  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 the
orders of the corresponding endomorphism algebra (which is an isogeny
invariant) that occur as endomorphism ring of a Drinfeld module in that isogeny
class? It is worth mentioning that this question is different from the ones
investigated by the authors Kuhn, Pink in [6] and Garai, Papikian in [3]. The
former authors rather provided an answer to the question, given a Drinfeld
module {\phi}, how does one efficiently compute the endomorphism ring of
{\phi}? The importance of our question resides in the fact that it might be
very helpful to better understand isogeny graphs of Drinfeld modules of higher
rank (r > 2) and may be reopen the debate concerning the application to
isogeny-based cryptography. We answer that question for the case whereby the
endomorphism algebra is a field by providing a necessary and sufficient
condition 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 those
orders occurring as endomorphism rings of rank 3 Drinfeld modules over a finite
field.
