The number of quartic 𝐷₄-fields with monogenic cubic resolvent ordered by conductor

Abstract

In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic D 4 D_4 -field. By a theorem of Wood, such quartic orders may be regarded as quartic D 4 D_4 -fields whose ring of integers has a monogenic cubic resolvent. We shall determine the asymptotic number of such objects when ordered by conductor. We shall also give a lower bound, which we suspect has the correct order of magnitude, and a slightly larger upper bound for the number of such objects when ordered by discriminant. A simplified version of the techniques used allows us to give a count for those elliptic curves with a marked rational 2-torsion point when ordered by discriminant.