On the number of monogenizations of a quartic order

TL;DR

This paper establishes new upper bounds on the number of essentially different generators and isomorphism classes of orders in quartic fields, significantly improving previous exponential bounds.

Contribution

It provides the first polynomial bounds on the number of monogenizations and isomorphism classes for quartic orders, refining earlier exponential estimates.

Findings

Fewer than 3000 essentially different generators for quartic orders

Fewer than 200 generators when discriminant is large

At most 10 classes of integral binary quartic forms for quartic orders

Abstract

We show that an order in a quartic field has fewer than $3000$ essentially different generators as a $\mathbb Z$-algebra (and fewer than $200$ if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of $2^{72}$. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most $10$ classes of integral binary quartic forms (and at most $7$ if the discriminant is sufficiently large). This significantly improves the previously best known bound of $2^{80}$.