def _compute(self):
self.k = self.label2nf(self.nflabel)
self._modulus = self.label2ideal(self.k, self.modlabel)
self.G = RayClassGroup(self.k, self._modulus)
self.H = self.G.dual_group()
#self.number = lmfdb_label2hecke(self.numlabel)
# make this canonical
self.modlabel = self.ideal2label(self._modulus)
self.credit = "Pari, SageMath"
self.codelangs = ('pari', 'sage')
self.parity = None
logger.debug('###### WebHeckeComputed')
def _compute(self):
self.k = self.label2nf(self.nflabel)
self._modulus = self.label2ideal(self.k, self.modlabel)
self.G = RayClassGroup(self.k, self._modulus)
self.H = self.G.dual_group()
#self.number = lmfdb_label2hecke(self.numlabel)
# make this canonical
self.modlabel = self.ideal2label(self._modulus)
self.credit = "Pari, SageMath"
self.codelangs = ('pari', 'sage')
self.parity = None
logger.debug('###### WebHeckeComputed')
class WebHecke(WebCharObject):
""" FIXME design issue: should underlying group elements be represented
by tuples or by representative ideals ?
for computations tuples are much better, this is also more compact.
"""
def _compute(self):
self.k = self.label2nf(self.nflabel)
self._modulus = self.label2ideal(self.k, self.modlabel)
self.G = RayClassGroup(self.k, self._modulus)
self.H = self.G.dual_group()
#self.number = lmfdb_label2hecke(self.numlabel)
# make this canonical
self.modlabel = self.ideal2label(self._modulus)
self.credit = "Pari, SageMath"
self.codelangs = ('pari', 'sage')
self.parity = None
logger.debug('###### WebHeckeComputed')
@property
def generators(self):
""" use representative ideals """
return self.textuple( map(self.ideal2tex, self.G.gen_ideals() ), tag=False )
""" labeling conventions are put here """
@staticmethod
def char2tex(c, val='\cdot',tag=True):
""" c is a Hecke character """
number = ','.join(map(str,c.exponents()))
s = r'\chi_{%s}(%s)'%(number,val)
if tag:
return r'\(%s\)'%s
else:
return s
def _char_desc(self, c, modlabel=None, prim=None):
""" c is a Hecke character of modulus self.modulus
unless modlabel is specified
"""
if modlabel == None:
modlabel = self.modlabel
numlabel = self.number2label( c.exponents() )
if prim == None:
prim = c.is_primitive()
return (modlabel, numlabel, self.char2tex(c), prim )
@staticmethod
def ideal2tex(ideal):
a,b = ideal.gens_two()
return "\(\langle %s, %s\\rangle\)"%(a._latex_(), b._latex_())
@staticmethod
def ideal2label(ideal):
"""
labeling convention for ideal f:
use two elements representation f = (n,b)
with n = f cap Z an integer
and b an algebraic element sum b_i a^i
label f as n.b1+b2*a^2+...bn*a^n
(dot between n and b, a is the field generator, use '+' and )
"""
a,b = ideal.gens_two()
s = '+'.join( '%sa%i'%(b,i) for i,b in enumerate(b.polynomial().list())
if b != 0 )
return "%s.%s"%(a,s.replace('+-','-').replace('/','o'))
@staticmethod
def label2ideal(k,label):
""" k = underlying number field """
if label.count('.'):
n, b = label.split(".")
else:
n, b = label, '0'
a = k.gen()
# FIXME: dangerous
n, b = evalpolelt(n,a,'a'), evalpolelt(b,a,'a')
n, b = k(n), k(b)
return k.ideal( (n,b) )
"""
underlying group contains ideal classes, but are represented
as exponent tuples on cyclic components (not canonical, but
more compact)
"""
#group2tex = ideal2tex
#group2label = ideal2label
#label2group = label2ideal
@staticmethod
def group2tex(x, tag=True):
if not isinstance(x, tuple):
x = x.exponents()
#s = '\cdot '.join('g_{%i}^{%i}'%(i,e) for i,e in enumerate(x) if e>0)
s = []
for i,e in enumerate(x):
if e > 0:
if e==1:
s.append('g_{%i}'%i)
else:
s.append('g_{%i}^{%i}'%(i,e))
s = '\cdot '.join(s)
if s == '': s = '1'
if tag: s = '\(%s\)'%s
return s
@staticmethod
def group2label(self,x):
return self.number2label(x.exponents())
def label2group(self,x):
""" x is either an element of k or a tuple of ints or an ideal """
if x.count('.'):
x = self.label2ideal(self.k,x)
elif x.count('a'):
a = self.k.gen()
x = evalpolelt(x,a,'a')
elif x.count(','):
x = tuple(map(int,x.split(',')))
return self.G(x)
@staticmethod
def number2label(number):
return '.'.join(map(str,number))
@staticmethod
def label2number(label):
return map(int,label.split('.'))
