How to split two-dimensional Jacobians: a geometric construction

Let $\pi\colon Y \to X$ be a branched cover of complex algebraic curves of respective genera $g(Y)=2$ and $g(X)=1$. The Jacobian of $Y$ is isogenous to the product of two elliptic curves: $\operatorname{Jac} Y \sim \operatorname{Jac} X \times \operatorname{Jac} W$. We present an explicit geometric construction of the complementary curve $W$. Furthermore, we establish a criterion to decide whether an algebraic correspondence of curves admits a push-out.