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We develop a method to compute the Ekedahl-Oort type of a curve C over afield k of characteristic p (which is the isomorphism type of the p-kernelgroup scheme J[p], where J is the Jacobian of C). Part of our method isgeneral, in that we introduce the new notion of a Hasse-Witt triple, whichre-encodes in a useful way the information contained in the Dieudonne module ofJ[p]. For complete intersection curves we then give a simple method to computethis Hasse-Witt triple. An implementation of this method is available in Magma.

Computing discrete invariants of varieties in positive characteristic. I. Ekedahl-Oort types of curves