the $j$ -invariant classifies elliptic curves up to isomorphism

In this entry, an isomorphism over $K$ should be understood in the sense of the entry isomorphism of varieties.

Theorem 1.

Let $K$ be a field, and let $\overline{K}$ be a fixed algebraic closure of $K$ .

1.

Two elliptic curves $E_{1}$ and $E_{2}$ are isomorphic (http://planetmath.org/IsomorphismOfVarieties) (over $\overline{K}$ ) if and only if they have the same $j$ -invariant, i.e. $j(E_{1})=j(E_{2})$ .
2.

Let $j_{0}\in\overline{K}$ be fixed. There exists an elliptic curve $E$ defined over the field $K(j_{0})$ such that $j(E)=j_{0}$ .

Proof.

For part $2$ :

•

For $j_{0}=0$ , the curve $E_{0}\colon y^{2}+y=x^{3}$ satisfies $j(E)=0$ ;
•

For $j_{0}=1728$ , the curve $E_{1728}\colon y^{2}=x^{3}+x$ satisfies $j(E_{1728})=1728$ ;
•

If $j_{0}\neq 0,1728$ consider the elliptic curve:

$E=E_{j_{0}}\colon y^{2}+xy=x^{3}-\frac{36}{j_{0}-1728}x-\frac{1}{j_{0}-1728}.$

It satisfies $j(E)=j_{0}$ and it is defined over $K(j_{0})$ .

∎

Title	the $j$ -invariant classifies elliptic curves up to isomorphism
Canonical name	TheJinvariantClassifiesEllipticCurvesUpToIsomorphism
Date of creation	2013-03-22 15:06:25
Last modified on	2013-03-22 15:06:25
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11G05
Classification	msc 14H52
Related topic	IsomorphismOfVarieties
Related topic	ArithmeticOfEllipticCurves