The ring of Eisenstein integers E = Z[w], w = exp(2πi/3) form a nice commutative ring. It's "nice" because it's a discrete subring of the complex numbers C with additive group a rank-2 lattice in C. (When you identify any two points z, w of C whenever z - w in E, the quotient of C becomes a torus T with a very specific geometry.) Namely, T = the result of starting from a regular hexagon and then sewing together corresponding points on the three pairs of opposite edges. Anyhow, the ideals of the ring E are all principal, so of the extremely simple form J = a E, for some a in E. (The condition a^2 ≠ a ensures that the ideal is non-trivial.) Now consider the *quotient ring* S = E / a E. If a = K + L w, then the number of points in S is |S| = |a|^2 = K*K - K*L + L*L. These {K*K - K*L + L*L} are the same numbers represented by K*K + K*L + L*L when K, L range over all integers. Assume gcd(K,L) = 1. Then: What is the quotient ring E / a E ??? The additive portion is just Z / (K*K + K*L + L*L) Z, nothing surprising. Question: ----- What is the multiplicative structure? ----- E.g., for K + L w = 3 + 4w, |S| = 37. What is the best way to describe the multiplication table? I know how to figure it out, but am looking for a general formula for the multiplication table of E / (K + Lw) E. NOTE: The elements of the quotient ring S = E / a E are naturally identified with the hexagonal tiling of the torus T defined as T = C / a E by the Voronoi regions of the points of S in T, which is kind of cool. —Dan