0', None, {'t': [28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], 'rvu': [[28, 131, 11, 85], [133, 46, 179, 20], [227, 148, 225, 197], [38, 221, 221, 126]]})
True
#The following illustrate the use of idxVals,
#i.e. p is only true under certain array rages
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', None, tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
False
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', idx_info = [{'st0': 0, 'st1': 0}, {'st0': 0, 'st1': 1}, {'st0': 2, 'st1': 2}, {'st0': 2, 'st1': 3}, {'st0': 3, 'st1': 0}, {'st0': 3, 'st1': 1}, {'st0': 3, 'st1': 2}, {'st0': 3, 'st1': 3}, {'st0': 0, 'st1': 2}, {'st0': 0, 'st1': 3}, {'st0': 1, 'st1': 0}, {'st0': 1, 'st1': 1}, {'st0': 1, 'st1': 2}, {'st0': 1, 'st1': 3}, {'st0': 2, 'st1': 0}, {'st0': 2, 'st1': 1}], tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
True
"""
"""
Note: sage_eval vs eval
sage_eval works on str of the format 'lambda x,y: 2*x+y'
whereas eval works on str of the format 2*x+y directly (no lambda)
Also, the result of sage_eval can apply on dicts whose keys are str
e.g. f(**{'x':2,'y':3})
whereas the result of eval applies on dict whose keys are variables
e.g. f(**{x:2,y:3})
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert (idx_info is None
or is_list(idx_info) and all(is_dict(v)
for v in idx_info)), indx_info
assert is_dict(tc), tc
assert all(is_str(k) for k in tc), tc.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
arrs = [v for v in vs if v in tc] #A,B
extfuns = [v for v in vs if v in ExtFun.efdict]
idxStr = [v for v in vs if v not in arrs + extfuns] #i,j,k
d_tc = dict([(v, tc[v]) for v in arrs])
d_extfun = dict([(v, ExtFun(v).get_fun()) for v in extfuns])
d_ = merge_dict([d_tc, d_extfun])
if idx_info is None: #obtain idxsVals from the pivot array
pivotContents = tc[arrs[0]]
idxVals = [idx for idx, _ in Miscs.travel(pivotContents)]
idx_info = [dict(zip(idxStr, idxV)) for idxV in idxVals]
ds = [merge_dict([d_, idx_info_]) for idx_info_ in idx_info]
try:
return all(f(**d) for d in ds)
except IndexError:
return False
except TypeError:
return False
except NameError as msg:
logger.warn(msg)
return False
def _element_constructor_(self, x):
"""
Convert ``x`` into ``self``.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: R(3) # indirect doctest
3
TESTS::
sage: F.<x,y,z> = FreeAlgebra(GF(5),3)
sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace')
sage: F(x) # indirect doctest
x
sage: F.1*L.2
y*z
sage: (F.1*L.2).parent() is F
True
::
sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K,3)
sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace')
sage: F.1+(z+1)*L.2
b + (z+1)*c
Check that :trac:`15169` is fixed::
sage: A.<x> = FreeAlgebra(CC)
sage: A(2)
2.00000000000000
We check that the string coercions work correctly over
inexact fields::
sage: F.<x,y> = FreeAlgebra(CC)
sage: F('2')
2.00000000000000
sage: F('x')
1.00000000000000*x
Check that it also converts factorizations::
sage: f = Factorization([(x,2),(y,3)]); f
1.00000000000000*x^2 * 1.00000000000000*y^3
sage: F(f)
1.00000000000000*x^2*y^3
"""
if isinstance(x, FreeAlgebraElement):
P = x.parent()
if P is self:
return x
if P is not self.base_ring():
return self.element_class(self, x)
elif hasattr(x,'letterplace_polynomial'):
P = x.parent()
if self.has_coerce_map_from(P): # letterplace versus generic
ngens = P.ngens()
M = self._indices
def exp_to_monomial(T):
out = []
for i in xrange(len(T)):
if T[i]:
out.append((i%ngens,T[i]))
return M(out)
return self.element_class(self, dict([(exp_to_monomial(T),c) for T,c in x.letterplace_polynomial().dict().iteritems()]))
# ok, not a free algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.all import sage_eval
G = self.gens()
d = {str(v): G[i] for i,v in enumerate(self.variable_names())}
return self(sage_eval(x, locals=d))
R = self.base_ring()
# coercion from free monoid
if isinstance(x, FreeMonoidElement) and x.parent() is self._indices:
return self.element_class(self, {x: R.one()})
# coercion from the PBW basis
if isinstance(x, PBWBasisOfFreeAlgebra.Element) \
and self.has_coerce_map_from(x.parent()._alg):
return self(x.parent().expansion(x))
# Check if it's a factorization
from sage.structure.factorization import Factorization
if isinstance(x, Factorization):
return self.prod(f**i for f,i in x)
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self, {})
return self.element_class(self, {self.one_basis(): x})
