# Binary quartic forms with bounded invariants and small Galois groups

## TL;DR

This paper classifies integral irreducible binary quartic forms with Galois groups as subgroups of the dihedral group of order eight, organizing them into families linked to quadratic forms and counting their equivalence classes.

## Contribution

It introduces a classification of such quartic forms based on quadratic form families and provides enumeration of their equivalence classes.

## Key findings

- Forms are organized into families indexed by quadratic forms.
- Enumeration of GL_2(Z)-equivalence classes for fixed quadratic forms.
- Connection between quartic forms and quadratic form discriminants.

## Abstract

In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.