## Algebraic curve mapping to elliptic curve - how to check whether this is possible?

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.

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Over a finite field there are good algorithms for computing the zeta function of $C$, from which, via the Tate conjecture, one can decide the existence of a morphism from $C$ to an elliptic curve. But computing such a morphism, once its existence is known, seems to be a hard problem. [Ref. : Handbook of elliptic and hyperelliptic curve cryptography.] So the "if" in your final sentence is not a small one, I believe. – inkspot Aug 30 at 16:39
I have two dubious ideas. If it's possible to find a map through $\mathbb P^1$ and classify the monodromy actions of the ramification groups at the ramified points, you can check if the map factors through an elliptic curve using group theory. If there is some kind of obvious "distinguished" $1$-form on this curve, which has an especially simple formula or good symmetry properties something, you can try to show it's the derivative of the map to an elliptic curve, but I'm not sure how you'd do that. – Will Sawin Aug 30 at 16:39