def test_conrey(nranges=[(20, 100), (200, 1000), (500, 500000)]):
for n, r in nranges:
for i in range(n):
q = randint(1, r)
print("n, r, i, q = {}, {}, {}, {}".format(n, r, i, q))
G = DirichletGroup_conrey(q)
inv = G.invariants()
cinv = tuple([Mod(g, q).multiplicative_order() for g in G.gens()])
try:
assert prod(inv) == G.order()
assert q <= 2 or G.zeta_order() == inv[0]
assert cinv == inv
except:
print('group error')
return q, inv, G
if q > 2:
m = 0
while gcd(m, q) != 1:
m = randint(1, q)
chi = G[m]
n = randint(1, q)
try:
assert chi.multiplicative_order() == Mod(
m, q).multiplicative_order()
if gcd(n, q) != 1:
try:
assert chi.logvalue(n) == -1
except:
print('non unit value error')
return chi, n, r
elif q < 10 ^ 6:
try:
ref = chi.sage_character()(n).n().real()
new = N(chi(n).real)
assert abs(ref - new) < 1e-5
except:
print('unit value error')
return chi, n
except:
print('char error')
return chi, n, r
class WebDirichlet(WebCharObject):
"""
For some applications (orbits, enumeration), Dirichlet characters may be
represented by a couple (modulus, number) without computing the Dirichlet
group.
"""
def _compute(self):
if self.modlabel:
self.modulus = m = int(self.modlabel)
self.H = DirichletGroup_conrey(m)
self.credit = 'SageMath'
self.codelangs = ('pari', 'sage')
def _char_desc(self, c, mod=None, prim=None):
""" usually num is the number, but can be a character """
if isinstance(c, DirichletCharacter_conrey):
if prim == None:
prim = c.is_primitive()
mod = c.modulus()
num = c.number()
elif mod == None:
mod = self.modulus
num = c
if prim == None:
prim = self.charisprimitive(mod,num)
return ( mod, num, self.char2tex(mod,num), prim)
def charisprimitive(self,mod,num):
if isinstance(self.H, DirichletGroup_conrey) and self.H.modulus()==mod:
H = self.H
else:
H = DirichletGroup_conrey(mod)
return H[num].is_primitive()
@property
def gens(self):
return map(int, self.H.gens())
@property
def generators(self):
#import pdb; pdb.set_trace()
#assert self.H.gens() is not None
return self.textuple(map(str, self.H.gens()))
""" for Dirichlet over Z, everything is described using integers """
@staticmethod
def char2tex(modulus, number, val='\cdot', tag=True):
c = r'\chi_{%s}(%s,%s)'%(modulus,number,val)
if tag:
return '\(%s\)'%c
else:
return c
group2tex = int
group2label = int
label2group = int
ideal2tex = int
ideal2label = int
label2ideal = int
""" numbering characters """
number2label = int
label2number = int
@property
def groupelts(self):
return map(self.group2tex, self.Gelts())
@cached_method
def Gelts(self):
res = []
m,n,k = self.modulus, 1, 1
while k < m and n <= self.maxcols:
if gcd(k,m) == 1:
res.append(k)
n += 1
k += 1
if n > self.maxcols:
self.coltruncate = True
return res
@staticmethod
def nextchar(m, n, onlyprimitive=False):
""" we know that the characters
chi_m(1,.) and chi_m(m-1,.)
always exist for m>1.
They are extremal for a given m.
"""
if onlyprimitive:
return WebDirichlet.nextprimchar(m, n)
if m == 1:
return 2, 1
if n == m - 1:
return m + 1, 1
k = n+1
while k < m:
if gcd(m, k) == 1:
return m, k
k += 1
raise Exception("nextchar")
@staticmethod
def prevchar(m, n, onlyprimitive=False):
""" Assume m>1 """
if onlyprimitive:
return WebDirichlet.prevprimchar(m, n)
if n == 1:
m, n = m - 1, m
if m <= 2:
return m, 1 # important : 2,2 is not a character
k = n-1
while k > 0:
if gcd(m, k) == 1:
return m, k
k -= 1
raise Exception("prevchar")
@staticmethod
def prevprimchar(m, n):
if m <= 3:
return 1, 1
if n > 2:
Gm = DirichletGroup_conrey(m)
while True:
n -= 1
if n <= 1: # (m,1) is never primitive for m>1
m, n = m - 1, m - 1
Gm = DirichletGroup_conrey(m)
if m <= 2:
return 1, 1
if gcd(m, n) != 1:
continue
# we have a character, test if it is primitive
chi = Gm[n]
if chi.is_primitive():
return m, n
@staticmethod
def nextprimchar(m, n):
if m < 3:
return 3, 2
if n < m - 1:
Gm = DirichletGroup_conrey(m)
while 1:
n += 1
if n >= m:
m, n = m + 1, 2
Gm = DirichletGroup_conrey(m)
if gcd(m, n) != 1:
continue
# we have a character, test if it is primitive
chi = Gm[n]
if chi.is_primitive():
return m, n
class WebDirichlet(WebCharObject):
"""
For some applications (orbits, enumeration), Dirichlet characters may be
represented by a couple (modulus, number) without computing the Dirichlet
group.