def render_hecke_algebras_webpage(**args):
data = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(
url_for('.render_hecke_algebras_webpage', label=lab), 301)
data = db.hecke_algebras.lookup(lab)
if data is None:
t = "Hecke algebra search error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash_error(
"%s is not a valid label for a Hecke algebra in the database.",
lab)
return render_template("hecke_algebras-error.html",
title=t,
properties=[],
bread=bread,
learnmore=learnmore_list())
info = {}
info.update(data)
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")),
('%s' % data['label'], ' ')]
credit = hecke_algebras_credit
info['level'] = int(data['level'])
info['weight'] = int(data['weight'])
info['num_orbits'] = int(data['num_orbits'])
dim_count = "not available"
proj = [
'orbit_label', 'hecke_op', 'num_hecke_op', 'Zbasis', 'discriminant',
'disc_fac', 'Qbasis', 'Qalg_gen'
]
orb = list(db.hecke_orbits.search({'parent_label': data['label']}, proj))
if orb:
#consistency check
if len(orb) != int(data['num_orbits']):
return search_input_error(info)
dim_count = 0
for v in orb:
ops = sage_eval(v['hecke_op'])
dim = int(matrix(ops[0]).nrows())
dim_count += dim
if dim > 4:
v['hecke_op_display'] = []
elif dim == 1:
v['hecke_op_display'] = [[i + 1, ops[i][0][0]]
for i in range(10)]
else:
v['hecke_op_display'] = [[i + 1, latex(matrix(ops[i]))]
for i in range(5)]
v['download_op'] = [
(lang,
url_for(".render_hecke_algebras_webpage_download",
orbit_label=v['orbit_label'],
lang=lang,
obj='operators')) for lang in ['gp', 'magma', 'sage']
]
if v['Zbasis'] is None:
for key in [
'Zbasis', 'discriminant', 'disc_fac', 'Qbasis',
'Qalg_gen'
]:
del v[key]
else:
v['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
v['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
if dim > 4:
v['gen_display'] = []
elif dim == 1:
v['gen_display'] = [v['Zbasis'][0][0]]
else:
v['gen_display'] = [
latex(matrix(dim, dim, v['Zbasis'][i]))
for i in range(dim)
]
v['inner_twists'] = "not available" # not yet in the database
v['download_gen'] = [
(lang,
url_for(".render_hecke_algebras_webpage_download",
orbit_label=v['orbit_label'],
lang=lang,
obj='gen')) for lang in ['gp', 'magma', 'sage']
]
info['orbits'] = orb
info['dim_alg'] = dim_count
info['l_adic'] = l_range
info['properties'] = [('Label', '%s' % info['label']),
('Level', '%s' % info['level']),
('Weight', '%s' % info['weight'])]
if info['num_orbits'] != 0:
info['friends'] = [('Newforms space ' + info['label'],
url_for("cmf.by_url_space_label",
level=info['level'],
weight=info['weight'],
char_orbit_label='a'))]
else:
info['friends'] = []
t = "Hecke algebra %s" % info['label']
return render_template("hecke_algebras-single.html",
info=info,
credit=credit,
title=t,
bread=bread,
properties=info['properties'],
learnmore=learnmore_list(),
friends=info['friends'],
KNOWL_ID='hecke_algebra.%s' % (info['label']))
def main():
if len(sys.argv) == 2:
print(prime_range(sage_eval(sys.argv[1])))
def _element_constructor_(self, x):
"""
Convert x into self.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: R(3) # indirect doctest
3
TESTS::
sage: F.<x,y,z> = FreeAlgebra(GF(5),3)
sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace')
sage: F(x) # indirect doctest
x
sage: F.1*L.2
y*z
sage: (F.1*L.2).parent() is F
True
::
sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K,3)
sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace')
sage: F.1+(z+1)*L.2
b + (z+1)*c
"""
if isinstance(x, FreeAlgebraElement):
P = x.parent()
if P is self:
return x
if not (P is self.base_ring()):
return self.element_class(self, x)
elif hasattr(x,'letterplace_polynomial'):
P = x.parent()
if self.has_coerce_map_from(P): # letterplace versus generic
ngens = P.ngens()
M = self.__monoid
def exp_to_monomial(T):
out = []
for i in xrange(len(T)):
if T[i]:
out.append((i%ngens,T[i]))
return M(out)
return self.element_class(self, dict([(exp_to_monomial(T),c) for T,c in x.letterplace_polynomial().dict().iteritems()]))
# ok, not a free algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.all import sage_eval
return sage_eval(x,locals=self.gens_dict())
F = self.__monoid
R = self.base_ring()
# coercion from free monoid
if isinstance(x, FreeMonoidElement) and x.parent() is F:
return self.element_class(self,{x:R(1)})
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self,{})
else:
return self.element_class(self,{F(1):x})
def render_hecke_algebras_webpage_l_adic(**args):
