Weierstrass equation of an elliptic curve

Recall that an 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 $E/K$ 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 $a_{1},a_{2},a_{3},a_{4},a_{6}$ 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 $x,y\in K(E)$ such that the map $\psi:E\to\mathbb{P}^{2}(K)$ sending $P$ to $[x(P),y(P),1]$ is an isomorphism        of $E/K$ 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 $a_{1},a_{2},a_{3},a_{4},a_{6}$ are constants in $K$.

Proposition 1.

Let $E/K$ 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 $a_{i}\in K$. Then:

1. 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:

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

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

2. 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 $2$ or $3$. 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 $A,B$ are elements of $K$.

Proposition 2.

Let $E/K$ be an elliptic curve and let

 $E_{1}:y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6},\quad E_{2}:y^{\prime 2% }+a_{1}x^{\prime}y^{\prime}+a_{3}y^{\prime}=x^{\prime 3}+a_{2}x^{\prime 2}+a_{% 4}x^{\prime}+a_{6}$

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

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

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

Title Weierstrass equation of an elliptic curve WeierstrassEquationOfAnEllipticCurve 2013-03-22 15:48:00 2013-03-22 15:48:00 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 11G05 msc 14H52 msc 11G07 Weierstrass model Weierstrass equation