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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:
$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$
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
$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$
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:
$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$
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:
    $\displaystyle x=u^2x'+r,\quad y=u^3y'+su^2x'+t$
    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:
$\displaystyle y^2=x^3+Ax+B$
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
$\displaystyle E_1: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, \quad E_2: y'^2+a_1x'y'+a_3y'=x'^3+a_2x'^2+a_4x'+a_6$
be two distinct Weierstrass models for $ E/K$. Then (by Prop. 1) there exists a change of variables $ (x,y)\mapsto (x',y')$ of the form:
$\displaystyle x=u^2x'+r,\quad y=u^3y'+su^2x'+t$
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, simple, projective plane, equation, point, genus, curve, nonsingular, field, elliptic curve
There are 10 references to this entry.

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 2881 times total.

Classification:
AMS MSC: 14H52 (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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