How to split two-dimensional Jacobians: a geometric construction

Let $\pi \text{\hspace{0.17em}⁣}:Y\to X$ 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: $\mathrm{Jac}Y\sim \mathrm{Jac}X\times \mathrm{Jac}W$. 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.