1. Cavendish Laboratory, Cambridge University, JJ Thomson Avenue, Cambridge CB3 OHE, UK
   
2. National High Magnetic Field Laboratory, Los Alamos National Laboratory, MS E536, Los Alamos, New Mexico 87545, USA
   
3. National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA
   
4. Department of Physics and Astronomy, University of British Columbia, Vancouver V6T 1Z4, Canada
   
5. Canadian Institute for Advanced Research, Toronto M5G 1Z8, Canada

Correspondence to: Suchitra E. Sebastian¹ Correspondence and requests for materials should be addressed to S.E.S. (Email: suchitra@phy.cam.ac.uk).

To understand the origin of superconductivity, it is crucial to ascertain the nature and origin of the primary carriers available to participate in pairing^{1, 2, 3, 4, 5, 6}. Recent quantum oscillation experiments on high-transition-temperature (high-T _c) copper oxide superconductors^{7, 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 superconductivity^{12, 13, 14, 15}. Here we report experiments on quantum oscillations in the magnetization (the de Haas-van Alphen effect) in superconducting YBa₂Cu₃O_6.51 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.