Let K, L, M be integers with gcd(K,L,M) = 1. They determine a connected Lie subgroup G = G(K,L,M) of the cubical 3-torus (ℝ/ℤ)<sup>3</sup> via

   G  =  {(x,y,z) ∊ (ℝ/ℤ)<sup>3</sup>  |  Kx + Ly + Mz = 0}

(where 0 denotes the identity element of ℝ/ℤ). 

G is a 2-torus and inherits a Riemannian metric of everywhere zero Gaussian curvature from (ℝ/ℤ)<sup>3</sup>, which belongs to a unique conformal equivalence class of Riemann surfaces of genus 1 (or at most two such classes, depending on the choice of orientation of G).

As is well-known, every Riemannian flat 2-torus is conformally equivalent to the quotient  ℂ/L for some lattice L  =  ⟨1, 𝛕⟩ for some 𝛕 in the set 

   X  =  {z ∊ ℂ  |  |z| ≥ 1 and |Re(𝛕)| ≤ 1/2}. 

Such a 𝛕 is determined uniquely unless i ≠ 𝛕 ∊ ∂X, in which case the boundary points 𝛕 and conj(-𝛕) correspond to the same class. Now suppose we have chosen an orientation for G.

In terms of (K,L,M), what is the parameter 𝛕 corresponding to the Riemannian flat torus G ?  Equivalently, what is the j-invariant of G ?