def check_order_parity(self, rec, verbose=False):
"""
check order and parity by constructing a Conrey character in Sage (use the first index in galois_orbit)
"""
# TIME about 30000s for full table
char = ConreyCharacter(rec['modulus'], rec['galois_orbit'][0])
parity = 1 if char.is_even() else -1
success = (parity == rec['parity']
and char.conductor() == rec['conductor']
and char.multiplicative_order() == rec['order'])
if verbose and not success:
print("Order-parity failure", parity, rec['parity'],
char.conductor(), rec['conductor'],
char.multiplicative_order(), rec['order'])
return success
class WebSmallDirichletCharacter(WebChar, WebDirichlet):
"""
Heritage: WebChar -> __init__()
WebDirichlet -> _compute()
"""
def _compute(self):
self.modulus = int(self.modlabel)
self.number = int(self.numlabel)
self.chi = ConreyCharacter(self.modulus, self.number)
self.codelangs = ('pari', 'sage')
@lazy_attribute
def conductor(self):
return self.chi.conductor()
@lazy_attribute
def indlabel(self):
if self.chi.indlabel is not None:
return self.chi.indlabel
else:
# Calling conductor computes the indlabel
self.chi.conductor()
return self.chi.indlabel
@lazy_attribute
def codeinit(self):
return {
'sage': [
'H = DirichletGroup(%i)'%(self.modulus),
'chi = H[%i]'%(self.number) ],
'pari': '[g,chi] = znchar(Mod(%i,%i))'%(self.number,self.modulus),
}
@lazy_attribute
def title(self):
return r"Dirichlet character %s" % (self.texname)
@lazy_attribute
def texname(self):
return self.char2tex(self.modulus, self.number)
@lazy_attribute
def codeisprimitive(self):
return { 'sage': 'chi.is_primitive()',
'pari': '#znconreyconductor(g,chi)==1 \\\\ if not primitive returns [cond,factorization]' }
@lazy_attribute
def codecond(self):
return { 'sage': 'chi.conductor()',
'pari': 'znconreyconductor(g,chi)' }
@lazy_attribute
def parity(self):
return (parity_string(-1),parity_string(1))[self.chi.is_even()]
@lazy_attribute
def codeparity(self):
return { 'sage': 'chi.is_odd()',
'pari': 'zncharisodd(g,chi)' }
def symbol_numerator(self):
""" chi is equal to a kronecker symbol if and only if it is real """
if self.order != 2:
return None
return symbol_numerator(self.conductor, self.chi.is_odd())
@lazy_attribute
def symbol(self):
return kronecker_symbol(self.symbol_numerator())
@lazy_attribute
def codesymbol(self):
m = self.symbol_numerator()
if m:
return { 'sage': 'kronecker_character(%i)'%m,
'pari': 'znchartokronecker(g,chi)'
}
return None
@lazy_attribute
def codegaloisorbit(self):
return { 'sage': ['chi.galois_orbit()'],
'pari': [ 'order = charorder(g,chi)',
'[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]' ]
}
class WebSmallDirichletCharacter(WebChar, WebDirichlet):
"""
Heritage: WebChar -> __init__()
WebDirichlet -> _compute()
"""
def _compute(self):
self.modulus = int(self.modlabel)
self.number = int(self.numlabel)
self.chi = ConreyCharacter(self.modulus, self.number)
self.credit = ''
self.codelangs = ('pari', 'sage')
@property
def conductor(self):
return self.chi.conductor()
@property
def previous(self): return None
@property
def next(self): return None
@property
def genvalues(self): return None
@property
def indlabel(self): return None
def value(self, *args): return None
def charsums(self, *args): return False
def gauss_sum(self, *args): return None
def jacobi_sum(self, *args): return None
def kloosterman_sum(self, *args): return None
@property
def codeinit(self):
return {
'sage': [ 'from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed',
'H = DirichletGroup_conrey(%i)'%(self.modulus),
'chi = H[%i]'%(self.number) ],
'pari': '[g,chi] = znchar(Mod(%i,%i))'%(self.number,self.modulus),
}
@property
def title(self):
return r"Dirichlet Character %s" % (self.texname)
@property
def texname(self):
return self.char2tex(self.modulus, self.number)
@property
def codeisprimitive(self):
return { 'sage': 'chi.is_primitive()',
'pari': '#znconreyconductor(g,chi)==1 \\\\ if not primitive returns [cond,factorization]' }
@property
def codecond(self):
return { 'sage': 'chi.conductor()',
'pari': 'znconreyconductor(g,chi)' }
@property
def parity(self):
return ('Odd', 'Even')[self.chi.is_even()]
@property
def codeparity(self):
return { 'sage': 'chi.is_odd()',
'pari': 'zncharisodd(g,chi)' }
@property
def galoisorbit(self):
order = self.order
mod, num = self.modulus, self.number
prim = self.isprimitive
#beware this **must** be a generator
orbit = ( power_mod(num, k, mod) for k in xsrange(1, order) if gcd(k, order) == 1) # use xsrange not xrange
return ( self._char_desc(num, prim=prim) for num in orbit )
def symbol_numerator(self):
""" chi is equal to a kronecker symbol if and only if it is real """
if self.order != 2:
return None
return symbol_numerator(self.conductor, self.chi.is_odd())
@property
def symbol(self):
return kronecker_symbol(self.symbol_numerator())
@property
def codesymbol(self):
m = self.symbol_numerator()
if m:
return { 'sage': 'kronecker_character(%i)'%m }
return None
@property
def codegaloisorbit(self):
return { 'sage': 'chi.sage_character().galois_orbit()',
'pari': [ 'order = charorder(g,chi)',
'[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]' ]
}