j-invariant

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass equation:

y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}

with coefficients $a_{i}\in\mathbb{Q}$ . Let:

$\displaystyle b_{2}$	$\displaystyle=$	$\displaystyle a_{1}^{2}+4a_{2},$
$\displaystyle b_{4}$	$\displaystyle=$	$\displaystyle 2a_{4}+a_{1}a_{3},$
$\displaystyle b_{6}$	$\displaystyle=$	$\displaystyle a_{3}^{2}+4a_{6},$
$\displaystyle b_{8}$	$\displaystyle=$	$\displaystyle a_{1}^{2}a_{6}+4a_{2}a_{6}-a_{1}a_{3}a_{4}+a_{3}^{2}a_{2}-a_{4}^% {2},$
$\displaystyle c_{4}$	$\displaystyle=$	$\displaystyle b_{2}^{2}-24b_{4},$
$\displaystyle c_{6}$	$\displaystyle=$	$\displaystyle-b_{2}^{3}+36b_{2}b_{4}-216b_{6}$

Definition 1.

1.

The discriminant of $E$ is defined to be

$\Delta=-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}$
2.

The j-invariant of $E$ is

$j=\frac{c_{4}^{3}}{\Delta}$
3.

The invariant differential is

$\omega=\frac{dx}{2y+a_{1}x+a_{3}}=\frac{dy}{3x^{2}+2a_{2}x+a_{4}-a_{1}y}$

Example:

If $E$ has a Weierstrass equation in the simplified form $y^{2}=x^{3}+Ax+B$ then

\Delta=-16(4A^{3}+27B^{2}),\quad j=-\frac{1728(4A)^{3}}{\Delta}

Note: The discriminant $\Delta$ coincides in this case with the usual notion of discriminant of the polynomial (http://planetmath.org/Discriminant) $x^{3}+Ax+B$ .

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