data = None
if 'orbit_label' in args and 'prime' in args:
lab = clean_input(args.get('orbit_label'))
if lab != args.get('orbit_label'):
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
try:
ell = int(args.get('prime'))
except ValueError:
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
data = db.hecke_ladic.lucky({'orbit_label': lab , 'ell': ell})
if data is None:
t = "Hecke Algebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for the ℓ-adic information for an Hecke Algebra orbit in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
proj = ['index','orbit_label','ell','idempotent','field','structure','properties','operators']
res = list(db.hecke_ladic.search({'level': data['level'],'weight': data['weight'],'orbit_label': data['orbit_label'], 'ell': data['ell']}, proj))
for f in res:
if f['idempotent'] != "":
dim = len(sage_eval(f['idempotent']))
l_max = sage_eval(f['idempotent'])[0][0].ndigits()
if dim > 4 or l_max > 5:
f['idempotent_display'] = []
elif dim == 1:
f['idempotent_display'] = sage_eval(f['idempotent'])[0][0]
else:
f['idempotent_display'] = latex(matrix(sage_eval(f['idempotent'])))
else:
f['idempotent_display']=latex(matrix([[1]]))
del f['idempotent']
f['download_id'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f['orbit_label'], index=f['index'], prime=f['ell'], lang=i, obj='idempotents')) for i in ['magma','sage']] # for 'gp' the code does not work, since p-adics are not implemented
field = f.pop('field')
if field is not None:
f['deg'] = field[1]
f['field_poly'] = field[2]
structure = f.pop('structure')
if structure is not None:
f['dim'] = structure[0]
f['num_gen'] = structure[1]
s2 = sage_eval(structure[2])
f['gens'] = [[int(s2.index(i)+1), str(i)] for i in s2]
f['rel'] = sage_eval(structure[3])
properties = f.pop('properties')
if properties is not None:
f['grading'] = properties[0]
f['gorenstein_def'] = properties[1]
f['gorenstein'] = "yes" if properties[1] == 0 else "no"
operators = f.pop('operators')
if operators is not None:
f['operators_mod_l'] = operators
f['num_hecke_op'] = len(operators)
f['size_op'] = size = sqrt(len(operators[0]))
if size > 4:
f['operators_mod_l_display'] = []
elif size == 1:
f['operators_mod_l_display'] = [[i+1, operators[i][0]] for i in range(10)]
else:
f['operators_mod_l_display'] = [[i+1, latex(matrix(size,size,operators[i]))] for i in range(5)]
f['download_op'] = [(lang, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f['orbit_label'], index=f['index'], prime=f['ell'], lang=lang, obj='operators')) for lang in ['magma','sage']] # for 'gp' the code does not work
info['num_l_adic_orbits'] = len(res)
info['l_adic_orbits'] = res
info['level']=int(data['level'])
info['weight']= int(data['weight'])
info['base_lab']=".".join([split(data['orbit_label'])[i] for i in [0,1,2]])
info['orbit_label']= str(data['orbit_label'])
info['ell']=int(data['ell'])
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % info['base_lab'], url_for('.render_hecke_algebras_webpage', label=info['base_lab'])), ('%s' % info['ell'], ' ')]
credit = hecke_algebras_credit
info['properties'] = [
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight']),
('Characteristic', '%s' %info['ell']),
('Orbit label', '%s' %info['orbit_label'])]
info['friends'] = [('Modular form ' + info['base_lab'], url_for("cmf.by_url_space_label", level=info['level'], weight=info['weight'], char_orbit_label='a'))]
t = "%s-adic and mod %s Data for the Hecke Algebra Orbit %s" % (info['ell'], info['ell'], info['orbit_label'])
return render_template("hecke_algebras_l_adic-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
def get_number_from(e):
if isinstance(e,unicode):
return S.sage_eval(e)
else:
return e
def _element_constructor_(self, x):
"""
Convert ``x`` into ``self``.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: R(3) # indirect doctest
3
TESTS::
sage: F.<x,y,z> = FreeAlgebra(GF(5),3)
sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace')
sage: F(x) # indirect doctest
x
sage: F.1*L.2
y*z
sage: (F.1*L.2).parent() is F
True
::
sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K,3)
sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace')
sage: F.1+(z+1)*L.2
b + (z+1)*c
Check that :trac:`15169` is fixed::
sage: A.<x> = FreeAlgebra(CC)
sage: A(2)
2.00000000000000
"""
if isinstance(x, FreeAlgebraElement):
P = x.parent()
if P is self:
return x
if not (P is self.base_ring()):
return self.element_class(self, x)
