Modular elliptic curve

Graphs of elliptic curves y² = x³ − x and y² = x³ − x + 1. If we consider these as curves over the rationals, then the modularity theorem asserts that they can be parametrized by a modular curve.

A modular elliptic curve is an elliptic curve E that admits a parametrization X₀(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.