We give an explicit characterization of all extreme rays of the cone of 5x5 copositive matrices. The results are based on the work of Baumert [L. D. Baumert, "Extreme copositive quadratic forms", PhD thesis, 1965], where an implicit characterization was given. We show that the class of extreme rays found by Baumert forms a 10-dimensional variety, which can be parametrized in a semi-trigonometric way.

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