elif hasattr(x, 'letterplace_polynomial'):
P = x.parent()
if self.has_coerce_map_from(P): # letterplace versus generic
ngens = P.ngens()
M = self._basis_keys
def exp_to_monomial(T):
out = []
for i in xrange(len(T)):
if T[i]:
out.append((i % ngens, T[i]))
return M(out)
return self.element_class(
self,
dict([(exp_to_monomial(T), c) for T, c in
x.letterplace_polynomial().dict().iteritems()]))
# ok, not a free algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.all import sage_eval
return sage_eval(x, locals=self.gens_dict())
R = self.base_ring()
# coercion from free monoid
if isinstance(x, FreeMonoidElement) and x.parent() is self._basis_keys:
return self.element_class(self, {x: R(1)})
# coercion from the PBW basis
if isinstance(x, PBWBasisOfFreeAlgebra.Element) \
and self.has_coerce_map_from(x.parent()._alg):
return self(x.parent().expansion(x))
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self, {})
return self.element_class(self, {self._basis_keys.one(): x})
def render_hecke_algebras_webpage_l_adic(**args):
data = None
if 'orbit_label' in args and 'prime' in args:
lab = clean_input(args.get('orbit_label'))
if lab != args.get('orbit_label'):
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
try:
ell = int(args.get('prime'))
except ValueError:
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
data = hecke_l_adic_db().find_one({'orbit_label': lab , 'ell': ell})
if data is None:
t = "Hecke Algebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for the ℓ-adic information for an Hecke Algebra orbit in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
res = hecke_l_adic_db().find({'level': data['level'],'weight': data['weight'],'orbit_label': data['orbit_label'], 'ell': data['ell']})
info['num_l_adic_orbits']=res.count()
res_clean = []
for f in res:
f_clean = {}
f_clean['index']=int(f['index'])
f_clean['orbit_label']=str(f['orbit_label'])
f_clean['ell']=int(f['ell'])
if f['idempotent'] != "":
f['dim']=len(sage_eval(f['idempotent']))
l_max= sage_eval(f['idempotent'])[0][0].ndigits()
if f['dim']>4 or l_max>5:
f_clean['idempotent_display']=[]
elif f['dim']==1:
f_clean['idempotent_display']=sage_eval(f['idempotent'])[0][0]
else:
f_clean['idempotent_display']=latex(matrix(sage_eval(f['idempotent'])))
else:
f_clean['idempotent_display']=latex(matrix([[1]]))
f_clean['download_id'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='idempotents')) for i in ['magma','sage']] # for 'gp' the code does not work, since p-adics are not implemented
try:
f_clean['deg']=int(f['field'][1])
f_clean['field_poly']=str(f['field'][2])
f_clean['dim']=int(f['structure'][0])
f_clean['num_gen']=int(f['structure'][1])
f_clean['gens']=[[int(sage_eval(f['structure'][2]).index(i)+1), str(i)] for i in sage_eval(f['structure'][2])]
f_clean['rel']=sage_eval(f['structure'][3])
f_clean['grading']=[int(i) for i in f['properties'][0]]
f_clean['gorenstein_def']=int(f['properties'][1])
if f_clean['gorenstein_def']==0:
f_clean['gorenstein']="yes"
else:
f_clean['gorenstein']="no"
f_clean['operators_mod_l']=[[int(i) for i in j] for j in f['operators']]
f_clean['num_hecke_op']=len(f_clean['operators_mod_l'])
f_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='operators')) for i in ['magma','sage']] # for 'gp' the code does not work
f_clean['size_op']=sqrt(len(f_clean['operators_mod_l'][0]))
if f_clean['size_op']>4:
f_clean['operators_mod_l_display']=[]
elif f_clean['size_op']==1:
f_clean['operators_mod_l_display']=[[i+1, f_clean['operators_mod_l'][i][0]] for i in range(0,10)]
else:
f_clean['operators_mod_l_display']=[[i+1,latex(matrix(f_clean['size_op'],f_clean['size_op'], f_clean['operators_mod_l'][i]))] for i in range(0,5)]
res_clean.append(f_clean)
except KeyError:
pass
info['l_adic_orbits']=res_clean
info['level']=int(data['level'])
info['weight']= int(data['weight'])
info['base_lab']=".".join([split(data['orbit_label'])[i] for i in [0,1,2]])
info['orbit_label']= str(data['orbit_label'])
info['ell']=int(data['ell'])
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % info['base_lab'], url_for('.render_hecke_algebras_webpage', label=info['base_lab'])), ('%s' % info['ell'], ' ')]
credit = hecke_algebras_credit
info['properties'] = [
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight']),
('Characteristic', '%s' %info['ell']),
('Orbit label', '%s' %info['orbit_label'])]
info['friends'] = [('Modular form ' + info['base_lab'], url_for("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
