Elliptic curve

Summary

An elliptic curve over a field $K$ is a smooth projective curve of genus one over $K$ together with a chosen point defined over $K$. Away from characteristics $2$ and $3$, it is given by an equation $y^2 = x^3 + ax + b$ with $4a^3 + 27b^2 \ne 0$. Its points form an abelian group under the chord-and-tangent law, which makes it the simplest abelian variety. Elliptic curves are central in number theory, for example in the proof of Fermat's Last Theorem, and in cryptography.