  Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a
positive integer. Under some technical assumptions, we give an algorithm to
span the isomorphism classes of principally polarized abelian varieties in the
isogeny class of $E^g$. The varieties are first described as hermitian lattices
over (not necessarily maximal) quadratic orders and then geometrically in terms
of their algebraic theta null point. We also show how to algebraically compute
Siegel modular forms of even weight given as polynomials in the theta constants
by a careful choice of an affine lift of the theta null point. We then use
these results to give an algebraic computation of Serre's obstruction for
principally polarized abelian threefolds isogenous to $E^3$ and of the Igusa
modular form in dimension $4$. We illustrate our algorithms with examples of
curves with many rational points over finite fields.
