Weierstrass equation

Builds on Field Projective space Elliptic curve

Definition

Let $K$ be a field. 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 adds one point $O = [0 : 1 : 0]$ . This closure is smooth exactly when $\Delta \ne 0$ , and then it is an elliptic curve 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.

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Elliptic curve concept
Field concept
Projective space concept
Weierstrass equation definition

Tag	`5WGZPE6249`
URI	`m2://5WGZPE6249`
Kind	definition
Writing	W0 stub
MSC	14H52, 11G05
Defines	Weierstrass equation
Uses	Field, Projective space, Elliptic curve

Weierstrass equation

Definition

Remarks

Reading path

Further reading

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