def gen_params_from_r(r,k):
"""
Description:
Finds a fundamental discriminant D to use as input to the Cocks-Pinch method
Input:
r - prime such that r % k == 1
k - embedding degree
Output:
r - prime such that r % k == 1
k - embedding degree
D - (negative) fundamental discriminant where D is a square mod r
"""
D = -Integer(Mod(int(random()*(1000)),r))
i = 0
while not kronecker(D,r) == 1: # expected number of iterations of the while loop is 2
D = -Integer(Mod(int(random()*(1000)),r))
i+=1
D = fundamental_discriminant(D)
if not (kronecker(D,r) == 1):
return r, k, 0
return r,k,D
def gen_params_from_r(r, k):
"""
Description:
Finds a fundamental discriminant D to use as input to the Cocks-Pinch method
Input:
r - prime such that r % k == 1
k - embedding degree
Output:
r - prime such that r % k == 1
k - embedding degree
D - (negative) fundamental discriminant where D is a square mod r
"""
D = -Integer(Mod(int(random() * (1000)), r))
i = 0
while not kronecker(
D, r) == 1: # expected number of iterations of the while loop is 2
D = -Integer(Mod(int(random() * (1000)), r))
i += 1
D = fundamental_discriminant(D)
if not (kronecker(D, r) == 1):
return r, k, 0
return r, k, D
def small_A_twist(E):
"""
Description:
Finds a curve isogenous to E that has small A in the curve equation y^2 = x^3 + A*x + B
Input:
E - elliptic curve
Output:
E' - elliptic curve isogenous to E that has small A in the curve equation y^2 = x^3 + A*x + B
"""
a = E.ainvs()[3]
q = E.base_field().order()
a = power_mod(Integer(a), -1, q)
if kronecker(a, q) == -1:
b = 2
while kronecker(b, q) == 1:
b += 1
a = a * b
assert kronecker(a, q) == 1
d = Mod(a, q).sqrt()
ainvs = [i for i in E.ainvs()]
ainvs[3] *= d**2
ainvs[4] *= d**3
return EllipticCurve(E.base_field(), ainvs)
def make_curve(k,D):
"""
Description:
Find all MNT curves with embedding degree k and fundamental discriminant D
Input:
k - embedding degree
D - (negative) fundamental discriminant
Output:
curves - list of the aforementioned elliptic curves;
each curve is represented as a tuple (q,t,r,k,D)
"""
assert k == 3 or k== 4 or k == 6, 'Invalid embedding degree'
assert fundamental_discriminant(D) == D, 'Invalid discriminant'
if k == 3:
curves = _mnt_subcall(k,D, 24,lambda x: ((x % 24 == 19) and kronecker(6,x) == 1), 6, 3, -3, lambda x: (x -3 )//6, lambda x: (x+3)//6, lambda l: 6*l - 1, lambda l: -6*l - 1, lambda l: 12*l*l - 1)
if k == 4:
curves = _mnt_subcall(k,D,-8, lambda x: ( (x % 8 == 3) and kronecker(-2, x) ==1) or (x == 2) or (x == 8), 3, 2, 1, lambda x: (x-2)//3, lambda x: (x-1)//3, lambda l: -l, lambda l: l+1, lambda l: l*l + l + 1)
if k == 6:
curves = _mnt_subcall(k,D,-8, lambda x: (x % 8 == 3) and kronecker(-2, x) == 1, 6, 5, 1, lambda x: (x+1)//6, lambda x: (x-1)//6, lambda l: 2*l + 1, lambda l: -2*l + 1, lambda l: 4*l*l + 1)
for c in curves:
assert is_valid_curve(c[0],c[1],c[2],c[3],c[4]), 'Invalid output'
assert k == c[3], 'Bug in code'
assert D == c[4], 'Bug in code'
return curves
def eta_argument(a, b, c, d):
res = (a + d) * c - b * d * (c * c - 1)
den = 24
if is_odd(c):
return res - 3 * c, den, kronecker(d, c)
else:
return res + 3 * d - 3 - 3 * c * d, den, kronecker(d, c)
def _action0(self,a,b,c,d):
r"""
Recall that the formula is valid only for c>0. Otherwise we have to use:
v(A)=v((-I)(-A))=sigma(-I,-A)v(-I)v(-A).
Then note that by the formula for sigma we have:
sigma(-I,SL2Z[a, b, c, d])=-1 if (c=0 and d<0) or c>0 and other wise it is =1.
