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j-Invariant

An invariant of an elliptic curve given in the form

y^2=x^3+ax+b

which is closely related to the elliptic discriminant and defined by

j(E)=(2^83^3a^3)/(4a^3+27b^2).

The determination of j as an algebraic integer in the quadratic field 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 Q(j) is the cube of an integer in Z.

See also

Elliptic Curve, Elliptic Discriminant, Frey Curve, j-Function

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 j-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.

Referenced on Wolfram|Alpha

j-Invariant

Cite this as:

Weisstein, Eric W. "j-Invariant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/j-Invariant.html