Current induced domain wall (DW) depinning of a narrow DW in out-of-plane magnetized (Pt/Co)_{3}/Pt multilayer elements is studied by magnetotransport. We find that for conventional measurements Joule heating effects conceal the real spin torque efficiency and so we use a measurement scheme at a constant sample temperature to unambiguously extract the spin torque contribution. From the variation of the depinning magnetic field with the current pulse amplitude we directly deduce the large nonadiabaticity factor in this material and we find that its amplitude is consistent with a momentum transfer mechanism.