  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 $2^{80}$.
