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Let $E$ be an ordinary elliptic curve over a finite field and $g$ be apositive integer. Under some technical assumptions, we give an algorithm tospan the isomorphism classes of principally polarized abelian varieties in theisogeny class of $E^g$. The varieties are first described as hermitian latticesover (not necessarily maximal) quadratic orders and then geometrically in termsof their algebraic theta null point. We also show how to algebraically computeSiegel modular forms of even weight given as polynomials in the theta constantsby a careful choice of an affine lift of the theta null point. We then usethese results to give an algebraic computation of Serre's obstruction forprincipally polarized abelian threefolds isogenous to $E^3$ and of the Igusamodular form in dimension $4$. We illustrate our algorithms with examples ofcurves with many rational points over finite fields.

Spanning the isogeny class of a power of an ordinary elliptic curve over  a finite field. Application to the number of rational points of curves of  genus $\leq 4$