"""
fak=1
if c<0:
a=-a; b=-b; c=-c; d=-d; fak=-self._fak
if c==0:
if a>0:
res = self._z**b
else:
res = self._fak*self._z**b
else:
if is_even(c):
arg = (a+d)*c-b*d*(c*c-1)+3*d-3-3*c*d
v=kronecker(c,d)
else:
arg = (a+d)*c-b*d*(c*c-1)-3*c
v=kronecker(d,c)
if not self._half_integral_weight:
# recall that we can use eta for any real weight
v=v**(2*self._weight)
arg=arg*(self._k_num)
res = v*fak*self._z**arg
if self._character:
res = res * self._character(d)
if self._is_dual:
res=res**-1
return res
def small_A_twist(E):
"""
Description:
Finds a curve isogenous to E that has small A in the curve equation y^2 = x^3 + A*x + B
Input:
E - elliptic curve
Output:
E' - elliptic curve isogenous to E that has small A in the curve equation y^2 = x^3 + A*x + B
"""
a = E.ainvs()[3]
q = E.base_field().order()
a = power_mod(Integer(a), -1, q)
if kronecker(a,q) == -1:
b = 2
while kronecker(b,q) == 1:
b += 1
a = a*b
assert kronecker(a,q) == 1
d = Mod(a,q).sqrt()
ainvs = [i for i in E.ainvs()]
ainvs[3]*= d**2
ainvs[4] *= d**3
return EllipticCurve(E.base_field(), ainvs)
def eta_argument(a,b,c,d):
res = (a+d)*c-b*d*(c*c-1)
den = 24
if is_odd(c):
return res-3*c,den,kronecker(d,c)
else:
return res+3*d-3-3*c*d,den,kronecker(d,c)
def _action0(self,a,b,c,d):
r"""
Recall that the formula is valid only for c>0. Otherwise we have to use:
v(A)=v((-I)(-A))=sigma(-I,-A)v(-I)v(-A).
Then note that by the formula for sigma we have:
sigma(-I,SL2Z[a, b, c, d])=-1 if (c=0 and d<0) or c>0 and other wise it is =1.
"""
fak=1
if c<0:
a=-a; b=-b; c=-c; d=-d; fak=-self._fak
if c==0:
if a>0:
res = self._z**b
else:
res = self._fak*self._z**b
else:
if is_even(c):
arg = (a+d)*c-b*d*(c*c-1)+3*d-3-3*c*d
v=kronecker(c,d)
else:
arg = (a+d)*c-b*d*(c*c-1)-3*c
v=kronecker(d,c)
if not self._half_integral_weight:
# recall that we can use eta for any real weight
v=v**(2*self._weight)
arg=arg*(self._k_num)
res = v*fak*self._z**arg
if self._character:
res = res * self._character(d)
if self._is_dual:
res=res**-1
return res
def sigma_rep(Delta, print_divisors=False):
s = 0
for DD in divisors(Delta):
if is_fundamental_discriminant(-DD):
D = -DD
for d in divisors(old_div(Delta, D)):
s += kronecker(D, d)
if print_divisors:
print(D, d, kronecker(D, d))
return s
def xi(self, A):
r""" The eight-root of unity in front of the Weil representation.
INPUT:
-''N'' -- integer
-''A'' -- element of PSL(2,Z)
EXAMPLES::
sage: A=SL2Z([41,77,33,62])
sage: WR.xi(A)
-zeta8^3]
sage: S,T=SL2Z.gens()
sage: WR.xi(S)
-zeta8^3
sage: WR.xi(T)
1
sage: A=SL2Z([-1,1,-4,3])
sage: WR.xi(A)
-zeta8^2
sage: A=SL2Z([0,1,-1,0])
sage: WR.xi(A)
-zeta8
"""
a = Integer(A[0, 0])
b = Integer(A[0, 1])
c = Integer(A[1, 0])
d = Integer(A[1, 1])
if (c == 0):
return 1
z = CyclotomicField(8).gen()
N = self._N
N2 = odd_part(N)
Neven = ZZ(2 * N).divide_knowing_divisible_by(N2)
c2 = odd_part(c)
Nc = gcd(Integer(2 * N), Integer(c))
cNc = ZZ(c).divide_knowing_divisible_by(Nc)
f1 = kronecker(-a, cNc)
f2 = kronecker(cNc, ZZ(2 * N).divide_knowing_divisible_by(Nc))
if (is_odd(c)):
s = c * N2
elif (c % Neven == 0):
s = (c2 + 1 - N2) * (a + 1)
else:
s = (c2 + 1 - N2) * (a + 1) - N2 * a * c2
r = -1 - QQ(N2) / QQ(gcd(c, N2)) + s
xi = f1 * f2 * z**r
return xi
def xi(self,A):
r""" The eight-root of unity in front of the Weil representation.
