def sqrt5_ideal(X):
"""
Return ideal in ring of integers of Q(sqrt(5)) defined by X.
INPUT:
- `X` -- ideal or list or element of F
OUTPUT:
- ideal
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_tables import F, sqrt5_ideal
sage: sqrt5_ideal(7)
Fractional ideal (7)
sage: sqrt5_ideal(F.0)
Fractional ideal (a)
sage: sqrt5_ideal([F.0, 2])
Fractional ideal (1)
"""
if not is_Ideal(X) or X.ring() != F:
return O_F.ideal(X)
return X
def fast_ideal(P):
return Prime(P) if is_Ideal(P) else P
def check_prime(K, P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
"""
if K is QQ:
if isinstance(P, (int, long, Integer)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError, "%s is not prime" % P
else:
if is_Ideal(P) and P.base_ring() is ZZ and P.is_prime():
return P.gen()
raise TypeError, "%s is not a prime ideal of %s" % (P, ZZ)
if not is_NumberField(K):
raise TypeError, "%s is not a number field" % K
if is_NumberFieldFractionalIdeal(P):
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
if is_NumberFieldElement(P):
if P in K:
P = K.ideal(P)
else:
raise TypeError, "%s is not an element of %s" % (P, K)
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
raise TypeError, "%s is not a valid prime of %s" % (P, K)
def check_prime(K, P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
"""
if K is QQ:
if isinstance(P, (int, long, Integer)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError, "%s is not prime" % P
else:
if is_Ideal(P) and P.base_ring() is ZZ and P.is_prime():
return P.gen()
raise TypeError, "%s is not a prime ideal of %s" % (P, ZZ)
if not is_NumberField(K):
raise TypeError, "%s is not a number field" % K
if is_NumberFieldFractionalIdeal(P):
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
if is_NumberFieldElement(P):
if P in K:
P = K.ideal(P)
else:
raise TypeError, "%s is not an element of %s" % (P, K)
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
raise TypeError, "%s is not a valid prime of %s" % (P, K)
def ideal(X):
if not is_Ideal(X):
return O_F.ideal(X)
return X
