Elements of small norm in Shanks’ cubic extensions of imaginary quadratic fields

Let be an imaginary quadratic number field with ring of integers Zk and let k(α) be the cubic extension of k generated by the polynomial ft(x)=x3−(t−1)x2−(t+2)x−1 with t Zk. In the present paper we characterize all elements γ Zk[α] with norms satisfying Nk(α)/k≤2t+1 for t≥14. This generalizes a corresponding result by Lemmermeyer and Pethő for Shanks’ cubic fields over the rationals.