Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois closure of $f(x,1)$ and $H(f) = \max\{|I(f)|^3, J(f)^2/4\}$ to be the *naive height* of $f$, defined by Bhargava and Shankar (see references below). Put $\nu(f)$ for the $\text{GL}_2(\mathbb{Z})$-equivalence class of $f$, under the action of $\text{GL}_2(\mathbb{Z})$ on the space of binary quartic forms. Consider the counting function

$$\displaystyle N(f; D_4, Z) = \#\{\nu(f): f \in \mathbb{Z}[x], \deg f = 4, H(f) < Z, \text{Gal}(f) \subset D_4 \}.$$

Is there a known asymptotic formula for $N(f; D_4, Z)$, or at least an asymptotic lower bound expected to be of the correct order of magnitude?

There are several related results. For example, if one is to count the number of *quartic fields* with Galois group $D_4$ instead, then it was shown by Cohen, Diaz y Diaz, and Olivier that there are roughly $cZ$ many such fields with absolute discriminant up to $Z$, where $c \approx 0.0523\cdots$. One should note that the density of quartic fields with Galois group $D_4$ among all quartic fields is positive, as was shown by Bhargava. This is in contrast to the case with polynomials, where $100\%$ of all quartic polynomials have Galois group $S_4$. It was further shown by Bhargava and Shankar that there are at most $O_{\varepsilon}\left(Z^{\frac{3}{4} + \epsilon}\right)$ number of quartic polynomials of "naive height" up to $Z$, where as the number of quartic polynomials of naive height up to $Z$ is $Z^{\frac{5}{6}}$. The naive height is defined using the $I,J$ invariants of quartic polynomials and is designed to mimic the discriminant but is easier to deal with. The order of magnitude should be the same for both counting methods.

References:

H. Cohen, F. Diaz y Diaz, and M. Olivier, *Enumerating quartic dihedral extensions of $\mathbb{Q}$*, Compositio Math. **133** (2002), 65–93.

M. Bhargava, *The density of discriminants of quartic rings and fields*, Annals of Mathematics **162** (2005), 1031-1063.

M. Bhargava, A. Shankar, *Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves*, Annals of Mathematics **181** (2015), 191-242.