# 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. 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. 2.

The j-invariant of $E$ is

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