@staticmethod
def label2nf(label):
return WebNumberField(label).K()
# FIXME: replace by calls to WebNF
#x = var('x')
#pol = evalpolelt(label,x,'x')
#return NumberField(pol,'a')
@property
def groupelts(self):
return map(self.group2tex, self.Gelts())
@cached_method
def Gelts(self):
res = []
c = 1
for x in self.G.iter_exponents():
res.append(x)
c += 1
if c > self.maxcols:
self.coltruncate = True
break
return res
def chargroup(self, mod):
return RayClassGroup(self.k, mod).dual_group()
class WebHecke(WebCharObject):
""" FIXME design issue: should underlying group elements be represented
by tuples or by representative ideals ?
for computations tuples are much better, this is also more compact.
"""
def _compute(self):
self.k = self.label2nf(self.nflabel)
self._modulus = self.label2ideal(self.k, self.modlabel)
self.G = RayClassGroup(self.k, self._modulus)
self.H = self.G.dual_group()
#self.number = lmfdb_label2hecke(self.numlabel)
# make this canonical
self.modlabel = self.ideal2label(self._modulus)
self.credit = "Pari, SageMath"
self.codelangs = ('pari', 'sage')
self.parity = None
logger.debug('###### WebHeckeComputed')
@property
def generators(self):
""" use representative ideals """
return self.textuple(map(self.ideal2tex, self.G.gen_ideals()),
tag=False)
""" labeling conventions are put here """
@staticmethod
def char2tex(c, val='\cdot', tag=True):
""" c is a Hecke character """
number = ','.join(map(str, c.exponents()))
s = r'\chi_{%s}(%s)' % (number, val)
if tag:
return r'\(%s\)' % s
else:
return s
def _char_desc(self, c, modlabel=None, prim=None):
""" c is a Hecke character of modulus self.modulus
unless modlabel is specified
"""
if modlabel == None:
modlabel = self.modlabel
numlabel = self.number2label(c.exponents())
if prim == None:
prim = c.is_primitive()
return (modlabel, numlabel, self.char2tex(c), prim)
@staticmethod
def ideal2tex(ideal):
a, b = ideal.gens_two()
return "\(\langle %s, %s\\rangle\)" % (a._latex_(), b._latex_())
@staticmethod
def ideal2label(ideal):
"""
labeling convention for ideal f:
use two elements representation f = (n,b)
with n = f cap Z an integer
and b an algebraic element sum b_i a^i
label f as n.b1+b2*a^2+...bn*a^n
(dot between n and b, a is the field generator, use '+' and )
"""
a, b = ideal.gens_two()
s = '+'.join('%sa%i' % (b, i)
for i, b in enumerate(b.polynomial().list()) if b != 0)
return "%s.%s" % (a, s.replace('+-', '-').replace('/', 'o'))
@staticmethod
def label2ideal(k, label):
""" k = underlying number field """
if label.count('.'):
n, b = label.split(".")
else:
n, b = label, '0'
a = k.gen()
# FIXME: dangerous
n, b = evalpolelt(n, a, 'a'), evalpolelt(b, a, 'a')
n, b = k(n), k(b)
return k.ideal((n, b))
"""
underlying group contains ideal classes, but are represented
as exponent tuples on cyclic components (not canonical, but
more compact)
"""
#group2tex = ideal2tex
#group2label = ideal2label
#label2group = label2ideal
@staticmethod
def group2tex(x, tag=True):
if not isinstance(x, tuple):
x = x.exponents()
#s = '\cdot '.join('g_{%i}^{%i}'%(i,e) for i,e in enumerate(x) if e>0)
s = []
for i, e in enumerate(x):
if e > 0:
if e == 1:
s.append('g_{%i}' % i)
else:
s.append('g_{%i}^{%i}' % (i, e))
s = '\cdot '.join(s)
if s == '': s = '1'
if tag: s = '\(%s\)' % s
return s
@staticmethod
def group2label(self, x):
return self.number2label(x.exponents())
def label2group(self, x):
""" x is either an element of k or a tuple of ints or an ideal """
if x.count('.'):
x = self.label2ideal(self.k, x)
elif x.count('a'):
a = self.k.gen()
x = evalpolelt(x, a, 'a')
elif x.count(','):
x = tuple(map(int, x.split(',')))
return self.G(x)
@staticmethod
def number2label(number):
return '.'.join(map(str, number))
@staticmethod
def label2number(label):
return map(int, label.split('.'))
@staticmethod
def label2nf(label):
return WebNumberField(label).K()
# FIXME: replace by calls to WebNF
#x = var('x')
#pol = evalpolelt(label,x,'x')
#return NumberField(pol,'a')
@property
def groupelts(self):
return map(self.group2tex, self.Gelts())
@cached_method
def Gelts(self):
res = []
c = 1
for x in self.G.iter_exponents():
res.append(x)
c += 1
if c > self.maxcols:
self.coltruncate = True
break
return res