The covering problem for the fields Q(−1)12 and Q(−3)12

AbstractLet k be a positive square free integer, N(−k)12 the ring of algebraic integers in Q(−k)12 and S the unit sphere in Cn, complex n-space. If A1,…, An are n linearly independent points of Cn then L = {u1Au + … + unAn} with ur ∈ N(−k)12 is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then θ(S, L) = V(S)d(L) is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3