Trace form via cyclotomic fields

Let n ≥ 3 be an integer, ζn a primitive nth root of unity, and K the cyclotomic field Q(ζn). In this paper, we present an explicit description of the integral trace form TrK(αxx), where α, x ∈ K and x is the complex conjugate of x. Furthermore, we present constructions of algebraic lattices via the twisted canonical homomorphism with optimal center density in dimensions 2, 4, 6, 8 and 12, which are rotated versions of the lattices Λn, for n = 2, 4, 6, 8 and K12.