Question: Let $C$ be an algebraic curve over some field (like the rationals) given by a plane projective model (possibly with singularities). Is there an easy way to see if this curve has a non-trivial rational map to an elliptic curve?
Criteria involving the Jacobian (something like $J(C)$ has a subgroup of codimension $1$) wouldn't help, as the curves I am interested in have a big genus ($>10$) and high degrees in both variables, so it is very unlikely that anything about their Jacobians can be computed.
Background: This question arose from an attempt to study rational points (over $\mathbb Q$) on certain curves which possibly are coverings of lower genus curves. In particular, if one could compute a covering map to an elliptic curve, one could efficiently look for rationals points.