The $2$-Class Tower of $\mathbb{Q}(\sqrt{-5460})$

TL;DR

This paper investigates whether the quadratic field ng(-5460) has a finite or infinite 2-class tower, providing computational evidence that it is finite, which impacts bounds on root-discriminants and related conjectures.

Contribution

The paper introduces new computational techniques and provides strong evidence that the 2-class tower of ng(-5460) is finite, addressing a critical case in the field.

Findings

Evidence suggests the 2-class tower is finite.

Properties of the Galois group are established.

Implications for root-discriminant bounds and Martinet's conjecture.

Abstract

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root-discriminants (if infinite) or else give a counter-example to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route.