the j-invariant classifies elliptic curves up to isomorphism

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

Theorem 1.

Let K be a field, and let K¯ be a fixed algebraic closureMathworldPlanetmath of K.

  1. 1.

    Two elliptic curvesMathworldPlanetmath E1 and E2 are isomorphic ( (over K¯) if and only if they have the same j-invariant, i.e. j(E1)=j(E2).

  2. 2.

    Let j0K¯ be fixed. There exists an elliptic curve E defined over the field K(j0) such that j(E)=j0.


For part 2:

  • For j0=0, the curve E0:y2+y=x3 satisfies j(E)=0;

  • For j0=1728, the curve E1728:y2=x3+x satisfies j(E1728)=1728;

  • If j00,1728 consider the elliptic curve:


    It satisfies j(E)=j0 and it is defined over K(j0).

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