Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as columns.

A moduli problem for elliptic curves is then just a contravariant functor $\textbf{Ell}\rightarrow\textbf{Sets}$.

For example, we usual level $N$ moduli problem is the functor sending $E/S$ to the set of isomorphisms $E[N]\stackrel{\sim}{\longrightarrow}(\mathbb{Z}/N\mathbb{Z})_S^2$.

There are a ton of these functors, mostly coming from various cohomology theories, but the only such functors I can think of that land in the category of finite sets all have to do with torsion points on the elliptic curve.

Does anyone have any examples of a functor from $\textbf{Ell}\rightarrow\textbf{FiniteSets}$ that doesn't have to do with torsion data?