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In addition to the typo, you displayed the correct transpose(AA) but you accidentally passed the un-transposed AA as the ieqs argument to Polyhedron. It should be:

sage: pol = Polyhedron(ieqs=transpose(AA))
sage: pol.Hrepresentation()
(An inequality (-1, -4, -8, -16, -32) x + 3125 >= 0,
 An inequality (0, -1, -4, -8, -16) x + 625 >= 0,
 An inequality (0, 0, -1, -4, -8) x + 125 >= 0,
 An inequality (0, 0, 0, -1, -4) x + 25 >= 0,
 An inequality (0, 0, 0, 0, -1) x + 5 >= 0,
 An inequality (1, 0, 0, 0, 0) x + 0 >= 0,
 An inequality (0, 0, 0, 0, 1) x + 0 >= 0,
 An inequality (0, 0, 0, 1, 0) x + 0 >= 0,
 An inequality (0, 0, 1, 0, 0) x + 0 >= 0,
 An inequality (0, 1, 0, 0, 0) x + 0 >= 0)
sage: len(pol.vertices())
32
sage: pol.vertices()
(A vertex at (3125, 0, 0, 0, 0),
 A vertex at (865, 505, 0, 5, 5),
 A vertex at (1625, 125, 125, 0, 0),
 A vertex at (1225, 325, 25, 25, 0),
 A vertex at (1385, 245, 65, 5, 5),
 A vertex at (1465, 205, 85, 0, 5),
 A vertex at (785, 545, 0, 0, 5),
 A vertex at (625, 625, 0, 0, 0),
 A vertex at (1025, 425, 0, 25, 0),
 A vertex at (2725, 0, 0, 25, 0),
 A vertex at (2885, 0, 0, 5, 5),
 A vertex at (2125, 0, 125, 0, 0),
 A vertex at (2525, 0, 25, 25, 0),
 A vertex at (2365, 0, 65, 5, 5),
 A vertex at (2285, 0, 85, 0, 5),
 A vertex at (2965, 0, 0, 0, 5),
 A vertex at (0, 0, 0, 0, 5),
 A vertex at (0, 0, 0, 0, 0),
 A vertex at (0, 505, 0, 5, 5),
 A vertex at (0, 125, 125, 0, 0),
 A vertex at (0, 325, 25, 25, 0),
 A vertex at (0, 245, 65, 5, 5),
 A vertex at (0, 205, 85, 0, 5),
 A vertex at (0, 545, 0, 0, 5),
 A vertex at (0, 625, 0, 0, 0),
 A vertex at (0, 425, 0, 25, 0),
 A vertex at (0, 0, 0, 25, 0),
 A vertex at (0, 0, 0, 5, 5),
 A vertex at (0, 0, 125, 0, 0),
 A vertex at (0, 0, 25, 25, 0),
 A vertex at (0, 0, 65, 5, 5),
 A vertex at (0, 0, 85, 0, 5))