Let $k$ be an algebraically closed field, let $\lambda \in k - \{0,1\}$ and let $C = k[t]/(t^n)$. Hartshorne's "Deformation Theory" chapter 1 exercise 4.9(c) asserts that the family $$y^2 = x(x-1)(x-(\lambda +t))$$ is a nontrivial family over $C$. The hint given is "the computation of $j$-invariant can be carried out over the ring $C$."
After a bit of searching online, it looks like the moduli space of elliptic curves over a ring becomes much more complicated than over a field (as in beyond the knowledge one acquires in a first course of algebraic geometry, which is what I currently have roughly).
My hope is that over $k[t]/(t^n)$, the problem would be only slightly more complicated than over $k$. But I have no idea where to even begin. If you could provide a reference, or a more detailed "hint" on how to define the $j$-invariant and use it to classify elliptic curves over $C$, that would be greatly appreciated.
