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The main goal of this article is to compute the class of the divisor of$\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of$\Omega\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteousformula and the theory of test curves. For this purpose, we study the locus ofmeromorphic differentials of the second kind, computing the dimension of themap of these loci to $\mathcal{M}_g$ and solving some enumerative problemsinvolving such differentials in low genus. A key tool of the proof is thecompactification of strata recently introduced byBainbridge-Chen-Gendron-Grushevsky-M\"oller.

On the locus of genus $3$ curves that admit meromorphic differentials with a zero of order $6$ and a pole of order $2$