j-invariant
Let E be an elliptic curve over ℚ with Weierstrass equation:
y2+a1xy+a3y=x3+a2x2+a4x+a6 |
with coefficients ai∈ℚ. Let:
b2 | = | a21+4a2, | ||
b4 | = | 2a4+a1a3, | ||
b6 | = | a23+4a6, | ||
b8 | = | a21a6+4a2a6-a1a3a4+a23a2-a24, | ||
c4 | = | b22-24b4, | ||
c6 | = | -b32+36b2b4-216b6 |
Definition 1.
-
1.
The discriminant of E is defined to be
Δ=-b22b8-8b34-27b26+9b2b4b6 -
2.
The j-invariant of E is
j=c34Δ -
3.
The invariant differential is
ω=dx2y+a1x+a3=dy3x2+2a2x+a4-a1y
Example:
If E has a Weierstrass equation in the simplified form y2=x3+Ax+B then
Δ=-16(4A3+27B2),j=-1728(4A)3Δ |
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 |