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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 smallperiod matrix $\Omega\in \mathbb{C}^{3\times 3}$ corresponding to a principallypolarized abelian threefold equipped with an automorphism of order $3$, returnsa Legendre-Rosenhain equation for a Picard curve with Jacobian isomorphic tothe 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, weapply the algorithm to obtain (numerically) all the isomorphism classes ofPicard curves with maximal complex multiplication attached to the sexticCM-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'sformula for the inverse Jacobian problem for hyperelliptic curves given in[Balakrishnan-Ionica-Lauter-Vincent, LMS J. Comput. Math., 19(suppl.A):283-300, 2016].

An inverse Jacobian algorithm for Picard curves