On the Length of the 2-Class Field Tower of Imaginary Quadratic Number Fields k with Cl2(k) Isomorphic to (2, 2, 2): A Heuristic Follow-Up

In this article we continue to study the 2-class field tower length of imaginary quadratic number fields with its 2-class group isomorphic to (2, 2, 2). We demonstrate that for fields k such that exactly three positive prime discriminants divide the discriminant of k, there are examples for which k has 2-class field tower length ≥ 3, as well as for which k has 2-class field tower length = 2. To obtain our examples we utilized PARI/GP to obtain our class groups to show length ≥ 3, and made use of an application of MAGMA to apply the p-group generation algorithm to show length = 2. Our results are significant as they demonstrate that for fields k of this type, when exactly three positive prime discriminants divide the discriminant of k, then k could have 2-class field tower length 2 or ≥ 3, inclusive of the possibility of having infinite 2-class field tower.