j-Invariant

An invariant of an Elliptic Curve given in the form $y^2=x^3+ax+b$ which is closely related to the Discriminant and defined by

$$j(E)\equiv{2^8 3^3 a^3\over 4a^3+27b^2}.$$

The determination of $j$ as an Algebraic Integer in the Quadratic Field $\Bbb{Q}(j)$ is discussed by Greenhill (1891), Weber (1902), Berwick (1928), Watson (1938), Gross and Zaiger (1985), and Dorman (1988). The norm of $j$ in $\Bbb{Q}(j)$ is the Cube of an Integer in $\Bbb{Z}$.

See also Discriminant (Elliptic Curve), Elliptic Curve, Frey Curve

References

Berwick, W. E. H. ``Modular Invariants Expressible in Terms of Quadratic and Cubic Irrationalities.'' Proc. London Math. Soc. 28, 53-69, 1928.

Dorman, D. R. ``Special Values of the Elliptic Modular Function and Factorization Formulae.'' J. reine angew. Math. 383, 207-220, 1988.

Greenhill, A. G. ``Table of Complex Multiplication Moduli.'' Proc. London Math. Soc. 21, 403-422, 1891.

Gross, B. H. and Zaiger, D. B. ``On Singular Moduli.'' J. reine angew. Math. 355, 191-220, 1985.

Watson, G. N. ``Ramanujans Vermutung über Zerfällungsanzahlen.'' J. reine angew. Math. 179, 97-128, 1938.

Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1979.