"""
def _compute(self):
if self.modlabel:
self.modulus = m = int(self.modlabel)
self.H = DirichletGroup_conrey(m)
self.credit = 'SageMath'
self.codelangs = ('pari', 'sage')
def _char_desc(self, c, mod=None, prim=None):
""" usually num is the number, but can be a character """
if isinstance(c, DirichletCharacter_conrey):
if prim == None:
prim = c.is_primitive()
mod = c.modulus()
num = c.number()
elif mod == None:
mod = self.modulus
num = c
if prim == None:
prim = self.charisprimitive(mod,num)
return ( mod, num, self.char2tex(mod,num), prim)
def charisprimitive(self,mod,num):
if isinstance(self.H, DirichletGroup_conrey) and self.H.modulus()==mod:
H = self.H
else:
H = DirichletGroup_conrey(mod)
return H[num].is_primitive()
@property
def gens(self):
return map(int, self.H.gens())
@property
def generators(self):
#import pdb; pdb.set_trace()
#assert self.H.gens() is not None
return self.textuple(map(str, self.H.gens()))
""" for Dirichlet over Z, everything is described using integers """
@staticmethod
def char2tex(modulus, number, val='\cdot', tag=True):
c = r'\chi_{%s}(%s,%s)'%(modulus,number,val)
if tag:
return '\(%s\)'%c
else:
return c
group2tex = int
group2label = int
label2group = int
ideal2tex = int
ideal2label = int
label2ideal = int
""" numbering characters """
number2label = int
label2number = int
@property
def groupelts(self):
return map(self.group2tex, self.Gelts())
@cached_method
def Gelts(self):
res = []
m,n,k = self.modulus, 1, 1
while k < m and n <= self.maxcols:
if gcd(k,m) == 1:
res.append(k)
n += 1
k += 1
if n > self.maxcols:
self.coltruncate = True
return res
@staticmethod
def nextchar(m, n, onlyprimitive=False):
""" we know that the characters
chi_m(1,.) and chi_m(m-1,.)
always exist for m>1.
They are extremal for a given m.
"""
if onlyprimitive:
return WebDirichlet.nextprimchar(m, n)
if m == 1:
return 2, 1
if n == m - 1:
return m + 1, 1
k = n+1
while k < m:
if gcd(m, k) == 1:
return m, k
k += 1
raise Exception("nextchar")
@staticmethod
def prevchar(m, n, onlyprimitive=False):
""" Assume m>1 """
if onlyprimitive:
return WebDirichlet.prevprimchar(m, n)
if n == 1:
m, n = m - 1, m
if m <= 2:
return m, 1 # important : 2,2 is not a character
k = n-1
while k > 0:
if gcd(m, k) == 1:
return m, k
k -= 1
raise Exception("prevchar")
@staticmethod
def prevprimchar(m, n):
if m <= 3:
return 1, 1
if n > 2:
Gm = DirichletGroup_conrey(m)
while True:
n -= 1
if n <= 1: # (m,1) is never primitive for m>1
m, n = m - 1, m - 1
Gm = DirichletGroup_conrey(m)
if m <= 2:
return 1, 1
if gcd(m, n) != 1:
continue
# we have a character, test if it is primitive
chi = Gm[n]
if chi.is_primitive():
return m, n
@staticmethod
def nextprimchar(m, n):
if m < 3:
return 3, 2
if n < m - 1:
Gm = DirichletGroup_conrey(m)
while 1:
n += 1
if n >= m:
m, n = m + 1, 2
Gm = DirichletGroup_conrey(m)
if gcd(m, n) != 1:
continue
# we have a character, test if it is primitive
chi = Gm[n]
if chi.is_primitive():
return m, n