Computing Elements of given Index in Totally Complex Cyclic Sextic Fields

In the present paper we show, that the problem of determining elements of given index in totally complex cyclic sextic fields (or in general, in sextic fields containing an imaginary quadratic and a real cubic subfield) can be reduced to the resolution of a cubic Thue inequality over . As an application of our results we consider the question of monogenity in an infinite two parametric family of totally complex cyclic sextic fields, composed of Shanksʹ simplist cubics with imaginary quadratic fields.