Spanning the isogeny class of a power of an elliptic curve

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