INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 360 Minimal number of generators: 61 Number of equivalence classes of cusps: 30 Genus: 16 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 2/9 3/8 1/2 4/7 5/6 1/1 6/5 13/9 3/2 34/21 7/4 2/1 15/7 23/10 7/3 5/2 8/3 3/1 10/3 17/5 7/2 4/1 9/2 5/1 6/1 19/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -5/11 1/3 1/2 -4/9 1/2 1/1 -3/7 0/1 1/1 -5/12 0/1 1/3 -2/5 0/1 1/2 -5/13 1/2 2/3 -3/8 1/2 1/1 -4/11 2/3 1/1 -5/14 4/5 1/1 -1/3 0/1 1/1 -4/13 1/1 4/3 -3/10 1/1 2/1 -2/7 1/1 1/0 -3/11 2/1 1/0 -1/4 0/1 1/0 -2/9 -1/1 0/1 -1/5 0/1 1/1 -1/6 1/1 1/0 -1/7 -1/1 1/0 0/1 0/1 1/0 1/6 0/1 1/1 2/11 1/1 2/1 1/5 1/1 1/0 2/9 1/0 3/13 -3/1 1/0 1/4 -1/1 1/0 3/11 -2/1 1/0 5/18 -2/1 1/0 2/7 -2/1 -1/1 1/3 -1/1 0/1 3/8 0/1 5/13 0/1 1/3 2/5 0/1 1/1 5/12 1/1 1/0 3/7 0/1 1/0 4/9 0/1 1/1 1/2 0/1 1/0 4/7 1/0 7/12 -4/1 1/0 3/5 -2/1 1/0 11/18 -2/1 -1/1 8/13 -2/1 1/0 5/8 -2/1 -1/1 7/11 -1/1 0/1 2/3 -1/1 1/0 5/7 -2/1 1/0 13/18 -2/1 -1/1 8/11 -2/1 1/0 3/4 -2/1 -1/1 10/13 -2/1 -3/2 7/9 -3/2 -1/1 11/14 -3/2 -4/3 4/5 -4/3 -1/1 5/6 -1/1 6/7 -1/1 -4/5 1/1 -1/1 0/1 6/5 -1/1 11/9 -1/1 -8/9 5/4 -1/1 -4/5 9/7 -1/1 -3/4 13/10 -3/4 -2/3 17/13 -3/4 -2/3 4/3 -1/1 -2/3 11/8 -2/3 -1/2 29/21 -1/1 -1/2 18/13 -1/1 -2/3 25/18 -3/4 -2/3 7/5 -2/3 -1/2 10/7 -3/5 -1/2 13/9 -1/2 3/2 -1/1 -1/2 11/7 -1/1 0/1 19/12 -2/1 -1/1 8/5 -1/1 -2/3 21/13 -3/4 -2/3 34/21 -2/3 47/29 -2/3 -5/8 13/8 -2/3 -1/2 5/3 -2/3 -1/2 7/4 -1/2 9/5 -1/2 -2/5 2/1 -1/2 0/1 15/7 -1/2 13/6 -1/2 -2/5 11/5 -1/2 -1/3 9/4 -1/3 0/1 16/7 -1/4 0/1 23/10 0/1 7/3 -1/2 0/1 19/8 -1/1 -1/2 31/13 -1/2 12/5 -1/2 -1/3 5/2 -1/3 0/1 8/3 0/1 11/4 0/1 1/1 3/1 -1/1 0/1 10/3 -2/1 -1/1 17/5 -1/1 24/7 -1/1 -6/7 7/2 -1/1 -2/3 25/7 -3/4 -2/3 43/12 -2/3 18/5 -2/3 -1/2 29/8 -1/1 -2/3 40/11 -2/3 -3/5 11/3 -2/3 -1/2 4/1 -1/1 -1/2 9/2 -1/2 14/3 -1/2 -3/7 5/1 -1/2 -1/3 21/4 -1/2 0/1 37/7 -1/2 16/3 -1/2 -2/5 11/2 -2/5 -1/3 6/1 -1/3 0/1 19/3 0/1 32/5 0/1 1/1 13/2 -1/1 0/1 7/1 -1/2 0/1 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(161,74,124,57) (-1/2,-5/11) -> (9/7,13/10) Hyperbolic Matrix(199,90,42,19) (-5/11,-4/9) -> (14/3,5/1) Hyperbolic Matrix(77,34,120,53) (-4/9,-3/7) -> (7/11,2/3) Hyperbolic Matrix(113,48,40,17) (-3/7,-5/12) -> (11/4,3/1) Hyperbolic Matrix(111,46,152,63) (-5/12,-2/5) -> (8/11,3/4) Hyperbolic Matrix(71,28,38,15) (-2/5,-5/13) -> (9/5,2/1) Hyperbolic