def supersingular_D(prime):
r"""
This function returns a fundamental discriminant `D` of an
imaginary quadratic field, where the given prime does not split
(see Silverman's Advanced Topics in the Arithmetic of Elliptic
Curves, page 184, exercise 2.30(d).)
INPUT:
- prime -- integer, prime
OUTPUT:
D -- integer, negative
EXAMPLES:
These examples return *supersingular discriminants* for 7,
15073 and 83401::
sage: supersingular_D(7)
-4
sage: supersingular_D(15073)
-15
sage: supersingular_D(83401)
-7
AUTHORS:
- David Kohel - [email protected]
- Iftikhar Burhanuddin - [email protected]
"""
if not (rings.Integer(prime).is_prime()):
raise ValueError("%s is not a prime" % prime)
# Making picking D more intelligent
D = -1
while True:
Dmod4 = rings.Mod(D, 4)
if Dmod4 in (0, 1) and (rings.kronecker(D, prime) != 1):
return D
D = D - 1
def supersingular_D(prime):
r"""
This function returns a fundamental discriminant `D` of an
imaginary quadratic field, where the given prime does not split
(see Silverman's Advanced Topics in the Arithmetic of Elliptic
Curves, page 184, exercise 2.30(d).)
INPUT:
- prime -- integer, prime
OUTPUT:
D -- integer, negative
EXAMPLES:
These examples return *supersingular discriminants* for 7,
15073 and 83401::
sage: supersingular_D(7)
-4
sage: supersingular_D(15073)
-15
sage: supersingular_D(83401)
-7
AUTHORS:
- David Kohel - [email protected]
- Iftikhar Burhanuddin - [email protected]
"""
if not (rings.Integer(prime).is_prime()):
raise ValueError("%s is not a prime" % prime)
#Making picking D more intelligent
D = -1
while True:
Dmod4 = rings.Mod(D, 4)
if Dmod4 in (0, 1) and (rings.kronecker(D, prime) != 1):
return D
D = D - 1
def supersingular_j(FF):
r"""
This function returns a supersingular j-invariant over the finite
field FF.
INPUT:
- ``FF`` -- finite field with p^2 elements, where p is a prime number
OUTPUT:
finite field element -- a supersingular j-invariant
defined over the finite field FF
EXAMPLES:
The following examples calculate supersingular j-invariants for a
few finite fields::
sage: supersingular_j(GF(7^2, 'a'))
6
Observe that in this example the j-invariant is not defined over
the prime field::
sage: supersingular_j(GF(15073^2,'a')) # optional - database
10630*a + 6033
sage: supersingular_j(GF(83401^2, 'a'))
67977
AUTHORS:
- David Kohel -- [email protected]
- Iftikhar Burhanuddin -- [email protected]
"""
if not (FF.is_field()) or not (FF.is_finite()):
raise ValueError("%s is not a finite field" % FF)
prime = FF.characteristic()
if not (rings.Integer(prime).is_prime()):
raise ValueError("%s is not a prime" % prime)
if not (rings.Integer(FF.cardinality())) == rings.Integer(prime ** 2):
raise ValueError("%s is not a quadratic extension" % FF)
if rings.kronecker(-1, prime) != 1:
j_invss = 1728 # (2^2 * 3)^3
elif rings.kronecker(-2, prime) != 1:
j_invss = 8000 # (2^2 * 5)^3
elif rings.kronecker(-3, prime) != 1:
j_invss = 0 # 0^3
elif rings.kronecker(-7, prime) != 1:
j_invss = 16581375 # (3 * 5 * 17)^3
elif rings.kronecker(-11, prime) != 1:
j_invss = -32768 # -(2^5)^3
elif rings.kronecker(-19, prime) != 1:
j_invss = -884736 # -(2^5 * 3)^3
elif rings.kronecker(-43, prime) != 1:
j_invss = -884736000 # -(2^6 * 3 * 5)^3
elif rings.kronecker(-67, prime) != 1:
j_invss = -147197952000 # -(2^5 * 3 * 5 * 11)^3
elif rings.kronecker(-163, prime) != 1:
j_invss = -262537412640768000 # -(2^6 * 3 * 5 * 23 * 29)^3
else:
D = supersingular_D(prime)
DBCP = HilbertClassPolynomialDatabase()
hc_poly = FF["x"](DBCP[D])
root_hc_poly_list = list(hc_poly.roots())
j_invss = root_hc_poly_list[0][0]
return FF(j_invss)
def supersingular_j(FF):
r"""
This function returns a supersingular j-invariant over the finite
field FF.
INPUT:
- ``FF`` -- finite field with p^2 elements, where p is a prime number
OUTPUT:
finite field element -- a supersingular j-invariant
defined over the finite field FF
EXAMPLES:
The following examples calculate supersingular j-invariants for a
few finite fields::
sage: supersingular_j(GF(7^2, 'a'))
6
Observe that in this example the j-invariant is not defined over
the prime field::
sage: supersingular_j(GF(15073^2,'a')) # optional - database
10630*a + 6033
sage: supersingular_j(GF(83401^2, 'a'))
67977
AUTHORS:
- David Kohel -- [email protected]
- Iftikhar Burhanuddin -- [email protected]
"""
if not (FF.is_field()) or not (FF.is_finite()):
raise ValueError("%s is not a finite field" % FF)
prime = FF.characteristic()
if not (rings.Integer(prime).is_prime()):
raise ValueError("%s is not a prime" % prime)
if not (rings.Integer(FF.cardinality())) == rings.Integer(prime**2):
raise ValueError("%s is not a quadratic extension" % FF)
if rings.kronecker(-1, prime) != 1:
j_invss = 1728 #(2^2 * 3)^3
elif rings.kronecker(-2, prime) != 1:
j_invss = 8000 #(2^2 * 5)^3
elif rings.kronecker(-3, prime) != 1:
j_invss = 0 #0^3
elif rings.kronecker(-7, prime) != 1:
j_invss = 16581375 #(3 * 5 * 17)^3
elif rings.kronecker(-11, prime) != 1:
j_invss = -32768 #-(2^5)^3
elif rings.kronecker(-19, prime) != 1:
j_invss = -884736 #-(2^5 * 3)^3
elif rings.kronecker(-43, prime) != 1:
j_invss = -884736000 #-(2^6 * 3 * 5)^3
elif rings.kronecker(-67, prime) != 1:
j_invss = -147197952000 #-(2^5 * 3 * 5 * 11)^3
elif rings.kronecker(-163, prime) != 1:
j_invss = -262537412640768000 #-(2^6 * 3 * 5 * 23 * 29)^3
else:
D = supersingular_D(prime)
DBCP = HilbertClassPolynomialDatabase()
hc_poly = FF['x'](DBCP[D])
root_hc_poly_list = list(hc_poly.roots())
j_invss = root_hc_poly_list[0][0]
return FF(j_invss)