  In this paper we give a module-theoretic description of the isomorphism
classes of abelian varieties $A$ isogenous to $B^r$, where the characteristic
polynomial $g$ of Frobenius of $B$ is an ordinary square-free $q$-Weil
polynomial, for a power $q$ of a prime $p$, or a square-free $p$-Weil
polynomial with no real roots. Under some extra assumptions on the polynomial
$g$ we give an explicit description of all the isomorphism classes which can be
computed in terms of fractional ideals of an order in a finite product of
number fields. In the ordinary case, we also give a module-theoretic
description of the polarizations of $A$.
