---
tag: 5WGZPE6249
kind: definition
name: Weierstrass equation
msc: [14H52, 11G05]
defines: Z3C8B6QTS6
uses: [04C5ZBG1WT, ZT9A6JK8QF, 427N6S6QQW]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_curve
---

## Definition

Let $K$ be a [[field|04C5ZBG1WT]].
A *Weierstrass equation* over $K$ is an equation
$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$$
with $a_1, a_2, a_3, a_4, a_6 \in K$.
Its associated quantities are
$$b_2 = a_1^2 + 4a_2, \quad b_4 = 2a_4 + a_1 a_3, \quad b_6 = a_3^2 + 4a_6,$$
$$b_8 = a_1^2 a_6 + 4a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2,$$
and its *discriminant* is
$$\Delta = -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6.$$

## Remarks

The projective closure of the curve in the [[projective plane|ZT9A6JK8QF]] adds one point $O = [0 : 1 : 0]$.
This closure is smooth exactly when $\Delta \ne 0$, and then it is an [[elliptic curve|427N6S6QQW]] with origin $O$.
If the characteristic of $K$ is not $2$ or $3$, a linear change of variables brings the equation to the short form $y^2 = x^3 + Ax + B$, for which $\Delta = -16(4A^3 + 27B^2)$.
Example: $y^2 = x^3 - x$ has $A = -1$ and $B = 0$, so $\Delta = -16 \cdot (-4) = 64 \ne 0$, and it defines an elliptic curve over $\Q$.
Non-example: $y^2 = x^3$ is a Weierstrass equation with $\Delta = 0$; the curve has a cusp at $(0, 0)$, where both partial derivatives of $y^2 - x^3$ vanish, so it is not an elliptic curve.