Matrix(37,14,140,53) (-5/13,-3/8) -> (1/4,3/11) Hyperbolic Matrix(71,26,172,63) (-3/8,-4/11) -> (2/5,5/12) Hyperbolic Matrix(205,74,36,13) (-4/11,-5/14) -> (11/2,6/1) Hyperbolic Matrix(169,60,138,49) (-5/14,-1/3) -> (11/9,5/4) Hyperbolic Matrix(31,10,34,11) (-1/3,-4/13) -> (6/7,1/1) Hyperbolic Matrix(33,10,188,57) (-4/13,-3/10) -> (1/6,2/11) Hyperbolic Matrix(155,46,64,19) (-3/10,-2/7) -> (12/5,5/2) Hyperbolic Matrix(121,34,32,9) (-2/7,-3/11) -> (11/3,4/1) Hyperbolic Matrix(89,24,152,41) (-3/11,-1/4) -> (7/12,3/5) Hyperbolic Matrix(115,26,84,19) (-1/4,-2/9) -> (4/3,11/8) Hyperbolic Matrix(55,12,142,31) (-2/9,-1/5) -> (5/13,2/5) Hyperbolic Matrix(81,14,52,9) (-1/5,-1/6) -> (3/2,11/7) Hyperbolic Matrix(25,4,106,17) (-1/6,-1/7) -> (3/13,1/4) Hyperbolic Matrix(77,10,100,13) (-1/7,0/1) -> (10/13,7/9) Hyperbolic Matrix(59,-8,96,-13) (0/1,1/6) -> (11/18,8/13) Hyperbolic Matrix(409,-76,296,-55) (2/11,1/5) -> (29/21,18/13) Hyperbolic Matrix(37,-8,162,-35) (1/5,2/9) -> (2/9,3/13) Parabolic Matrix(449,-124,344,-95) (3/11,5/18) -> (13/10,17/13) Hyperbolic Matrix(499,-140,360,-101) (5/18,2/7) -> (18/13,25/18) Hyperbolic Matrix(51,-16,16,-5) (2/7,1/3) -> (3/1,10/3) Hyperbolic Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(217,-92,92,-39) (5/12,3/7) -> (7/3,19/8) Hyperbolic Matrix(245,-108,152,-67) (3/7,4/9) -> (8/5,21/13) Hyperbolic Matrix(107,-48,136,-61) (4/9,1/2) -> (11/14,4/5) Hyperbolic Matrix(57,-32,98,-55) (1/2,4/7) -> (4/7,7/12) Parabolic Matrix(191,-116,28,-17) (3/5,11/18) -> (13/2,7/1) Hyperbolic Matrix(245,-152,108,-67) (8/13,5/8) -> (9/4,16/7) Hyperbolic Matrix(297,-188,188,-119) (5/8,7/11) -> (11/7,19/12) Hyperbolic Matrix(91,-64,64,-45) (2/3,5/7) -> (7/5,10/7) Hyperbolic Matrix(499,-360,140,-101) (5/7,13/18) -> (7/2,25/7) Hyperbolic Matrix(409,-296,76,-55) (13/18,8/11) -> (16/3,11/2) Hyperbolic Matrix(449,-344,124,-95) (3/4,10/13) -> (18/5,29/8) Hyperbolic Matrix(271,-212,124,-97) (7/9,11/14) -> (13/6,11/5) Hyperbolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic Matrix(61,-72,50,-59) (1/1,6/5) -> (6/5,11/9) Parabolic Matrix(107,-136,48,-61) (5/4,9/7) -> (11/5,9/4) Hyperbolic Matrix(553,-724,152,-199) (17/13,4/3) -> (40/11,11/3) Hyperbolic Matrix(481,-664,92,-127) (11/8,29/21) -> (5/1,21/4) Hyperbolic Matrix(911,-1266,562,-781) (25/18,7/5) -> (47/29,13/8) Hyperbolic Matrix(325,-466,136,-195) (10/7,13/9) -> (31/13,12/5) Hyperbolic