Note on a polynomial of Emma Lehmer

Recently, Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor $p = {t^4} + 5{t^3} + 15{t^2} + 25t + 25$ as translates of Gaussian periods. Later, Schoof and Washington showed that these units were fundamental units. In this note, we observe that Lehmer’s family comes from the covering of modular curves ${X_1}(25) \to {X_0}(25)$. This gives a conceptual explanation for the existence of Lehmer’s units: they are modular units (which have been studied extensively). By relating Lehmer’s construction with ours, one finds expressions for certain Gauss sums as values of modular units on ${X_1}(25)$.