  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 $\mathbb P^2_{\mathbb C}.$