Matrix(233,-340,98,-143) (13/9,3/2) -> (19/8,31/13) Hyperbolic Matrix(323,-512,94,-149) (19/12,8/5) -> (24/7,7/2) Hyperbolic Matrix(1429,-2312,882,-1427) (21/13,34/21) -> (34/21,47/29) Parabolic Matrix(59,-96,8,-13) (13/8,5/3) -> (7/1,1/0) Hyperbolic Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(149,-314,28,-59) (2/1,15/7) -> (37/7,16/3) Hyperbolic Matrix(369,-796,70,-151) (15/7,13/6) -> (21/4,37/7) Hyperbolic Matrix(481,-1102,134,-307) (16/7,23/10) -> (43/12,18/5) Hyperbolic Matrix(379,-876,106,-245) (23/10,7/3) -> (25/7,43/12) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(171,-578,50,-169) (10/3,17/5) -> (17/5,24/7) Parabolic Matrix(399,-1448,62,-225) (29/8,40/11) -> (32/5,13/2) Hyperbolic Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic Matrix(115,-722,18,-113) (6/1,19/3) -> (19/3,32/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(161,74,124,57) -> Matrix(7,-2,-10,3) Matrix(199,90,42,19) -> Matrix(5,-2,-12,5) Matrix(77,34,120,53) -> Matrix(1,0,-2,1) Matrix(113,48,40,17) -> Matrix(1,0,-2,1) Matrix(111,46,152,63) -> Matrix(5,-2,-2,1) Matrix(71,28,38,15) -> Matrix(1,0,-4,1) Matrix(37,14,140,53) -> Matrix(1,0,-2,1) Matrix(71,26,172,63) -> Matrix(3,-2,2,-1) Matrix(205,74,36,13) -> Matrix(3,-2,-10,7) Matrix(169,60,138,49) -> Matrix(9,-8,-10,9) Matrix(31,10,34,11) -> Matrix(1,0,-2,1) Matrix(33,10,188,57) -> Matrix(1,-2,2,-3) Matrix(155,46,64,19) -> Matrix(1,-2,-2,5) Matrix(121,34,32,9) -> Matrix(1,0,-2,1) Matrix(89,24,152,41) -> Matrix(1,-4,0,1) Matrix(115,26,84,19) -> Matrix(1,2,-2,-3) Matrix(55,12,142,31) -> Matrix(1,0,2,1) Matrix(81,14,52,9) -> Matrix(1,0,-2,1) Matrix(25,4,106,17) -> Matrix(1,-2,0,1) Matrix(77,10,100,13) -> Matrix(3,2,-2,-1) Matrix(59,-8,96,-13) -> Matrix(1,-2,0,1) Matrix(409,-76,296,-55) -> Matrix(1,0,-2,1) Matrix(37,-8,162,-35) -> Matrix(1,-4,0,1) Matrix(449,-124,344,-95) -> Matrix(3,4,-4,-5) Matrix(499,-140,360,-101) -> Matrix(3,4,-4,-5) Matrix(51,-16,16,-5) -> Matrix(1,0,0,1) Matrix(49,-18,128,-47) -> Matrix(1,0,4,1) Matrix(217,-92,92,-39) -> Matrix(1,0,-2,1) Matrix(245,-108,152,-67) -> Matrix(3,-2,-4,3) Matrix(107,-48,136,-61) -> Matrix(3,-4,-2,3) Matrix(57,-32,98,-55) -> Matrix(1,-4,0,1) Matrix(191,-116,28,-17) -> Matrix(1,2,-2,-3) Matrix(245,-152,108,-67) -> Matrix(1,2,-4,-7) Matrix(297,-188,188,-119) -> Matrix(1,0,0,1) Matrix(91,-64,64,-45) -> Matrix(1,4,-2,-7) Matrix(499,-360