What is "equianharmonic"

a. (context maths English) equally anharmonic; specifically, whose anharmonic ratio is a cube root of unity n. (cx geometry English) A set of four points whose cross ratio (or anharmonic ratio) is a cube root of 1.

In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g = 0 and g = 1. This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.)

In the equianharmonic case, the minimal half period ω is real and equal to

$$\frac{\Gamma^3(1/3)}{4\pi}$$

where Γ is the Gamma function. The half period is

$$\omega_1=\tfrac{1}{2}(-1+\sqrt3i)\omega_2.$$

Here the period lattice is a real multiple of the Eisenstein integers.

The constants e, e and e are given by

e = 4e,   e = 4,   e = 4e.

The case g = 0, g = a may be handled by a scaling transformation.

Category:Modular forms Category:Elliptic curves