Identifying the conformal equivalence class of a 2-torus subgroup of the cubical 3-torus 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)3 via
G  =  {(x,y,z) ∊ (ℝ/ℤ)3  |  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)3, 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, ⟩ for some  in the set
X  =  {z ∊ ℂ  |  Im(z) > 0  and  |z| ≥ 1  and  |Re()| ≤ 1/2}.
Such a  is determined uniquely unless i ≠  ∊ ∂X, in which case the boundary points  and conj(-) correspond to the same class. Now suppose we have chosen an orientation for G.
In terms of (K,L,M), what is the parameter  corresponding to the Riemannian flat torus G ?  Equivalently, what is the j-invariant of G ?
 A: This may be easier than I thought.  Assume WLOG that |K| ≥ |L| ≥ |M| with gcd(K,L,M) = 1 as above.
Then the subgroup H  ⊂  ℤ3 of integer translations of ℝ3 that preserves the 2-plane P, defined by
P  =  {(x,y,z) ∊ ℝ3  |  Kx + Ly + Mz  =  0}
(where 0 denotes 0) is generated by v = (-L,K,0) and w = (0,-M,L):
H  =  ⟨v, w⟩  ⊂  P
as a subgroup of P, which itself is a subgroup of ℝ3.  We can assume the counterclockwise angle in P from v to w is less then pi radians, and this condition will determine the orientation on P.
But the 2-torus in question is the quotient P / H.  Therefore the parallelogram generated by v = (-L,K,0) and w = (0,-M,L) is a fundamental domain in P for this group action.
Therefore the parameter  is determined a) by cos(angle(1,))  =  cos(angle(v,w)), where
cos(angle(v,w)  =  -KM / (√(K^2 + L^2) √(L^2 + M^2))
and b) by the condition that ||  =  ||v|| / ||w||, where
||v|| / ||w||  =  √(K^2+L^2) / √(L^2+M^2).

The above assumes that either division by zero is a valid operation, or else that the case (K,L,M) = (1,0,0) is excluded a priori as trivial.
