j-invariant (original) (raw)

Let E be an elliptic curve over ℚ with Weierstrass equation:

with coefficients ai∈ℚ. Let:

Definition 1.

3. 1. The invariant differential is

      ω=d⁢x2⁢y+a1⁢x+a3=d⁢y3⁢x2+2⁢a2⁢x+a4-a1⁢y

Example:

If E has a Weierstrass equation in the simplified formy2=x3+A⁢x+B then

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