  Itinerant ferromagnetism is one of the most studied quantum phase
transitions, the transition point and the nature of this phase transition being
widely discussed. In dilute Fermi liquids, this analysis has been carried out
up to second-order in the gas parameter, where the results for any spin
degeneracy are universal in terms of only the s-wave scattering length $a_0$.
We extend this analysis to third-order where energies depend, not only on
$a_0$, but also on the s-wave effective range $r_0$ and the p-wave scattering
length $a_1$. The introduction in the theory of these new parameters changes
the transition point, with respect to the second-order estimation, and also can
modify the nature of the phase transition itself. We analyze these interaction
effects on the phase transition for different spin values. The emerging phase
diagram shows that the type of ferromagnetic transition changes dramatically as
a function of $r_0$ and $a_1$ and, importantly, that this classification is not
solely determined by the spin value, as happens at second order.
