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 ${E}_{0}:{y}^{2}+y={x}^{3}$ satisfies $j(E)=0$;

•
For ${j}_{0}=1728$, the curve ${E}_{1728}:{y}^{2}={x}^{3}+x$ satisfies $j({E}_{1728})=1728$;

•
If ${j}_{0}\ne 0,1728$ consider the elliptic curve:
$$E={E}_{{j}_{0}}:{y}^{2}+xy={x}^{3}\frac{36}{{j}_{0}1728}x\frac{1}{{j}_{0}1728}.$$ It satisfies $j(E)={j}_{0}$ and it is defined over $K({j}_{0})$.
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Title  the $j$invariant classifies elliptic curves up to isomorphism 

Canonical name  TheJinvariantClassifiesEllipticCurvesUpToIsomorphism 
Date of creation  20130322 15:06:25 
Last modified on  20130322 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 