INPUT:
-''N'' -- integer
-''A'' -- element of PSL(2,Z)
EXAMPLES::
sage: A=SL2Z([41,77,33,62])
sage: WR.xi(A)
-zeta8^3]
sage: S,T=SL2Z.gens()
sage: WR.xi(S)
-zeta8^3
sage: WR.xi(T)
1
sage: A=SL2Z([-1,1,-4,3])
sage: WR.xi(A)
-zeta8^2
sage: A=SL2Z([0,1,-1,0])
sage: WR.xi(A)
-zeta8
"""
a=Integer(A[0,0]); b=Integer(A[0,1])
c=Integer(A[1,0]); d=Integer(A[1,1])
if(c==0):
return 1
z=CyclotomicField(8).gen()
N=self._N
N2=odd_part(N)
Neven=ZZ(2*N).divide_knowing_divisible_by(N2)
c2=odd_part(c)
Nc=gcd(Integer(2*N),Integer(c))
cNc=ZZ(c).divide_knowing_divisible_by(Nc)
f1=kronecker(-a,cNc)
f2=kronecker(cNc,ZZ(2*N).divide_knowing_divisible_by(Nc))
if(is_odd(c)):
s=c*N2
elif( c % Neven == 0):
s=(c2+1-N2)*(a+1)
else:
s=(c2+1-N2)*(a+1)-N2*a*c2
r=-1-QQ(N2)/QQ(gcd(c,N2))+s
xi=f1*f2*z**r
return xi
def class_nr_pos_def_qf(D):
r"""
Compute the class number of positive definite quadratic forms.
For fundamental discriminants this is the class number of Q(sqrt(D)),
otherwise it is computed using: Cohen 'A course in Computational Algebraic Number Theory', p. 233
"""
if D>0:
return 0
D4 = D % 4
if D4 == 3 or D4==2:
return 0
K = QuadraticField(D)
if is_fundamental_discriminant(D):
return K.class_number()
else:
D0 = K.discriminant()
Df = ZZ(D).divide_knowing_divisible_by(D0)
if not is_square(Df):
raise ArithmeticError("Did not get a discrimimant * square! D={0} disc(D)={1}".format(D,D0))
D2 = sqrt(Df)
h0 = QuadraticField(D0).class_number()
w0 = _get_w(D0)
w = _get_w(D)
#print "w,w0=",w,w0
#print "h0=",h0
h = 1
for p in prime_divisors(D2):
h = QQ(h)*(1-kronecker(D0,p)/QQ(p))
#print "h=",h
#print "fak=",
h=QQ(h*h0*D2*w)/QQ(w0)
return h
def class_nr_pos_def_qf(D):
r"""
Compute the class number of positive definite quadratic forms.
For fundamental discriminants this is the class number of Q(sqrt(D)),
otherwise it is computed using: Cohen 'A course in Computational Algebraic Number Theory', p. 233
"""
if D>0:
return 0
D4 = D % 4
if D4 == 3 or D4==2:
return 0
K = QuadraticField(D)
if is_fundamental_discriminant(D):
return K.class_number()
else:
D0 = K.discriminant()
Df = ZZ(D).divide_knowing_divisible_by(D0)
if not is_square(Df):
raise ArithmeticError,"DId not get a discrinimant * square! D={0} disc(D)={1}".format(D,D0)
D2 = sqrt(Df)
h0 = QuadraticField(D0).class_number()
w0 = _get_w(D0)
w = _get_w(D)
#print "w,w0=",w,w0
#print "h0=",h0
h = 1
for p in prime_divisors(D2):
h = QQ(h)*(1-kronecker(D0,p)/QQ(p))
#print "h=",h
#print "fak=",
h=QQ(h*h0*D2*w)/QQ(w0)
return h
def test_curve(q, t, r, k, D, E):
"""
Description:
Tests that E is an elliptic curve over F_q with trace t, a subgroup of order r with embedding degree k, and fundamental discriminant D
Input:
q - size of prime field
t - trace of Frobenius
r - size of prime order subgroup
k - embedding degree
D - (negative) fundamental discriminant
Output:
bool - true iff E is an elliptic curve over F_q with trace t, a subgroup of order r with embedding degree k, and fundamental discriminant D
"""
bool = True
bool = bool and (power_mod(q, k, r) == 1) #q^k -1 ==0 mod r
bool = bool and (E.trace_of_frobenius() == t)
bool = bool and (kronecker(
(t * t - 4 * q) * Integer(D).inverse_mod(q), q) == 1)
bool = bool and (E.cardinality() == q + 1 - t)
