We consider abelian varieties over a finite field which are ordinary, or over a prime field, and which have commutative endomorphism algebra.
Works of Deligne and Centeleghe-Stix allow us to describe these abelian varieties in terms of fractional ideals of an order in an étale algebra. I will explain how such descriptions can be exploited to explictly compute the abelian varieties up to isomorphism.
Moreover, results by Howe give us a way to compute principal polarisations of the abelian varieties in the ordinary case. In a joint work with Bergström and Karemaker we extend these results to the prime field case.