We present fully covered phased light curves for 56 Jovian Trojan asteroids as observed by the K2 mission of the Kepler space telescope. This set of objects has been monitored during Campaign 6 and represents a nearly unbiased subsample of the population of small solar system bodies. We derived precise periods and amplitudes for all Trojans, and found their distributions to be compatible with the previous statistics. We point out, however, that ground-based rotation periods are often unreliable above 20 h, and we find an overabundance of rotation periods above 60 h compared with other minor planet populations. From amplitude analysis we derive a rate of binarity of 20 ± 5%. Our spin rate distribution confirms the previously obtained spin barrier of ~5 h and the corresponding ~0.5 g cm-3 cometary-like density limit, also suggesting a high internal porosity for Jovian Trojans. One of our targets, asteroid 65227 exhibits a double rotation period, which can either be due to binarity or the outcome of a recent collision.