A tire has a carcass reinforcement (1) which, when viewed in meridian secn, is wound in each bead B about a bead wire (2) coated with a rubber mix, passing from the heel to the toe of the bead B, the upturn (10) being located in a profile (3) of rubber mix in the form of a wedge defined by two sides (31 and 32) coming from an apex A located beneath the section of the coated bead wire (2). Advantageously the upturn (10) surrounds in its entirety the contour of the profile (3), forming either a first radially inner side (32), a lateral side (30), and a final radially outer side (31) or a first radially outer side (31), a lateral side (30), and a final radially inner side (32). The tire may form with different rims J high-performance assemblies in the event of travel at low or zero pressure, permitting simple mounting of the tires and optionally of bearing supports S for the tire tread on the rims J.