#### Vol. 302, No. 1, 2019

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Binary quartic forms with bounded invariants and small Galois groups

### Cindy (Sin Yi) Tsang and Stanley Yao Xiao

Vol. 302 (2019), No. 1, 249–291
##### Abstract

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 $f\left(x,y\right)$ of nonzero discriminant. Then, we shall enumerate the ${GL}_{2}\left(ℤ\right)$-equivalence classes of all such forms associated to a fixed $f\left(x,y\right)$.

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##### Keywords
binary quartic forms, coregular spaces, arithmetic statistics
##### Mathematical Subject Classification 2010
Primary: 11E76, 11R45
##### Milestones
Received: 27 February 2018
Revised: 21 October 2018
Accepted: 26 January 2019
Published: 5 November 2019
##### Authors
 Cindy (Sin Yi) Tsang Yau Mathematical Sciences Center Tsinghua University Beijing China School of Mathematics (Zhuhai) Sun Yat-Sen University Tangjiawan, Zhuhai Guangdong China Stanley Yao Xiao Mathematical Institute University of Oxford United Kingdom Department of Mathematics University of Toronto Canada