Klein's Absolute Invariant

$$J(q)\equiv {4\over 27} {[1-\lambda(q)+\lambda^2(q)]^3\over \... ...2(q)[1-\lambda(q)]^2} = {[E_4(q)]^3\over[E_4(q)]^3-[E_6(q)^2]}$$

(Cohn 1994), where $q\equiv e^{i\pi t}$ is the Nome, $\lambda(q)$ is the Elliptic Lambda Function

$$\lambda(q)\equiv k^2(q) = \left[{\vartheta _2(q)\over \vartheta _3(q)}\right]^4,$$

$\vartheta _i(q)$ is a Theta Function, and the $E_i(q)$ are Ramanujan-Eisenstein Series. $J(t)$ is Gamma-Modular.

See also Elliptic Lambda Function, j-Function, Pi, Ramanujan-Eisenstein Series, Theta Function

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 115 and 179, 1987.

Cohn, H. Introduction to the Construction of Class Fields. New York: Dover, p. 73, 1994.

Weisstein, E. W. ``j-Function.'' Mathematica notebook jFunction.m.