Binary quartic forms with bounded invariants and small Galois groups

Cindy, Tsang, Stanley Yao Xiao

Published 2017-02-23Version 1

In this paper, we enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of integral binary quartic forms which are fixed under substitution by a particular matrix in $\operatorname{GL}_2(\mathbb{R})$ which is proportional over $\mathbb{R}$ to an integer matrix. In particular, whenever such a form $F$ is irreducible, the Galois group of the splitting field of $F$ is isomorphic to a subgroup of the dihedral group $\mathcal{D}_4$ of order eight. We also give a new criterion for when the negative Pell's equation $x^2 - Dy^2 = -1$ is soluble in integers $x$ and $y$.