In her paper "Quartic rings associated to binary quartic forms" (https://doi.org/10.1093/imrn/rnr070), Wood showed that $\operatorname{GL}_2(\mathbb{Z})$-classes of integral binary quartic forms are in discriminant-preserving bijection to pairs $(Q,C)$ where $Q$ is a quartic ring and $C$ is a monogenic cubic resolvent ring of $Q$. This allows the cubic resolvent $C$ of a quartic ring $Q$ to be interpreted in terms of the cubic resolvent polynomial of the corresponding to the binary form $F$, given by

$$ (1) \text{ } \displaystyle \mathcal{C}_F(x) = x^3 - \frac{I(F)}{x}x - \frac{J(F)}{27},$$


$$\displaystyle I(F) = 12 a_4 a_0 - 3 a_3 a_1 + a_2^2,$$ $$\displaystyle J(F) = 72 a_4 a_2 a_0 + 9 a_3 a_2 a_1 - 27 a_4 a_1^2 - 27 a_0 a_3^2 - 2 a_2^3$$

are the $I$ and $J$-invariants of the binary quartic form $F$.

Now suppose $Q$ is a $D_4$-quartic ring, so that the Galois closure of the fraction field $K_Q$ of $Q$ has Galois group isomorphic to the dihedral group $D_4$. Denote by $K_Q^\dagger$ for the Galois closure of $K_Q$. Then $K_Q^\dagger$ contains exactly three pairwise non-isomorphic quadratic fields $k_1, k_2, k_3$.

Put $N(k_1, k_2, k_3; \mathcal{C}, D)$ for the number of (isomorphism classes of) $D_4$-quartic rings $Q$ of discriminant $D$ with monogenic cubic resolvent $\mathcal{C}$ (so that it corresponds to a polynomial as in (1)) and such that $K_Q^\dagger$ contains the pairwise distinct quadratic fields $k_1, k_2, k_3$. Can one give an estimate for $N(k_1, k_2, k_3; \mathcal{C}, D)$, in particular an upper bound that depends only on $D$?


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