  Let k be a field of characteristic different from 2. There can be an
obstruction for an indecomposable principally polarized abelian threefold (A,a)
over k to be a Jacobian over k. It can be computed in terms of the rationality
of the square root of the value of a certain Siegel modular form. We show how
to do this explicitly for principally polarized abelian threefolds which are
the third power of an elliptic curve with complex multiplication. We use our
numeric results to prove or refute the existence of some optimal curves of
genus 3.
