Digital Library of Mathematical Functions
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23 Weierstrass Elliptic and Modular FunctionsModular Functions

§23.19 Interrelations

23.19.1 λ(τ)=16(η2(2τ)η(12τ)η3(τ))8,
23.19.2 J(τ)=427(1-λ(τ)+λ2(τ))3(λ(τ)(1-λ(τ)))2,
23.19.3 J(τ)=g23g23-27g32,

where g2,g3 are the invariants of the lattice 𝕃 with generators 1 and τ; see §23.3(i).

Also, with Δ defined as in (23.3.4),

23.19.4 Δ=(2π)12η24(τ).