,140,-101) -> Matrix(3,4,-4,-5) Matrix(409,-296,76,-55) -> Matrix(1,0,-2,1) Matrix(449,-344,124,-95) -> Matrix(3,4,-4,-5) Matrix(271,-212,124,-97) -> Matrix(1,2,-4,-7) Matrix(61,-50,72,-59) -> Matrix(7,8,-8,-9) Matrix(61,-72,50,-59) -> Matrix(7,8,-8,-9) Matrix(107,-136,48,-61) -> Matrix(5,4,-14,-11) Matrix(553,-724,152,-199) -> Matrix(11,8,-18,-13) Matrix(481,-664,92,-127) -> Matrix(3,2,-8,-5) Matrix(911,-1266,562,-781) -> Matrix(11,8,-18,-13) Matrix(325,-466,136,-195) -> Matrix(7,4,-16,-9) Matrix(233,-340,98,-143) -> Matrix(1,0,0,1) Matrix(323,-512,94,-149) -> Matrix(3,4,-4,-5) Matrix(1429,-2312,882,-1427) -> Matrix(23,16,-36,-25) Matrix(59,-96,8,-13) -> Matrix(3,2,-8,-5) Matrix(57,-98,32,-55) -> Matrix(7,4,-16,-9) Matrix(149,-314,28,-59) -> Matrix(3,2,-8,-5) Matrix(369,-796,70,-151) -> Matrix(5,2,-8,-3) Matrix(481,-1102,134,-307) -> Matrix(9,2,-14,-3) Matrix(379,-876,106,-245) -> Matrix(1,2,-2,-3) Matrix(49,-128,18,-47) -> Matrix(1,0,4,1) Matrix(171,-578,50,-169) -> Matrix(7,8,-8,-9) Matrix(399,-1448,62,-225) -> Matrix(3,2,-2,-1) Matrix(37,-162,8,-35) -> Matrix(7,4,-16,-9) Matrix(115,-722,18,-113) -> Matrix(1,0,4,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 12 Degree of the the map X: 12 Degree of the the map Y: 60 Permutation triple for Y: ((1,6,24,51,39,58,36,49,25,7,2)(3,12,40,59,56,50,31,8,30,13,4)(5,18,43,38,28,27,10,9,35,32,19)(14,41,48,17,47,54,44,52,23,37,42)(15,26,22,21,29,34,33,55,57,45,16); (1,4,16,46,29,8,7,28,47,17,5)(3,10,23,6,22,26,25,42,32,31,11)(9,34,56,54,24,53,49,48,40,45,35)(12,38,21,41,60,44,15,18,50,58,39)(13,30,52,57,36,27,20,19,51,33,14); (1,2,8,32,45,52,60,41,33,9,3)(4,14,25,53,24,23,30,29,38,43,15)(5,20,27,7,26,44,56,59,48,21,6)(10,36,50,34,46,16,40,39,19,42,37)(11,31,18,17,49,57,55,51,54,28,12)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 60 Minimal number of generators: 11 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 10 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/1 6/5 7/4 2/1 8/3 3/1 4/1 9/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/0 1/4 -1/1 1/0 1/3 -1/1 0/1 1/2 0/1 1/0 2/3 -1/1 1/0 3/4 -2/1 -1/1 1/1 -1/1 0/1 6/5 -1/1 5/4 -1/1 -4/5 4/3 -1/1 -2/3 3/2 -1/1 -1/2 5/3 -2/3 -1/2 7/4 -1/2 2/1 -1/2 0/1 5/2 -1/3 0/1 8/3 0/1 3/1 -1/1 0/1 4/1 -1/1 -1/2 9/2 -1/2 5/1 -1/2 -1/3 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(9,-1,1,0) (0/1,1/4) -> (5/1,1/0) Hyperbolic