An annular combustion chamber for a turbomachine, including an external wall and an internal wall which are oriented substantially axially relative to the rotation axis of the turbomachine, the combustion chamber being closed upstream by a chamber end wall oriented substantially radially, the chamber being supplied with compressed air coming from a compressor via a nozzle, the output direction of which is offset radially relative to the mid-axis of the combustion chamber, the chamber end wall including cooling air supply holes inclined to the direction normal to the chamber end wall. The number of holes oriented radially in the direction opposite to that where the outlet of the nozzle is located is greater than the number of holes oriented radially in the direction of the outlet of the nozzle.