bool = bool and (E.cardinality() % r == 0)
return bool
def test_curve(q,t,r,k,D,E):
"""
Description:
Tests that E is an elliptic curve over F_q with trace t, a subgroup of order r with embedding degree k, and fundamental discriminant D
Input:
q - size of prime field
t - trace of Frobenius
r - size of prime order subgroup
k - embedding degree
D - (negative) fundamental discriminant
Output:
bool - true iff E is an elliptic curve over F_q with trace t, a subgroup of order r with embedding degree k, and fundamental discriminant D
"""
bool = True
bool = bool and (power_mod(q, k, r) == 1) #q^k -1 ==0 mod r
bool = bool and (E.trace_of_frobenius() == t)
bool = bool and (kronecker((t*t-4*q) * Integer(D).inverse_mod(q),q) == 1)
bool = bool and (E.cardinality() == q+1-t)
bool = bool and (E.cardinality() % r ==0)
return bool
def eta_conjugated(a,b,c,d,l):
r"""
Gives eta(V_l A V_l^-1) with A=(a,b,c,d) for c>0
"""
assert c>=0 and (c%l)==0
if l<>1:
cp = QQ(c)/QQ(l)
bp = QQ(b)*QQ(l)
else:
cp=c; bp=b
if c==0:
l*bp,1
res = (a+d)*cp-bp*d*(cp*cp-1)
if is_odd(c):
return res-3*cp,kronecker(d,cp)
else:
return res+3*d-3-3*cp*d,kronecker(cp,d)
def eta_conjugated(a,b,c,d,l):
r"""
Gives eta(V_l A V_l^-1) with A=(a,b,c,d) for c>0
"""
assert c>=0 and (c%l)==0
if l<>1:
cp = QQ(c)/QQ(l)
bp = QQ(b)*QQ(l)
else:
cp=c; bp=b
if c==0:
l*bp,1
res = (a+d)*cp-bp*d*(cp*cp-1)
if is_odd(c):
return res-3*cp,kronecker(d,cp)
else:
return res+3*d-3-3*cp*d,kronecker(cp,d)
def make_curve(k, D):
"""
Description:
Find all MNT curves with embedding degree k and fundamental discriminant D
Input:
k - embedding degree
D - (negative) fundamental discriminant
Output:
curves - list of the aforementioned elliptic curves;
each curve is represented as a tuple (q,t,r,k,D)
"""
assert k == 3 or k == 4 or k == 6, 'Invalid embedding degree'
assert fundamental_discriminant(D) == D, 'Invalid discriminant'
if k == 3:
curves = _mnt_subcall(
k, D, 24, lambda x: (
(x % 24 == 19) and kronecker(6, x) == 1), 6, 3, -3, lambda x:
(x - 3) // 6, lambda x: (x + 3) // 6, lambda l: 6 * l - 1,
lambda l: -6 * l - 1, lambda l: 12 * l * l - 1)
if k == 4:
curves = _mnt_subcall(
k, D, -8, lambda x:
((x % 8 == 3) and kronecker(-2, x) == 1) or (x == 2) or (x == 8),
3, 2, 1, lambda x: (x - 2) // 3, lambda x: (x - 1) // 3,
lambda l: -l, lambda l: l + 1, lambda l: l * l + l + 1)
if k == 6:
curves = _mnt_subcall(k, D, -8, lambda x:
(x % 8 == 3) and kronecker(-2, x) == 1, 6, 5, 1,
lambda x: (x + 1) // 6, lambda x: (x - 1) // 6,
lambda l: 2 * l + 1, lambda l: -2 * l + 1,
lambda l: 4 * l * l + 1)
for c in curves:
assert is_valid_curve(c[0], c[1], c[2], c[3], c[4]), 'Invalid output'
assert k == c[3], 'Bug in code'
assert D == c[4], 'Bug in code'
return curves
def compare_formulas_2(D, k):
d1 = old_div(RR(abs(D)), RR(6))
if D < 0:
D = -D
s1 = RR(
sqrt(abs(D)) * sum([
log(d) for d in divisors(D) if is_fundamental_discriminant(-d)
and kronecker(-d, old_div(D, d)) == 1
]))
d2 = RR((old_div(2, (sqrt(3) * pi))) * s1)
return d1 - d2, d2, RR(2 * sqrt(D) * log(D) / pi)
def _action(self,A):
[a,b,c,d]=A
v=kronecker(c,d)*self._one ### I want the type of the result always be a number field element
if(d % 4 == 3):
v=-v*self._i
elif (c % 4 <> 0):
raise ValueError,"Only use theta multiplier for 4|c!"
