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[parent] Weierstrass equation of an elliptic curve (Definition)

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 1   Let $K$ be an arbitrary field. A Weierstrass equation for an elliptic curve $E/K$ is an equation of the form: $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ 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 Riemann Roch theorem for curves:

Theorem 1   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 $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ where $a_1, a_2, a_3,a_4,a_6$ are constants in $K$ .

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

Proposition 1   Let $E/K$ be an elliptic curve given by a Weierstrass model of the form: $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ with $a_i\in K$ . Then:
  1. The only change of variables $(x,y)\mapsto (x',y')$ preserving the projective point $[0,1,0]$ and which also result in a Weierstrass equation, are of the form: $$x=u^2x'+r,\quad y=u^3y'+su^2x'+t$$ with $u,r,s,t\in K$ and $u\neq 0$ .
  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 1   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 $K$ .

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 $E/K$ be an elliptic curve and let $$ E_1: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, \quad E_2: y'^2+a_1x'y'+a_3y'=x'^3+a_2x'^2+a_4x'+a_6$$ be two distinct Weierstrass models for $E/K$ . Then (by Prop. [*]) there exists a change of variables $(x,y)\mapsto (x',y')$ of the form: $$x=u^2x'+r,\quad y=u^3y'+su^2x'+t$$ 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 this entry) and $\Delta(E_1)=u^{12}\Delta(E_2)$ , where $\Delta(E_i)$ is the discriminant (as defined in here).




"Weierstrass equation of an elliptic curve" is owned by alozano.
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Other names:  Weierstrass model
Also defines:  Weierstrass equation

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minimal model for an elliptic curve (Definition) by alozano
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Cross-references: discriminant, invariant, characteristic, variables, proposition, projective curve, isomorphism, map, rational functions, application, simple, projective plane, equation, point, genus, curve, nonsingular, field, elliptic curve
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This is version 3 of Weierstrass equation of an elliptic curve, born on 2006-03-23, modified 2006-03-23.
Object id is 7762, canonical name is WeierstrassEquationOfAnEllipticCurve.
Accessed 5001 times total.

Classification:
AMS MSC14H52 (Algebraic geometry :: Curves :: Elliptic curves)
 11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)
 11G07 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over local fields)

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