Equianharmonic

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title: "Equianharmonic" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["modular-forms", "elliptic-curves", "elliptic-functions"] topic_path: "general/modular-forms" source: "https://en.wikipedia.org/wiki/Equianharmonic" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0 and g3 = 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 ω2 is real and equal to

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

where \Gamma 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 e1, e2 and e3 are given by

: e_1=4^{-1/3}e^{(2/3)\pi i},\qquad e_2=4^{-1/3},\qquad e_3=4^{-1/3}e^{-(2/3)\pi i}.

The case g2 = 0, g3 = a may be handled by a scaling transformation.

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