if self._character<>None:
v = self._character(d)*v
if self._is_dual:
v = v**-1
return v
def _action(self,A):
[a,b,c,d]=A
v=kronecker(c,d)*self._one ### I want the type of the result always be a number field element
if(d % 4 == 3):
v=-v*self._i
elif (c % 4 <> 0):
raise ValueError,"Only use theta multiplier for 4|c!"
if self._character<>None:
v = self._character(d)*v
if self._is_dual:
v = v**-1
return v
def test_promise(r,k,D):
"""
Description:
Tests that r,k,D is a valid input to the Cocks-Pinch method
Input:
r - prime
k - embedding degree
D - (negative) funadmental discriminant
Output:
bool - true iff (r,k,D) is a valid input to the Cocks-Pinch method
"""
bool = (kronecker(D,r) == 1) # D is a square mod r
bool = bool and ( (r-1) % k ==0) # k | r-1
bool = bool and (D == fundamental_discriminant(D)) # check that D is a fundamental discriminant
return bool
def test_promise(r, k, D):
"""
Description:
Tests that r,k,D is a valid input to the Cocks-Pinch method
Input:
r - prime
k - embedding degree
D - (negative) funadmental discriminant
Output:
bool - true iff (r,k,D) is a valid input to the Cocks-Pinch method
"""
bool = (kronecker(D, r) == 1) # D is a square mod r
bool = bool and ((r - 1) % k == 0) # k | r-1
bool = bool and (D == fundamental_discriminant(D)
) # check that D is a fundamental discriminant
return bool
def _generate_s(bases, k2, k3):
s = []
for b in bases:
s_b = set()
for p in range(1, 4 * b, 2):
if kronecker(b, p) == -1:
s_b.add(p)
s.append(s_b)
for i in range(len(s)):
mod = 4 * bases[i]
inv2 = pow(k2, -1, mod)
inv3 = pow(k3, -1, mod)
s2 = set()
s3 = set()
for z in s[i]:
s2.add(inv2 * (z + k2 - 1) % mod)
s3.add(inv3 * (z + k3 - 1) % mod)
s[i] &= s2 & s3
return s
def _dimension_formula(self,k,eps=1,cuspidal=1):
ep = 0
N = self._N
if (2*k) % 4 == 1: ep = 1
if (2*k) % 4 == 3: ep = -1
if ep==0: return 0,0
if eps==-1:
ep = -ep
twok = ZZ(2*k)
K0 = 1
sqf = ZZ(N).divide_knowing_divisible_by(squarefree_part(N))
if sqf>12:
b2 = max(sqf.divisors())
else:
b2 = 1
b = sqrt(b2)
if ep==1:
K0 = floor(QQ(b+2)/QQ(2))
else:
# print "b=",b
K0 = floor(QQ(b-1)/QQ(2))
if is_even(N):
e2 = ep*kronecker(2,twok)/QQ(4)
else:
e2 = 0
N2 = odd_part(N)
N22 = ZZ(N).divide_knowing_divisible_by(N2)
k3 = kronecker(3,twok)
if gcd(3,N)>1:
if eps==1:
e3 = -ep*kronecker(-3,4*k+ep-1)/QQ(3)
else:
e3 = -1*ep*kronecker(-3,4*k+ep+1)/QQ(3)
#e3 = -1/3*ep
else:
f1 = kronecker(3,2*N22)*kronecker(-12,N2) - ep
f2 = kronecker(-3,twok+1)
e3 = f1*f2/QQ(6)
ID = QQ(N+ep)*(k-1)/QQ(12)
P = 0
for d in ZZ(4*N).divisors():
dm4=d % 4
if dm4== 2 or dm4 == 1:
h = 0
elif d == 3:
h = QQ(1)/QQ(3)
elif d == 4:
h = QQ(1)/QQ(2)
else:
h = class_nr_pos_def_qf(-d)
if self._verbose>1:
print "h({0})={1}".format(d,h)
if h<>0:
P= P + h
P = QQ(P)/QQ(4)
if self._verbose>0:
print "P=",P
