Abelian varieties with no power isogenous to a Jacobian

Let X be a curve of genus at least 4 that is very general or very general hyperelliptic. We classify all the ways in which a power (JX)^k of the Jacobian of X can be isogenous to a product of Jacobians of curves. As an application, we show that if A is a very general principally polarized abelian variety of dimension at least 4 or the intermediate Jacobian of a very general cubic threefold, then no power A^k is isogenous to a product of Jacobians of curves. This confirms various cases of the Coleman–Oort conjecture. We further deduce from our results some progress on the question of whether the integral Hodge conjecture fails for A as above.