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The question we propose to answer throughout this paper is the following:Given an isogeny class of Drinfeld modules over a finite field, what are theorders of the corresponding endomorphism algebra (which is an isogenyinvariant) that occur as endomorphism ring of a Drinfeld module in that isogenyclass? It is worth mentioning that this question is different from the onesinvestigated by the authors Kuhn, Pink in [6] and Garai, Papikian in [3]. Theformer authors rather provided an answer to the question, given a Drinfeldmodule {\phi}, how does one efficiently compute the endomorphism ring of{\phi}? The importance of our question resides in the fact that it might bevery helpful to better understand isogeny graphs of Drinfeld modules of higherrank (r > 2) and may be reopen the debate concerning the application toisogeny-based cryptography. We answer that question for the case whereby theendomorphism algebra is a field by providing a necessary and sufficientcondition for a given order to be the endomorphism ring of a Drinfeld module.We apply our result to rank r = 3 Drinfeld modules and explicitly compute thoseorders occurring as endomorphism rings of rank 3 Drinfeld modules over a finitefield.

Orders occurring as endomorphism ring of a Drinfeld module in some isogeny classes of Drinfeld modules of higher ranks