A family of cyclic quartic fields arising from modular curves

Abstract

We study a family of cyclic quartic fields arising from the covering of modular curves X 1 ( 16 ) → X 0 ( 16 ) {X_1}(16) \to {X_0}(16) . An integral basis and a fundamental system of units are found. It is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field.