To understand the origin of superconductivity, it is crucial to ascertain the nature and origin of the primary carriers available to participate in pairing1, 2, 3, 4, 5, 6. Recent quantum oscillation experiments on high-transition-temperature (high-Tc) copper oxide superconductors7, 8, 9, 10 have revealed the existence of a Fermi surface akin to that in normal metals, comprising fermionic carriers that undergo orbital quantization11. The unexpectedly small size of the observed carrier pocket, however, leaves open a variety of possibilities for the existence or form of any underlying magnetic order, and its relation to d-wave superconductivity12, 13, 14, 15. Here we report experiments on quantum oscillations in the magnetization (the de Haas-van Alphen effect) in superconducting YBa2Cu3O6.51 that reveal more than one carrier pocket. In particular, we find evidence for the existence of a much larger pocket of heavier mass carriers playing a thermodynamically dominant role in this hole-doped superconductor. Importantly, characteristics of the multiple pockets within this more complete Fermi surface impose constraints on the wavevector of any underlying order and the location of the carriers in momentum space. These constraints enable us to construct a possible density-wave model with spiral or related modulated magnetic order, consistent with experimental observations.