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We show that the structure of the enumeration degrees De has a finite automorphism base consisting of finitely many total elements below the first jump of its least element. As a consequence we obtain that the rigidity of the structure of the enumeration degrees is implied by the rigidity of the local structures of the Σ 0 2 enumeration degrees, the ∆ 0 2 Turing degrees and the computably enumerable Turing degrees.doi:10.1090/conm/690/13862 fatcat:kfnacq43ajdnjaefrpy7geswcy