Convex foliations of degree 5 on the complex projective plane

We show that, up to automorphisms of P²_C, there are fourteen homogeneous convex foliations of degree 55 on P²_C. We establish some properties of the Fermat foliation F^d₀ of degree d≥2 and of the Hilbert modular foliation F⁵_H of degree 55. As a consequence, we obtain that every reduced convex foliation of degree 5 on P²_C is linearly conjugated to one of the two foliations F⁵₀ or F⁵_H, which is a partial answer to a question posed in 2013 by D. Marín 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≥2 on P²_C.