def __truediv__(self, I):
"""
Form the quotient by an integral ideal.
EXAMPLES::
sage: QQ / ZZ
Q/Z
"""
from sage.rings.ideal import Ideal_generic
from sage.groups.additive_abelian.qmodnz import QmodnZ
if I is ZZ:
return QmodnZ(1)
elif isinstance(I, Ideal_generic) and I.base_ring() is ZZ:
return QmodnZ(I.gen())
else:
return super(RationalField, self).__truediv__(I)
def value_module_qf(self):
r"""
Return `\QQ / n\ZZ` with `n\ZZ = (V,W) + \ZZ \{ (w,w) | w \in W \}`.
This is where the torsion quadratic form takes values.
EXAMPLES::
sage: A2 = Matrix(ZZ, 2, 2, [2,-1,-1,2])
sage: L = IntegralLattice(2*A2)
sage: D = L.discriminant_group()
sage: D
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1/3]
sage: D.value_module_qf()
Q/2Z
"""
return QmodnZ(self._modulus_qf)
def value_module(self):
r"""
Return `\QQ / m\ZZ` with `m = (V, W)`.
This is where the inner product takes values.
EXAMPLES::
sage: A2 = Matrix(ZZ, 2, 2, [2,-1,-1,2])
sage: L = IntegralLattice(2*A2)
sage: D = L.discriminant_group()
sage: D
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1/3]
sage: D.value_module()
Q/Z
"""
return QmodnZ(self._modulus)