On the Narrow 2-Class Field Tower of Some Real Quadratic Number Fields: Lengths Heuristics Follow-Up

Abstract

In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow $2$-class field over $\mathrm{k}$, and $[\mathrm{G}_i]$ denote the lower central series of $\mathrm{G}$, we give heuristic evidence that the length of the narrow $2$-class field tower of $\mathrm{k}$ is equal to $2$ when $\mathrm{G}/\mathrm{G}_3$ is of type $64.150$ (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert $2$-class field for these fields.