Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields

Abstract

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes of abelian varieties over a finite field $\mathbb{F}_q$ which belong to an isogeny class determined by a characteristic polynomial $h$ of Frobenius when $h$ is ordinary, or $q$ is prime and $h$ has no real roots.