t = "%s-adic and mod %s Data for the Hecke Algebra Orbit %s" % (info['ell'], info['ell'], info['orbit_label'])
return render_template("hecke_algebras_l_adic-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
def render_hecke_algebras_webpage(**args):
data = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(url_for('.render_hecke_algebras_webpage', label=lab), 301)
data = hecke_db().find_one({'label': lab })
if data is None:
t = "Hecke Algebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for a Hecke Algebras in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % data['label'], ' ')]
credit = hecke_algebras_credit
f = hecke_db().find_one({'level': data['level'],'weight': data['weight'],'num_orbits': data['num_orbits']})
info['level']=int(f['level'])
info['weight']= int(f['weight'])
info['num_orbits']= int(f['num_orbits'])
dim_count = "not available"
orb = hecke_orbits_db().find({'parent_label': f['label']}).sort([('orbit', ASC)])
if orb.count()!=0:
#consistency check
if orb.count()!= int(f['num_orbits']):
return search_input_error(info)
dim_count=0
res_clean = []
for v in orb:
v_clean = {}
v_clean['orbit_label'] = v['orbit_label']
v_clean['hecke_op'] = v['hecke_op']
v_clean['gen'] = v['gen']
v_clean['dim'] = int(matrix(sage_eval(v_clean['hecke_op'])[0]).dimensions()[0])
dim_count = dim_count + v_clean['dim']
if v_clean['dim']>4:
v_clean['hecke_op_display'] = []
elif v_clean['dim']==1:
v_clean['hecke_op_display'] = [[i+1, (sage_eval(v_clean['hecke_op'])[i])[0][0]] for i in range(0,10)]
else:
v_clean['hecke_op_display'] = [[i+1, latex(matrix(sage_eval(v_clean['hecke_op'])[i]))] for i in range(0,5)]
v_clean['num_hecke_op'] = v['num_hecke_op']
v_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='operators')) for i in ['gp', 'magma','sage']]
if 'Zbasis' in v.keys():
v_clean['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
if v_clean['dim']>4:
v_clean['gen_display'] = []
elif v_clean['dim']==1:
v_clean['gen_display'] = [v_clean['Zbasis'][0][0]]
else:
v_clean['gen_display'] = [latex(matrix(v_clean['dim'],v_clean['dim'], v_clean['Zbasis'][i])) for i in range(0,v_clean['dim'])]
v_clean['discriminant'] = int(v['discriminant'])
v_clean['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
v_clean['Qbasis'] = [int(i) for i in v['Qbasis']]
v_clean['Qalg_gen'] = [int(i) for i in v['Qalg_gen']]
if 'inner_twists' in v.keys():
v_clean['inner_twists'] = [str(i) for i in v['inner_twists']]
else:
v_clean['inner_twists'] = "not available"
v_clean['download_gen'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='gen')) for i in ['gp', 'magma','sage']]
res_clean.append(v_clean)
info['orbits'] = res_clean
info['dim_alg'] = dim_count
info['l_adic'] = l_range
info['properties'] = [
('Label', '%s' %info['label']),
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight'])]
if info['num_orbits']!=0:
info['friends'] = [('Newforms space ' + info['label'], url_for("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
else:
info['friends'] = []
t = "Hecke Algebra %s" % info['label']
return render_template("hecke_algebras-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
def as_vector(self):
return sage_eval(self.get_value())
def _element_constructor_(self, x):
"""
Convert x into self.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: R(3) # indirect doctest
3
TESTS::
sage: F.<x,y,z> = FreeAlgebra(GF(5),3)
sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace')
sage: F(x) # indirect doctest
x
sage: F.1*L.2
y*z
sage: (F.1*L.2).parent() is F
True
::
sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K,3)
sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace')
sage: F.1+(z+1)*L.2
b + (z+1)*c
"""
if isinstance(x, FreeAlgebraElement):
P = x.parent()
if P is self:
return x
if not (P is self.base_ring()):
return self.element_class(self, x)
elif hasattr(x, 'letterplace_polynomial'):
P = x.parent()
if self.has_coerce_map_from(P): # letterplace versus generic
ngens = P.ngens()
M = self.__monoid
def exp_to_monomial(T):
out = []
for i in xrange(len(T)):
if T[i]:
out.append((i % ngens, T[i]))
return M(out)
return self.element_class(
self,
dict([(exp_to_monomial(T), c) for T, c in
x.letterplace_polynomial().dict().iteritems()]))
# ok, not a free algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.all import sage_eval
return sage_eval(x, locals=self.gens_dict())
F = self.__monoid
R = self.base_ring()
