Weierstrass equation

Summary

A Weierstrass equation is a cubic equation $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ with coefficients in a field. When its discriminant is nonzero, it defines an elliptic curve, with the point at infinity as the origin. Every elliptic curve can be written in this form, so Weierstrass equations are the standard way to compute with elliptic curves. The name comes from the differential equation of the Weierstrass elliptic function, which yields such an equation over $\C$.