Decomposing Jacobians Via Galois covers

Let $phi : X o Y$ be a (possibly ramified) cover, with $X$ and $Y$ of strictly positive genus. We develop tools to identify the Prym variety of $phi$, up to isogeny, as the Jacobian of a quotient curve $C$ of the Galois closure of a composition $X o Y o P^1$ of $phi$ with a well-chosen map $Y o P^1$ that identifies branch points of $phi$. To our knowledge, this method recovers all previously obtained descriptions of Prym varieties as Jacobians. It also finds new decompositions, and for some of these, including one where $X$ has genus $3$, $Y$ has genus $1$ and $phi$ is a degree $3$ map totally ramified over $2$ points, we find an algebraic equation of the curve $C$