Weierstrass equation of an elliptic curve

Recall that an elliptic curve over a field $K$ is a projective nonsingular curve $E$ defined over $K$ of genus $1$ together with a point $O\in E$ defined over $K$.

Definition.

Let $K$ be an arbitrary field. A Weierstrass equation for an elliptic curve $E\mathrm{/}K$ is an equation of the form:

$$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$$

where $a_{\mathrm{1}}\mathrm{,}{a}_{\mathrm{2}}\mathrm{,}{a}_{\mathrm{3}}\mathrm{,}{a}_{\mathrm{4}}\mathrm{,}{a}_{\mathrm{6}}$ are constants in $K$.

All elliptic curves have a Weierstrass model in ${\mathbb{P}}^2(K)$, the projective plane over $K$. This is a simple application of the http://planetmath.org/node/RiemannRochTheoremRiemann Roch theorem for curves:

Theorem.

Let $E$ be an elliptic curve defined over a field $K$. Then there exists rational functions $x\mathrm{,}y\mathrm{\in }K\mathit{}\mathrm{(}E\mathrm{)}$ such that the map $\psi \mathrm{:}E\mathrm{\to }{\mathrm{P}}^{\mathrm{2}}\mathit{}\mathrm{(}K\mathrm{)}$ sending $P$ to $\mathrm{[}x\mathit{}\mathrm{(}P\mathrm{)}\mathrm{,}y\mathit{}\mathrm{(}P\mathrm{)}\mathrm{,}\mathrm{1}\mathrm{]}$ is an isomorphism of $E\mathrm{/}K$ to the projective curve given by

$$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$$

where $a_{\mathrm{1}}\mathrm{,}{a}_{\mathrm{2}}\mathrm{,}{a}_{\mathrm{3}}\mathrm{,}{a}_{\mathrm{4}}\mathrm{,}{a}_{\mathrm{6}}$ are constants in $K$.

Moreover, the following proposition specifies any possible change of variables.

Proposition 1.

Let $E\mathrm{/}K$ be an elliptic curve given by a Weierstrass model of the form:

$$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$$

with $a_i\mathrm{\in }K$. Then:

1. The only change of variables $(x,y)\mapsto (x^{\prime },y^{\prime })$ preserving the projective point $[0,1,0]$ and which also result in a Weierstrass equation, are of the form:

   $$x=u^2{x}^{\prime }+r,y=u^3{y}^{\prime }+s{u}^2{x}^{\prime }+t$$

   with $u,r,s,t\in K$ and $u\ne 0$.

2. Any two Weierstrass equations for $E/K$ differ by a change of variables of the form given in $(1)$.

Once we have one Weierstrass model for a given elliptic curve $E/K$, and as long as the characteristic of $K$ is not $2$ or $3$, there exists a change of variables (of the form given in the previous proposition) which simplifies the model considerably.

Corollary.

Let $K$ be a field of characteristic different from $\mathrm{2}$ or $\mathrm{3}$. Let $E$ be an elliptic curve defined over $K$. Then there exists a Weierstrass model for $E$ of the form:

$$y^2=x^3+Ax+B$$

where $A\mathrm{,}B$ are elements of $K$.

Finally, remember that the $j$-invariant of an elliptic curve is invariant under isomorphism, but the discriminant depends on the model chosen.

Proposition 2.

Let $E\mathrm{/}K$ be an elliptic curve and let

$$E_1:y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6,E_2:y^{\prime 2}+a_1{x}^{\prime }{y}^{\prime }+a_3{y}^{\prime }=x^{\prime 3}+a_2{x}^{\prime 2}+a_4{x}^{\prime }+a_6$$

be two distinct Weierstrass models for $E\mathrm{/}K$. Then (by Prop. 1) there exists a change of variables $\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{\mapsto }\mathrm{(}{x}^{\mathrm{\prime }}\mathrm{,}{y}^{\mathrm{\prime }}\mathrm{)}$ of the form:

$$x=u^2{x}^{\prime }+r,y=u^3{y}^{\prime }+s{u}^2{x}^{\prime }+t$$

with $u\mathrm{,}r\mathrm{,}s\mathrm{,}t\mathrm{\in }K$ and $u\mathrm{\ne }\mathrm{0}$. Moreover, $j\mathit{}\mathrm{(}{E}_{\mathrm{1}}\mathrm{)}\mathrm{=}j\mathit{}\mathrm{(}{E}_{\mathrm{2}}\mathrm{)}$, i.e. the $j$ invariants are equal ($j\mathit{}\mathrm{(}E\mathrm{)}$ is defined in http://planetmath.org/node/JInvariantthis entry) and $\mathrm{\Delta }\mathit{}\mathrm{(}{E}_{\mathrm{1}}\mathrm{)}\mathrm{=}{u}^{\mathrm{12}}\mathit{}\mathrm{\Delta }\mathit{}\mathrm{(}{E}_{\mathrm{2}}\mathrm{)}$, where $\mathrm{\Delta }\mathit{}\mathrm{(}{E}_i\mathrm{)}$ is the discriminant (as defined in http://planetmath.org/node/JInvarianthere).

Title	Weierstrass equation of an elliptic curve
Canonical name	WeierstrassEquationOfAnEllipticCurve
Date of creation	2013-03-22 15:48:00
Last modified on	2013-03-22 15:48:00
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Definition
Classification	msc 11G05
Classification	msc 14H52
Classification	msc 11G07
Synonym	Weierstrass model
Defines	Weierstrass equation