Let $m>1$ and $\mathfrak {d} \neq 0$ be integers such that $v_{p}(\mathfrak {d}) \neq m$ for any prime p. We construct a matrix $A(\mathfrak {d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak {d}$ with the following property: For any tame $ \mathbb {Z}/m \mathbb {Z}$ -number field K of discriminant $\mathfrak {d}$ , the matrix $A(\mathfrak {d})$ represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.

In the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.