On parameters of orders of quartic fields and essential pairs -- extended version

TL;DR

This paper introduces an algorithm to find ternary quadratic form pairs that parametrize quartic field orders, and explores essential pairs as a new technique for deriving parameters and integral bases from number field data.

Contribution

It presents a novel algorithm for identifying parametrizations of quartic orders and investigates essential pairs as a practical tool for computing integral bases.

Findings

Essential pairs are common for maximal quartic orders.

Essential pairs provide a simple way to obtain integral bases.

The algorithm effectively finds parametrizations for given quartic orders.

Abstract

Classes of pairs of ternary quadratic forms parametrize quartic rings by a result of Bhargava. We give an algorithm for finding a pair of ternary quadratic forms that parametrize a given order of a quartic field. We examine a new technique, essential pairs, for obtaining parameters of orders of quartic fields from a number field database. Essential pairs for maximal orders of quartic fields are very common and provide a simple means of obtaining an integral basis for the ring of integers.