# coercion from free monoid
if isinstance(x, FreeMonoidElement) and x.parent() is F:
return self.element_class(self, {x: R(1)})
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self, {})
else:
return self.element_class(self, {F(1): x})
def eval_lambda(f, idx_info, tc):
"""
Evaluate array expression p, e.g. p: A[i,j,k]=B[2i+3j+k]
if idx_info is specified then only test p on the idxs from idx_info
Assumes:
the first array in lambda is the pivot
lambda A,B,i,j,k: ... => i,j,k belong to A
inv = 'lambda B,C,D,i,j: B[i][j]=C[D[2i+3]]'
returns true if inv is true on tc
Examples:
sage: var('a,b,c,i,j')
(a, b, c, i, j)
sage: InvArray.eval_lambda('lambda a,b,c,i,j: a[i][j]==2*b[i]+c[j]', None, {'a':[[4,-5],[20,11]],'b':[1,9],'c':[2,-7]})
True
sage: InvArray.eval_lambda('lambda c,a,b,xor,i: c[i] == xor(a[i],b[i])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 334, 79, 281]})
False
sage: InvArray.eval_lambda('lambda c,a,b,xor,i1: c[i1] == xor(a[i1],b[i1])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 134, 79, 241]})
True
sage: InvArray.eval_lambda('lambda rvu, t, rvu1, rvu0: (rvu[rvu0][rvu1]) + (-t[4*rvu0 + rvu1]) == 0', None, {'t': [28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], 'rvu': [[28, 131, 11, 85], [133, 46, 179, 20], [227, 148, 225, 197], [38, 221, 221, 126]]})
True
#The following illustrate the use of idxVals,
#i.e. p is only true under certain array rages
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', None, tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
False
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', idx_info = [{'st0': 0, 'st1': 0}, {'st0': 0, 'st1': 1}, {'st0': 2, 'st1': 2}, {'st0': 2, 'st1': 3}, {'st0': 3, 'st1': 0}, {'st0': 3, 'st1': 1}, {'st0': 3, 'st1': 2}, {'st0': 3, 'st1': 3}, {'st0': 0, 'st1': 2}, {'st0': 0, 'st1': 3}, {'st0': 1, 'st1': 0}, {'st0': 1, 'st1': 1}, {'st0': 1, 'st1': 2}, {'st0': 1, 'st1': 3}, {'st0': 2, 'st1': 0}, {'st0': 2, 'st1': 1}], tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
True
"""
"""
Note: sage_eval vs eval
sage_eval works on str of the format 'lambda x,y: 2*x+y'
whereas eval works on str of the format 2*x+y directly (no lambda)
Also, the result of sage_eval can apply on dicts whose keys are str
e.g. f(**{'x':2,'y':3})
whereas the result of eval applies on dict whose keys are variables
e.g. f(**{x:2,y:3})
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert (idx_info is None or
is_list(idx_info) and all(is_dict(v) for v in idx_info)), indx_info
assert is_dict(tc), tc
assert all(is_str(k) for k in tc), tc.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
arrs = [v for v in vs if v in tc] #A,B
extfuns = [v for v in vs if v in ExtFun.efdict]
idxStr = [v for v in vs if v not in arrs+extfuns] #i,j,k
d_tc = dict([(v,tc[v]) for v in arrs])
d_extfun= dict([(v,ExtFun(v).get_fun()) for v in extfuns])
d_ = merge_dict([d_tc,d_extfun])
if idx_info is None: #obtain idxsVals from the pivot array
pivotContents = tc[arrs[0]]
idxVals = [idx for idx,_ in Miscs.travel(pivotContents)]
idx_info = [dict(zip(idxStr,idxV)) for idxV in idxVals]
ds = [merge_dict([d_, idx_info_]) for idx_info_ in idx_info]
try:
return all(f(**d) for d in ds)
except IndexError:
return False
except TypeError:
return False
except NameError as msg:
logger.warn(msg)
return False
def _element_constructor_(self, x):
"""
Convert ``x`` into ``self``.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: R(3) # indirect doctest
3
TESTS::
sage: F.<x,y,z> = FreeAlgebra(GF(5),3)
sage: L.<x,y,z> = FreeAlgebra(ZZ,3,implementation='letterplace')
sage: F(x) # indirect doctest
x
sage: F.1*L.2
y*z
sage: (F.1*L.2).parent() is F
True
::
sage: K.<z> = GF(25)
sage: F.<a,b,c> = FreeAlgebra(K,3)
sage: L.<a,b,c> = FreeAlgebra(K,3, implementation='letterplace')
sage: F.1+(z+1)*L.2
b + (z+1)*c
Check that :trac:`15169` is fixed::
sage: A.<x> = FreeAlgebra(CC)
sage: A(2)
2.00000000000000
"""
if isinstance(x, FreeAlgebraElement):
P = x.parent()
if P is self:
return x
if not (P is self.base_ring()):
return self.element_class(self, x)
elif hasattr(x,'letterplace_polynomial'):
P = x.parent()
if self.has_coerce_map_from(P): # letterplace versus generic
ngens = P.ngens()
M = self._basis_keys
def exp_to_monomial(T):
out = []
for i in xrange(len(T)):
if T[i]:
out.append((i%ngens,T[i]))
return M(out)
return self.element_class(self, dict([(exp_to_monomial(T),c) for T,c in x.letterplace_polynomial().dict().iteritems()]))
