Weierstrass equation of an elliptic curve


Recall that an elliptic curveMathworldPlanetmath over a field K is a projective nonsingularPlanetmathPlanetmath curve E defined over K of genus 1 together with a point OE 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:

y2+a1xy+a3y=x3+a2x2+a4x+a6

where a1,a2,a3,a4,a6 are constants in K.

All elliptic curves have a Weierstrass model in 2(K), the projective planeMathworldPlanetmath 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,yK(E) such that the map ψ:EP2(K) sending P to [x(P),y(P),1] is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of E/K to the projective curve given by

y2+a1xy+a3y=x3+a2x2+a4x+a6

where a1,a2,a3,a4,a6 are constants in K.

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

Proposition 1.

Let E/K be an elliptic curve given by a Weierstrass model of the form:

y2+a1xy+a3y=x3+a2x2+a4x+a6

with aiK. Then:

  1. 1.

    The only change of variables (x,y)(x,y) preserving the projective point [0,1,0] and which also result in a Weierstrass equation, are of the form:

    x=u2x+r,y=u3y+su2x+t

    with u,r,s,tK and u0.

  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:

y2=x3+Ax+B

where A,B are elements of K.

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

Proposition 2.

Let E/K be an elliptic curve and let

E1:y2+a1xy+a3y=x3+a2x2+a4x+a6,E2:y2+a1xy+a3y=x3+a2x2+a4x+a6

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

x=u2x+r,y=u3y+su2x+t

with u,r,s,tK and u0. Moreover, j(E1)=j(E2), i.e. the j invariants are equal (j(E) is defined in http://planetmath.org/node/JInvariantthis entry) and Δ(E1)=u12Δ(E2), where Δ(Ei) 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