Let $k$ be an algebraically closed of characteristic $\neq 2,3$. The $j$-invariant induces a bijection

$\{\text{elliptic curves over } k\}/\cong \longrightarrow k.$

Perhaps this is a silly question: I'm curious if there is any meaningful geometric interpretation of the induced ring structure on the set of isomorphism-classes of elliptic curves. The resulting operations can be written down in Weierstrass forms, but the formulas are, of course, not illuminating at all. The zero element is $y^2+y=x^3$. It would be even more interesting what structures there are on the category of elliptic curves over $k$, so without modding out isomorphisms.