def as_permutation(self):
r"""
Return the element of the permutation group G (isomorphic to the
abelian group A) associated to a in A.
EXAMPLES::
sage: G = AbelianGroup(3,[2,3,4],names="abc"); G
Multiplicative Abelian group isomorphic to C2 x C3 x C4
sage: a,b,c=G.gens()
sage: Gp = G.permutation_group(); Gp
Permutation Group with generators [(6,7,8,9), (3,4,5), (1,2)]
sage: a.as_permutation()
(1,2)
sage: ap = a.as_permutation(); ap
(1,2)
sage: ap in Gp
True
"""
from sage.libs.gap.libgap import libgap
G = self.parent()
A = libgap.AbelianGroup(G.gens_orders())
phi = libgap.IsomorphismPermGroup(A)
gens = libgap.GeneratorsOfGroup(A)
L2 = libgap.Product([geni**Li for geni, Li in zip(gens, self.list())])
pg = libgap.Image(phi, L2)
return G.permutation_group()(pg)
def norm_of_galois_extension(self):
r"""
Return the norm as a Galois extension of `\QQ`, which is
given by the product of all galois_conjugates.
EXAMPLES::
sage: E(3).norm_of_galois_extension()
1
sage: E(6).norm_of_galois_extension()
1
sage: (E(2) + E(3)).norm_of_galois_extension()
3
sage: parent(_)
Integer Ring
"""
obj = self._obj
k = obj.Conductor().sage()
return libgap.Product(libgap([obj.GaloisCyc(i) for i in range(k) if k.gcd(i) == 1])).sage()
