The study of trace forms of number fields is a fertile area of research in algebraic number theory. The integral trace form of a totally real or totally complex number field F represents the quadratic form of the lattice associated to OF, the ring of integers of F. In this work, we study the trace form TrQ(_p)/Q(x__) of the cyclotomic field Q(_p) and determine its minimum when x is a nonzero element of a certain Z-submodule Mt of OQ(_p). The main contribution is a new and concise formula for computing the above minimum. As an application, we show how one can easily determine lower bounds on the minimum distances of the associated lattices, and hence on their packing densities.