j-invariant

Summary

The $j$-invariant is the function $j = 1728\, E_4^3 / (E_4^3 - E_6^2)$ on the upper half-plane, built from Eisenstein series. It is invariant under the modular group, holomorphic on $\mathbb{H}$, and has a simple pole at infinity. It gives a bijection from $\mathrm{SL}_2(\Z) \backslash \mathbb{H}$ onto $\C$. Two elliptic curves over $\C$ are isomorphic exactly when they have the same $j$-invariant.