# ok, not a free algebra element (or should not be viewed as one).
if isinstance(x, basestring):
from sage.all import sage_eval
return sage_eval(x,locals=self.gens_dict())
R = self.base_ring()
# coercion from free monoid
if isinstance(x, FreeMonoidElement) and x.parent() is self._basis_keys:
return self.element_class(self,{x:R(1)})
# coercion from the PBW basis
if isinstance(x, PBWBasisOfFreeAlgebra.Element) \
and self.has_coerce_map_from(x.parent()._alg):
return self(x.parent().expansion(x))
# coercion via base ring
x = R(x)
if x == 0:
return self.element_class(self,{})
return self.element_class(self,{self._basis_keys.one():x})
def render_hecke_algebras_webpage(**args):
data = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(url_for('.render_hecke_algebras_webpage', label=lab), 301)
data = db.hecke_algebras.lookup(lab)
if data is None:
t = "Hecke Algebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for a Hecke Algebras in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % data['label'], ' ')]
credit = hecke_algebras_credit
info['level']=int(data['level'])
info['weight']= int(data['weight'])
info['num_orbits']= int(data['num_orbits'])
dim_count = "not available"
proj = ['orbit_label','hecke_op','num_hecke_op','Zbasis','discriminant','disc_fac','Qbasis','Qalg_gen']
orb = list(db.hecke_orbits.search({'parent_label': data['label']}, proj))
if orb:
#consistency check
if len(orb) != int(data['num_orbits']):
return search_input_error(info)
dim_count=0
for v in orb:
ops = sage_eval(v['hecke_op'])
dim = int(matrix(ops[0]).nrows())
dim_count += dim
if dim > 4:
v['hecke_op_display'] = []
elif dim == 1:
v['hecke_op_display'] = [[i+1, ops[i][0][0]] for i in range(10)]
else:
v['hecke_op_display'] = [[i+1, latex(matrix(ops[i]))] for i in range(5)]
v['download_op'] = [(lang, url_for(".render_hecke_algebras_webpage_download", orbit_label=v['orbit_label'], lang=lang, obj='operators')) for lang in ['gp', 'magma','sage']]
if v['Zbasis'] is None:
for key in ['Zbasis','discriminant','disc_fac','Qbasis','Qalg_gen']:
del v[key]
else:
v['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
v['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
if dim > 4:
v['gen_display'] = []
elif dim == 1:
v['gen_display'] = [v['Zbasis'][0][0]]
else:
v['gen_display'] = [latex(matrix(dim,dim,v['Zbasis'][i])) for i in range(dim)]
v['inner_twists'] = "not available" # not yet in the database
v['download_gen'] = [(lang, url_for(".render_hecke_algebras_webpage_download", orbit_label=v['orbit_label'], lang=lang, obj='gen')) for lang in ['gp', 'magma','sage']]
info['orbits'] = orb
info['dim_alg'] = dim_count
info['l_adic'] = l_range
info['properties'] = [
('Label', '%s' %info['label']),
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight'])]
if info['num_orbits']!=0:
info['friends'] = [('Newforms space ' + info['label'], url_for("cmf.by_url_space_label", level=info['level'], weight=info['weight'], char_orbit_label='a'))]
else:
info['friends'] = []
t = "Hecke Algebra %s" % info['label']
return render_template("hecke_algebras-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
def render_hecke_algebras_webpage_l_adic(**args):
data = None
if 'orbit_label' in args and 'prime' in args:
lab = clean_input(args.get('orbit_label'))
if lab != args.get('orbit_label'):
base_lab = ".".join([split(lab)[i] for i in [0, 1, 2]])
return redirect(
url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
try:
ell = int(args.get('prime'))
except ValueError:
base_lab = ".".join([split(lab)[i] for i in [0, 1, 2]])
return redirect(
url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
data = db.hecke_ladic.lucky({'orbit_label': lab, 'ell': ell})
if data is None:
t = "Hecke algebra search error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash_error(
"%s is not a valid label for the ℓ-adic information for an Hecke algebra orbit in the database.",
lab)
return render_template("hecke_algebras-error.html",
title=t,
properties=[],
bread=bread,
learnmore=learnmore_list())
info = {}
info.update(data)
proj = [
'index', 'orbit_label', 'ell', 'idempotent', 'field', 'structure',
'properties', 'operators'
]
res = list(
db.hecke_ladic.search(
{
'level': data['level'],
'weight': data['weight'],
'orbit_label': data['orbit_label'],
'ell': data['ell']
}, proj))
for f in res:
if f['idempotent'] != "":
dim = len(sage_eval(f['idempotent']))
l_max = sage_eval(f['idempotent'])[0][0].ndigits()
if dim > 4 or l_max > 5:
f['idempotent_display'] = []
elif dim == 1:
f['idempotent_display'] = sage_eval(f['idempotent'])[0][0]
else:
f['idempotent_display'] = latex(
matrix(sage_eval(f['idempotent'])))
else:
f['idempotent_display'] = latex(matrix([[1]]))
del f['idempotent']
f['download_id'] = [
(i,
url_for(".render_hecke_algebras_webpage_ell_download",
orbit_label=f['orbit_label'],
index=f['index'],
prime=f['ell'],
lang=i,
obj='idempotents')) for i in ['magma', 'sage']
] # for 'gp' the code does not work, since p-adics are not implemented
field = f.pop('field')
if field is not None:
f['deg'] = field[1]
f['field_poly'] = field[2]
structure = f.pop('structure')
if structure is not None:
f['dim'] = structure[0]
f['num_gen'] = structure[1]
s2 = sage_eval(structure[2])
f['gens'] = [[int(s2.index(i) + 1), str(i)] for i in s2]
f['rel'] = sage_eval(structure[3])
properties = f.pop('properties')
if properties is not None:
f['grading'] = properties[0]
f['gorenstein_def'] = properties[1]
f['gorenstein'] = "yes" if properties[1] == 0 else "no"
operators = f.pop('operators')
if operators is not None:
f['operators_mod_l'] = operators
f['num_hecke_op'] = len(operators)
f['size_op'] = size = sqrt(len(operators[0]))
if size > 4:
f['operators_mod_l_display'] = []
elif size == 1:
f['operators_mod_l_display'] = [[i + 1, operators[i][0]]
for i in range(10)]
else:
f['operators_mod_l_display'] = [[
i + 1, latex(matrix(size, size, operators[i]))
] for i in range(5)]
f['download_op'] = [
(lang,
url_for(".render_hecke_algebras_webpage_ell_download",
orbit_label=f['orbit_label'],
index=f['index'],
prime=f['ell'],
lang=lang,
obj='operators')) for lang in ['magma', 'sage']
] # for 'gp' the code does not work
info['num_l_adic_orbits'] = len(res)
info['l_adic_orbits'] = res
info['level'] = int(data['level'])
info['weight'] = int(data['weight'])
info['base_lab'] = ".".join(
[split(data['orbit_label'])[i] for i in [0, 1, 2]])
info['orbit_label'] = str(data['orbit_label'])
info['ell'] = int(data['ell'])
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")),
('%s' % info['base_lab'],
url_for('.render_hecke_algebras_webpage',
label=info['base_lab'])), ('%s' % info['ell'], ' ')]
credit = hecke_algebras_credit
info['properties'] = [('Level', '%s' % info['level']),
('Weight', '%s' % info['weight']),
('Characteristic', '%s' % info['ell']),
('Orbit label', '%s' % info['orbit_label'])]
info['friends'] = [('Modular form ' + info['base_lab'],
url_for("cmf.by_url_space_label",
level=info['level'],
weight=info['weight'],
char_orbit_label='a'))]
t = "%s-adic and mod %s data for the Hecke algebra orbit %s" % (
info['ell'], info['ell'], info['orbit_label'])
return render_template("hecke_algebras_l_adic-single.html",
info=info,
credit=credit,
title=t,
bread=bread,
properties=info['properties'],
learnmore=learnmore_list(),
friends=info['friends'],
KNOWL_ID='hecke_algebra_l_adic.%s' %
(info['orbit_label']))
def render_hecke_algebras_webpage(**args):
data = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(url_for('.render_hecke_algebras_webpage', label=lab), 301)
data = hecke_db().find_one({'label': lab })
if data is None:
t = "Hecke Agebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for a Hecke Algebras in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % data['label'], ' ')]
credit = hecke_algebras_credit
f = hecke_db().find_one({'level': data['level'],'weight': data['weight'],'num_orbits': data['num_orbits']})
info['level']=int(f['level'])
info['weight']= int(f['weight'])
info['num_orbits']= int(f['num_orbits'])
dim_count = "not available"
orb = hecke_orbits_db().find({'parent_label': f['label']}).sort([('orbit', ASC)])
if orb.count()!=0:
#consistency check
if orb.count()!= int(f['num_orbits']):
return search_input_error(info)
dim_count=0
res_clean = []
for v in orb:
v_clean = {}
v_clean['orbit_label'] = v['orbit_label']
v_clean['hecke_op'] = v['hecke_op']
v_clean['gen'] = v['gen']
v_clean['dim'] = int(matrix(sage_eval(v_clean['hecke_op'])[0]).dimensions()[0])
dim_count = dim_count + v_clean['dim']
if v_clean['dim']>4:
v_clean['hecke_op_display'] = []
elif v_clean['dim']==1:
v_clean['hecke_op_display'] = [[i+1, (sage_eval(v_clean['hecke_op'])[i])[0][0]] for i in range(0,10)]
else:
v_clean['hecke_op_display'] = [[i+1, latex(matrix(sage_eval(v_clean['hecke_op'])[i]))] for i in range(0,5)]
v_clean['num_hecke_op'] = v['num_hecke_op']
v_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='operators')) for i in ['gp', 'magma','sage']]
if 'Zbasis' in v.keys():
v_clean['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
if v_clean['dim']>4:
v_clean['gen_display'] = []
elif v_clean['dim']==1:
v_clean['gen_display'] = [v_clean['Zbasis'][0][0]]
else:
v_clean['gen_display'] = [latex(matrix(v_clean['dim'],v_clean['dim'], v_clean['Zbasis'][i])) for i in range(0,v_clean['dim'])]
v_clean['discriminant'] = int(v['discriminant'])
v_clean['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
v_clean['Qbasis'] = [int(i) for i in v['Qbasis']]
v_clean['Qalg_gen'] = [int(i) for i in v['Qalg_gen']]
if 'inner_twists' in v.keys():
v_clean['inner_twists'] = [str(i) for i in v['inner_twists']]
else:
v_clean['inner_twists'] = "not available"
v_clean['download_gen'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='gen')) for i in ['gp', 'magma','sage']]
res_clean.append(v_clean)
info['orbits'] = res_clean
info['dim_alg'] = dim_count
info['l_adic'] = l_range
info['properties'] = [
('Label', '%s' %info['label']),
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight'])]
if info['num_orbits']!=0:
info['friends'] = [('Newforms space ' + info['label'], url_for("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
else:
info['friends'] = []
t = "Hecke Algebra %s" % info['label']
return render_template("hecke_algebras-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])