The number of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor

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 $D_4$-field. By a theorem of M. Wood, such quartic orders may be regarded as quartic $D_4$-fields whose ring of integers has a monogenic cubic resolvent. We shall give the asymptotic number of such objects when ordered by conductor, as well as estimate the asymptotic number of such objects when ordered by discriminant.