On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)

We show that an order in a quartic field has fewer than 3000 essentially different generators as a $\mathbb Z$-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of $2^{72}$. <br> Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large). This significantly improves the previously best known bound of $2^{80}$.