  From the study of long-range-interacting systems to the simulation of gauge
fields, open-shell Lanthanide atoms with their large magnetic moment and narrow
optical transitions open novel directions in the field of ultracold quantum
gases. As for other atomic species, the magneto-optical trap (MOT) is the
working horse of experiments but its operation is challenging, due to the large
electronic spin of the atoms. Here we present an experimental study of
narrow-line Dysprosium MOTs. We show that the combination of radiation pressure
and gravitational forces leads to a spontaneous polarization of the electronic
spin. The spin composition is measured using a Stern-Gerlach separation of spin
levels, revealing that the gas becomes almost fully spin-polarized for large
laser frequency detunings. In this regime, we reach the optimal operation of
the MOT, with samples of typically $3\times 10^8$ atoms at a temperature of
15\,$\mu$K. The spin polarization reduces the complexity of the radiative
cooling description, which allows for a simple model accounting for our
measurements. We also measure the rate of density-dependent atom losses,
finding good agreement with a model based on light-induced Van der Waals
forces. A minimal two-body loss rate $\beta\sim 2\times10^{-11}\,$cm$^{3}$/s is
reached in the spin-polarized regime. Our results constitute a benchmark for
the experimental study of ultracold gases of magnetic Lanthanide atoms.
