Modular elliptic curve

A modular elliptic curve is an elliptic curve that admits a parametrisation by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, formerly known as the Shimura-Taniyama conjecture asserts that every elliptic curve defined over the rational numbers is modular.