Modular elliptic curve

A modular elliptic curve is an elliptic curve E that admits a parametrisation X₀(N) → E 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.

Equivalent formulations

Equivalent to the above formulation of modularity is that the L-series of E agrees with the L-series of a normalized eigenform.