Elliptic curve

Builds on Field

Definition

Let $K$ be a field. An elliptic curve over $K$ is a pair $(E, O)$ with two properties. $E$ is a smooth projective curve over $K$ of genus $1$ , and $O \in E(K)$ is a point of $E$ defined over $K$ .

Equivalently, it is a curve in the projective plane given by a Weierstrass equation with $a_1, \ldots, a_6 \in K$ : $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$ Its discriminant $\Delta$ must be nonzero, and $O = [0 : 1 : 0]$ is the point at infinity. If the characteristic of $K$ is not $2$ or $3$ , every elliptic curve has an equation $y^2 = x^3 + ax + b$ with $a, b \in K$ , and the condition $\Delta \ne 0$ becomes $4a^3 + 27b^2 \ne 0$ .

Remarks

Example: $E \colon y^2 = x^3 - x$ over $\Q$ . Here $a = -1$ , $b = 0$ , and $4a^3 + 27b^2 = -4 \ne 0$ , so the cubic $x^3 - x = x(x-1)(x+1)$ has three distinct roots and $E$ is smooth.

Non-example: $y^2 = x^3 + x^2$ has $\Delta = 0$ . Its projective closure has a node at $(0, 0)$ , where both partial derivatives of $y^2 - x^3 - x^2$ vanish, so the smoothness condition fails.

The condition $4a^3 + 27b^2 \ne 0$ says that $x^3 + ax + b$ has no repeated root in an algebraic closure of $K$ . For $y^2 = x^3 + ax + b$ one has $\Delta = -16(4a^3 + 27b^2)$ . The set $E(K)$ of points over $K$ , including $O$ , is an abelian group with identity $O$ under the chord-and-tangent group law.

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Field concept
Elliptic curve definition

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Tag	`SJ2BVG9F4W`
URI	`m2://SJ2BVG9F4W`
Kind	definition
Writing	W0 stub
MSC	14H52, 11G05, 11G07
Defines	Elliptic curve
Uses	Field
Changed	2 November 2026

Elliptic curve

Definition

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