Yup-and she's their defacto leader (if it's not outright mentioned). Celestia personally summons Twilight and wants to train her. What the hell else would she be training her for? It's kind of lazily implied "magic stuff", but that doesn't explain a ton of the episodes where she's having the importance of love and friends hammered home (i.e. good qualities that Celestia seems to want embodied in a leader).