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This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n = 3 or n = 5, and that it is equal to 3 (respectively 9) if n = 9 (respectively if n = 15) and to 1 for all remaining odd orders. Some precise representative elements of the classes are given.

The number of conjugacy classes of elements of the Cremona group of some given finite order