j-invariant

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

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