P=P + QQ(ep)*kronecker(-4,N)/QQ(8)
if eps==-1:
P = -P
if self._verbose>0:
print "P=",P
# P = -2*N**2 + N*(twok+10-ep*3) +(twok+10)*ep-1
if self._verbose>0:
print "ID=",ID
P = P - QQ(1)/QQ(2*K0)
# P = QQ(P)/QQ(24) - K0
# P = P - K0
res = ID + P + e2 + e3
if self._verbose>1:
print "twok=",twok
print "K0=",K0
print "ep=",ep
print "e2=",e2
print "e3=",e3
print "P=",P
if cuspidal==0:
res = res + K0
return res #,ep
def _dimension_formula(self,k,eps=1,cuspidal=1):
ep = 0
N = self._N
if (2*k) % 4 == 1: ep = 1
if (2*k) % 4 == 3: ep = -1
if ep==0: return 0,0
if eps==-1:
ep = -ep
twok = ZZ(2*k)
K0 = 1
sqf = ZZ(N).divide_knowing_divisible_by(squarefree_part(N))
if sqf>12:
b2 = max(sqf.divisors())
else:
b2 = 1
b = sqrt(b2)
if ep==1:
K0 = floor(QQ(b+2)/QQ(2))
else:
# print "b=",b
K0 = floor(QQ(b-1)/QQ(2))
if is_even(N):
e2 = ep*kronecker(2,twok)/QQ(4)
else:
e2 = 0
N2 = odd_part(N)
N22 = ZZ(N).divide_knowing_divisible_by(N2)
k3 = kronecker(3,twok)
if gcd(3,N)>1:
if eps==1:
e3 = -ep*kronecker(-3,4*k+ep-1)/QQ(3)
else:
e3 = -1*ep*kronecker(-3,4*k+ep+1)/QQ(3)
#e3 = -1/3*ep
else:
f1 = kronecker(3,2*N22)*kronecker(-12,N2) - ep
f2 = kronecker(-3,twok+1)
e3 = f1*f2/QQ(6)
ID = QQ(N+ep)*(k-1)/QQ(12)
P = 0
for d in ZZ(4*N).divisors():
dm4=d % 4
if dm4== 2 or dm4 == 1:
h = 0
elif d == 3:
h = QQ(1)/QQ(3)
elif d == 4:
h = QQ(1)/QQ(2)
else:
h = class_nr_pos_def_qf(-d)
if self._verbose>1:
print("h({0})={1}".format(d,h))
if h!=0:
P= P + h
P = QQ(P)/QQ(4)
if self._verbose>0:
print("P={0}".format(P))
P=P + QQ(ep)*kronecker(-4,N)/QQ(8)
if eps==-1:
P = -P
if self._verbose>0:
print("P={0}".format(P))
# P = -2*N**2 + N*(twok+10-ep*3) +(twok+10)*ep-1
if self._verbose>0:
print("ID={0}".format(ID))
P = P - QQ(1)/QQ(2*K0)
# P = QQ(P)/QQ(24) - K0
# P = P - K0
res = ID + P + e2 + e3
if self._verbose>1:
print("twok={0}".format(twok))
print("K0={0}".format(K0))
print("ep={0}".format(ep))
print("e2={0}".format(e2))
print("e3={0}".format(e3))
print("P={0}".format(P))
if cuspidal==0:
res = res + K0
return res #,ep
def compare_formulas_2(D, k):
d1 = old_div(RR(abs(D)), RR(6))
if D < 0:
D = -D
s1 = RR(
sqrt(abs(D)) * sum([
log(d) for d in divisors(D) if is_fundamental_discriminant(-d)
and kronecker(-d, old_div(D, d)) == 1
]))
d2 = RR((old_div(2, (sqrt(3) * pi))) * s1)
return d1 - d2, d2, RR(2 * sqrt(D) * log(D) / pi)
A3 = lambda N: sum(
[kronecker(-N, x) for x in Zmod(N) if x**2 + x + Zmod(N)(1) == 0])
def compare_formulas_2a(D, k):
d1 = dimension_new_cusp_forms(kronecker_character(D), k)
if D < 0:
D = -D
d2 = RR(1 / pi * sqrt(D) * sum([
log(d) * sigma(old_div(D, d), 0) for d in divisors(D) if Zmod(d)
(old_div(D, d)).is_square() and is_fundamental_discriminant(-d)
]))
return d1 - d2
def compare_formulas_3(D, k):
d1 = dimension_cusp_forms(kronecker_character(D), k)