Convex foliations of degree 5 on the complex projective plane

We show that up to automorphisms of $\mathbb P^2_{\mathbb C}$ there are $14$ homogeneous convex foliations of degree $5$ on $\mathbb P^2_{\mathbb C}.$ We establish some properties of the Fermat foliation $\mathcal F_{0}^{d}$ of degree $d\geq2$ and of the Hilbert modular foliation $\mathcal{F}_H^{5}$ of degree $5.$ As a consequence, we obtain that every reduced convex foliation of degree $5$ on $\mathbb P^2_{\mathbb C}$ is linearly conjugated to one of the two foliations $\mathcal F_{0}^{5}$ or $\mathcal{F}_H^{5},$ which is a partial answer to a question posed in $2013$ by D. Mar\'{\i}n and J.V. Pereira. We end with two conjectures about the Camacho-Sad indices along the line at infinity at the non radial singularities of the homogeneous convex foliations of degree $d\geq2$ on $\mathbb P^2_{\mathbb C}.$