j-Invariant

An invariant of an elliptic curve given in the form

which is closely related to the elliptic discriminant and defined by

The determination of  as an algebraic integer in the quadratic field  is discussed by Greenhill (1891), Weber (1902), Berwick (1928), Watson (1938), Gross and Zaiger (1985), and Dorman (1988). The norm of  in  is the cube of an integer in .

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.

Stepanov, S. A. "The -Invariant." §7.2 in Codes on Algebraic Curves. New York: Kluwer, pp. 178-180, 1999.

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.