Platinum monatomic nanowires were predicted to spontaneously develop magnetism, involving a sizable orbital moment via spin orbit coupling, and a colossal magnetic anisotropy. We present here a fully-relativistic (spin-orbit coupling included) pseudo-potential density functional calculation of electronic and magnetic properties, and of Landauer ballistic conductance of Pt model nanocontacts consisting of short nanowire segments suspended between Pt leads or tips, reprented by bulk planes. Even if short, and despite the nonmagnetic Pt leads, the nanocontact is found to be locally magnetic with magnetization strictly parallel to its axis. Especially under strain, the energy barrier to flip the overall spin direction is predicted to be tens of meV high, and thus the corresponding blocking temperatures large, suggesting the use of static Landauer ballistic electrical conductance calculations. We carry out such calculations, to find that inclusion of spin-orbit coupling and of magnetism lowers the ballistic conductance by about $15\div20$% relative to the nonmagnetic case, yielding $ G\sim 2 G_0$ ($G_0=2e^2/h$), in good agreement with break junction results. The spin filtering properties of this highly unusual spontaneously magnetic nanocontact are also analysed.