  We study the inverse Jacobian problem for the case of Picard curves over
$\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small
period matrix $\Omega\in \mathbb{C}^{3\times 3}$ corresponding to a principally
polarized abelian threefold equipped with an automorphism of order $3$, returns
a Legendre-Rosenhain equation for a Picard curve with Jacobian isomorphic to
the given abelian variety.
  Our method corrects a formula obtained by Koike-Weng in [Math. Comp.,
74(249):499-518, 2005] which is based on a theorem of Siegel. As a result, we
apply the algorithm to obtain (numerically) all the isomorphism classes of
Picard curves with maximal complex multiplication attached to the sextic
CM-fields with class number at most $4$. In particular, we obtain
(conjecturally) the complete list of CM Picard curves defined over
$\mathbb{Q}$.
  In the appendix, Vincent gives a correction to the generalization of Takase's
formula for the inverse Jacobian problem for hyperelliptic curves given in
[Balakrishnan-Ionica-Lauter-Vincent, LMS J. Comput. Math., 19(suppl.
A):283-300, 2016].
