1404 Ajax, provisional designation 1936 QW, is a large, carbonaceous Jupiter trojan, about 82 kilometers in diameter. It was discovered by German astronomer Karl Reinmuth at Heidelberg Observatory on 17 August 1936. The C-type body belongs to the Greek Camp of Trojan asteroids. It orbits the Sun in the L4 Lagrangian point of the Sun-Jupiter system at a distance of 4.7–5.9 AU once every 12 years and 3 months (4,462 days). It has a rotation period of 29.4 hours and an albedo in the range of 0.05 to 0.07, as measured by the IRAS, Akari, WISE and NEOWISE surveys.