Esempio n. 1
0
    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()