Matrix(25,-7,18,-5) (1/4,1/3) -> (4/3,3/2) Hyperbolic Matrix(16,-7,7,-3) (1/3,1/2) -> (2/1,5/2) Hyperbolic Matrix(21,-13,13,-8) (1/2,2/3) -> (3/2,5/3) Hyperbolic Matrix(25,-18,7,-5) (2/3,3/4) -> (3/1,4/1) Hyperbolic Matrix(24,-19,19,-15) (3/4,1/1) -> (5/4,4/3) Hyperbolic Matrix(31,-36,25,-29) (1/1,6/5) -> (6/5,5/4) Parabolic Matrix(29,-49,16,-27) (5/3,7/4) -> (7/4,2/1) Parabolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(9,-1,1,0) -> Matrix(1,0,-2,1) Matrix(25,-7,18,-5) -> Matrix(1,2,-2,-3) Matrix(16,-7,7,-3) -> Matrix(1,0,-2,1) Matrix(21,-13,13,-8) -> Matrix(1,2,-2,-3) Matrix(25,-18,7,-5) -> Matrix(1,2,-2,-3) Matrix(24,-19,19,-15) -> Matrix(3,4,-4,-5) Matrix(31,-36,25,-29) -> Matrix(3,4,-4,-5) Matrix(29,-49,16,-27) -> Matrix(3,2,-8,-5) Matrix(25,-64,9,-23) -> Matrix(1,0,2,1) Matrix(19,-81,4,-17) -> Matrix(3,2,-8,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 2 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 0/1 2 1 1/1 (-1/1,0/1) 0 11 6/5 -1/1 4 1 4/3 (-1/1,-2/3) 0 11 3/2 (-1/1,-1/2) 0 11 7/4 -1/2 2 1 2/1 (-1/2,0/1) 0 11 8/3 0/1 2 1 3/1 (-1/1,0/1) 0 11 4/1 (-1/1,-1/2) 0 11 9/2 -1/2 2 1 1/0 (-1/2,0/1) 0 11 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(11,-12,10,-11) (1/1,6/5) -> (1/1,6/5) Reflection Matrix(19,-24,15,-19) (6/5,4/3) -> (6/5,4/3) Reflection Matrix(18,-25,5,-7) (4/3,3/2) -> (3/1,4/1) Glide Reflection Matrix(13,-21,8,-13) (3/2,7/4) -> (3/2,7/4) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(7,-16,3,-7) (2/1,8/3) -> (2/1,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(-1,9,0,1) (9/2,1/0) -> (9/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,4,1) (-1/1,1/0) -> (-1/2,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,2,1) (-1/1,1/1) -> (-1/1,0/1) Matrix(11,-12,10,-11) -> Matrix(-1,0,2,1) (1/1,6/5) -> (-1/1,0/1) Matrix(19,-24,15,-19) -> Matrix(5,4,-6,-5) (6/5,4/3) -> (-1/1,-2/3) Matrix(18,-25,5,-7) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(13,-21,8,-13) -> Matrix(3,2,-4,-3) (3/2,7/4) -> (-1/1,-1/2) Matrix(15,-28,8,-15) -> Matrix(-1,0,4,1) (7/4,2/1) -> (-1/2,0/1) Matrix(7,-16,3,-7) -> Matrix(-1,0,4,1) (2/1,8/3) -> (-1/2,0/1) Matrix(17,-48,6,-17) -> Matrix(-1,0,2,1) (8/3,3/1) -> (-1/1,0/1) Matrix(17,-72,4,-17) -> Matrix(3,2,-4,-3) (4/1,9/2) -> (-1/1,-1/2) Matrix(-1,9,0,1) -> Matrix(-1,0,4,1) (9/2,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.