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

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the

-invariant classifies elliptic curves up to isomorphism

(Theorem)

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

Theorem 1 Let be a field, and let $\overline{K}$ be a fixed algebraic closure of .

Two elliptic curves and are isomorphic (over $\overline{K}$ ) if and only if they have the same -invariant, i.e. .
Let $j_0\in \overline{K}$ be fixed. There exists an elliptic curve defined over the field such that .

Proof. For part

For , the curve $E_0\colon y^2+y=x^3$ satisfies ;
For , 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:
$\displaystyle E=E_{j_0} \colon y^2+xy=x^3-\frac{36}{j_0-1728}x-\frac{1}{j_0-1728}.$
It satisfies and it is defined over .

$\qedsymbol$

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-invariant classifies elliptic curves up to isomorphism" is owned by alozano.

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-invariant classifies elliptic curves up to isomorphism, born on 2005-03-01, modified 2005-03-01.
Object id is 6839, canonical name is JInvariantClassifiesEllipticCurvesUpToIsomorphism.
Accessed 1458 times total.

Classification:

AMS MSC:	14H52 (Algebraic geometry :: Curves :: Elliptic curves)
	11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)

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