A curve X over Q is modular if it is dominated by X₁(N) for some N; if in addition the image of its jacobian in J₁(N) is contained in the new subvariety of J₁(N), then X is called a new modular curve.

Elliptic curves are among the more beguiling objects in modern mathematics. They don’t seem complicated, but they form an expressway between the math that many people learn in high school and research ...