Computing abelian varieties over finite fields isogenous to a power

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 $A$.