# 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. 1.

$E_{1}$ 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. 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 TheJinvariantClassifiesEllipticCurvesUpToIsomorphism 2013-03-22 15:06:25 2013-03-22 15:06:25 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 11G05 msc 14H52 IsomorphismOfVarieties ArithmeticOfEllipticCurves