Based on what the Constitution says, the presidency has very ambiguous powers. Much of what the Constitution says about the presidency is open to interpretation.
For example, the Constitution says that the “executive power is vested in” the presidency. But it does not define what the executive power is. It says that the president "shall take care that the laws be faithfully executed." But it does not say exactly what the president may or may not do in order to accomplish this. It says the president is the commander-in-chief of the military but does not exactly lay out what that allows him (or someday her) to do.
Of course, the Constitution does give some specific powers like the power to veto and to appoint judges. But the bulk of the presidency’s powers are granted by ambiguous statements like the ones above.