Taniyama-Shimura Conjecture - any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve
X0(N)
for some integer N; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer q-expansion, followed if need be by an isogeny.
I think (sorry wikipedia thinks lol)
