Let E be an elliptic curveMathworldPlanetmath over with Weierstrass equation:


with coefficients ai. Let:

b2 = a12+4a2,
b4 = 2a4+a1a3,
b6 = a32+4a6,
b8 = a12a6+4a2a6-a1a3a4+a32a2-a42,
c4 = b22-24b4,
c6 = -b23+36b2b4-216b6
Definition 1.
  1. 1.

    The discriminantPlanetmathPlanetmathPlanetmathPlanetmath of E is defined to be

  2. 2.

    The j-invariant of E is

  3. 3.

    The invariant differential is



If E has a Weierstrass equation in the simplified form y2=x3+Ax+B then


Note: The discriminant Δ coincides in this case with the usual notion of discriminant of the polynomial (http://planetmath.org/Discriminant) x3+Ax+B.

Title j-invariant
Canonical name Jinvariant
Date of creation 2013-03-22 13:49:54
Last modified on 2013-03-22 13:49:54
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 9
Author alozano (2414)
Entry type Definition
Classification msc 14H52
Synonym discriminant
Synonym j-invariant
Synonym j invariant
Related topic EllipticCurve
Related topic BadReduction
Related topic ModularDiscriminant
Related topic Discriminant
Related topic ArithmeticOfEllipticCurves
Defines j-invariant
Defines discriminant of an elliptic curve
Defines invariant differential