---
tag: 427N6S6QQW
kind: concept
name: Elliptic curve
msc: [14H52, 11G05, 11G07]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_curve
same_as:
  wikidata: Q268493
  wikipedia: Elliptic curve
---

## Summary

An elliptic curve over a [[field|04C5ZBG1WT]] $K$ 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|P7V1BQJPSS]], 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.

## Explanations

### L1 human

Over the real numbers, an elliptic curve looks like a smooth curve in the plane, in one or two pieces, symmetric about the $x$-axis.
Its key feature is that one can "add" two points on it: draw the line through them, take the third intersection point, and reflect it in the $x$-axis.
For example, $y^2 = x^3 - x$ is an elliptic curve, since $4 \cdot (-1)^3 + 27 \cdot 0^2 = -4 \ne 0$.
The curve $y^2 = x^3$ is not one, because it has a sharp cusp at the origin.
Despite the name, an elliptic curve is not an ellipse; the name comes from elliptic integrals.