Special units in real cyclic sextic fields

We study the real cyclic sextic fields generated by a root w of ${(X - 1)^6} - ({t^2} + 108){({X^2} + X)^2}$, $t \in {\mathbf {Z}} - \{ 0, \pm 6, \pm 26\}$ . We show that, when ${t^2} + 108$ is square-free (except for powers of 2 and 3), and $t \ne 0$, $\pm 10$, $\pm 54$, then w is a generator of the module of relative units. The details of the proofs are given in [3].