Elliptic curve
Number theory
A smooth projective curve of genus 1 together with a chosen rational point on it. Concretely, the solutions of a nonsingular cubic such as y² = x³ + ax + b, plus a point at infinity.
What is usually left out
A smooth plane cubic with no rational point on it is not an elliptic curve; the chosen base point is part of the definition. What makes these objects central is that their points form an abelian group under a geometric addition rule, which is what both arithmetic geometry and elliptic-curve cryptography exploit.
Related terms
In the atlas
- Birch and Swinnerton-Dyer conjecture — Number theory, open