On the number of monogenizations of a quartic order

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 $2^{72}$. 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 $2^{80}$.