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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/7 -1/3 -1/5 0/1 1/7 3/13 1/3 5/11 1/2 3/5 7/9 1/1 9/7 3/2 5/3 2/1 11/5 5/2 8/3 3/1 49/15 10/3 4/1 13/3 14/3 5/1 6/1 7/1 8/1 9/1 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 1/1 -5/11 1/2 -9/20 4/5 1/1 -4/9 1/1 3/2 -3/7 1/0 -11/26 -1/1 -2/3 -8/19 0/1 1/0 -5/12 0/1 1/1 -12/29 1/2 1/1 -7/17 1/0 -9/22 0/1 1/1 -2/5 1/1 1/0 -3/8 -2/1 -1/1 -4/11 -1/2 0/1 -5/14 0/1 1/3 -1/3 1/0 -4/13 -1/4 0/1 -3/10 0/1 1/3 -8/27 0/1 1/4 -5/17 1/2 -2/7 1/1 1/0 -1/4 -1/1 0/1 -4/17 0/1 1/4 -3/13 1/2 -5/22 0/1 1/1 -2/9 0/1 1/0 -3/14 2/3 1/1 -7/33 1/0 -4/19 1/2 1/1 -1/5 1/0 -3/16 -1/1 0/1 -5/27 -1/2 -2/11 -1/2 0/1 -1/6 -1/1 0/1 -2/13 0/1 1/2 -3/20 1/1 2/1 -1/7 1/0 -2/15 0/1 1/2 -1/8 0/1 1/1 0/1 0/1 1/0 1/7 1/0 1/6 -1/1 0/1 1/5 1/0 2/9 -1/2 0/1 3/13 0/1 4/17 0/1 1/4 1/4 0/1 1/1 2/7 0/1 1/2 3/10 0/1 1/1 1/3 1/0 2/5 0/1 1/0 3/7 -1/2 4/9 -1/4 0/1 5/11 0/1 6/13 0/1 1/6 1/2 0/1 1/1 7/13 1/2 6/11 3/4 1/1 5/9 3/2 9/16 2/1 3/1 4/7 2/1 1/0 3/5 1/0 8/13 -4/1 1/0 5/8 -3/1 -2/1 2/3 -1/1 1/0 3/4 -1/3 0/1 7/9 0/1 11/14 0/1 1/11 4/5 0/1 1/4 5/6 0/1 1/3 1/1 1/0 5/4 -1/3 0/1 9/7 0/1 13/10 0/1 1/7 4/3 0/1 1/2 7/5 1/2 17/12 2/3 1/1 10/7 1/1 1/0 3/2 1/1 2/1 14/9 3/2 2/1 11/7 5/2 30/19 23/8 3/1 49/31 3/1 19/12 3/1 16/5 27/17 7/2 8/5 4/1 1/0 5/3 1/0 12/7 -6/1 1/0 19/11 -9/2 7/4 -4/1 -3/1 2/1 -1/1 1/0 11/5 -1/1 20/9 -1/1 -1/2 9/4 -1/1 0/1 16/7 0/1 1/0 7/3 1/0 12/5 -1/1 -1/2 29/12 -1/1 -2/3 17/7 -1/2 22/9 -1/2 0/1 5/2 -1/1 0/1 13/5 1/2 34/13 1/1 5/4 21/8 1/1 2/1 8/3 0/1 1/0 19/7 1/2 30/11 0/1 1/2 11/4 2/3 1/1 3/1 1/0 13/4 -10/3 -3/1 49/15 -3/1 36/11 -3/1 -29/10 23/7 -11/4 10/3 -5/2 -2/1 17/5 1/0 7/2 -3/1 -2/1 4/1 -3/2 -1/1 13/3 -1/1 22/5 -1/1 -11/12 9/2 -1/1 -4/5 14/3 -1/1 -3/4 5/1 -1/2 6/1 0/1 1/0 13/2 0/1 1/1 7/1 1/0 15/2 -4/1 -3/1 8/1 -2/1 1/0 9/1 1/0 1/0 -1/1 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(61,28,-268,-123) (-1/2,-5/11) -> (-3/13,-5/22) Hyperbolic Matrix(173,78,-814,-367) (-5/11,-9/20) -> (-3/14,-7/33) Hyperbolic Matrix(521,234,118,53) (-9/20,-4/9) -> (22/5,9/2) Hyperbolic Matrix(59,26,-202,-89) (-4/9,-3/7) -> (-5/17,-2/7) Hyperbolic Matrix(835,354,526,223) (-3/7,-11/26) -> (19/12,27/17) Hyperbolic Matrix(109,46,-718,-303) (-11/26,-8/19) -> (-2/13,-3/20) Hyperbolic Matrix(219,92,388,163) (-8/19,-5/12) -> (9/16,4/7) Hyperbolic Matrix(985,408,408,169) (-5/12,-12/29) -> (12/5,29/12) Hyperbolic Matrix(271,112,-1280,-529) (-12/29,-7/17) -> (-7/33,-4/19) Hyperbolic Matrix(161,66,-866,-355) (-7/17,-9/22) -> (-3/16,-5/27) Hyperbolic Matrix(485,198,218,89) (-9/22,-2/5) -> (20/9,9/4) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(49,18,-226,-83) (-3/8,-4/11) -> (-2/9,-3/14) Hyperbolic Matrix(199,72,152,55) (-4/11,-5/14) -> (13/10,4/3) Hyperbolic Matrix(97,34,174,61) (-5/14,-1/3) -> (5/9,9/16) Hyperbolic Matrix(185,58,118,37) (-1/3,-4/13) -> (14/9,11/7) Hyperbolic Matrix(183,56,232,71) (-4/13,-3/10) -> (11/14,4/5) Hyperbolic Matrix(47,14,-366,-109) (-3/10,-8/27) -> (-2/15,-1/8) Hyperbolic Matrix(865,256,544,161) (-8/27,-5/17) -> (27/17,8/5) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(41,10,86,21) (-1/4,-4/17) -> (6/13,1/2) Hyperbolic Matrix(43,10,-314,-73) (-4/17,-3/13) -> (-1/7,-2/15) Hyperbolic Matrix(249,56,40,9) (-5/22,-2/9) -> (6/1,13/2) Hyperbolic Matrix(163,34,302,63) (-4/19,-1/5) -> (7/13,6/11) Hyperbolic Matrix(117,22,218,41) (-1/5,-3/16) -> (1/2,7/13) Hyperbolic Matrix(315,58,38,7) (-5/27,-2/11) -> (8/1,9/1) Hyperbolic Matrix(117,20,76,13) (-2/11,-1/6) -> (3/2,14/9) Hyperbolic Matrix(37,6,154,25) (-1/6,-2/13) -> (4/17,1/4) Hyperbolic Matrix(187,28,20,3) (-3/20,-1/7) -> (9/1,1/0) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(171,-22,70,-9) (0/1,1/7) -> (17/7,22/9) Hyperbolic Matrix(305,-46,126,-19) (1/7,1/6) -> (29/12,17/7) Hyperbolic Matrix(137,-24,40,-7) (1/6,1/5) -> (17/5,7/2) Hyperbolic Matrix(211,-46,78,-17) (1/5,2/9) -> (8/3,19/7) Hyperbolic Matrix(79,-18,338,-77) (2/9,3/13) -> (3/13,4/17) Parabolic Matrix(51,-14,62,-17) (1/4,2/7) -> (4/5,5/6) Hyperbolic Matrix(223,-66,98,-29) (2/7,3/10) -> (9/4,16/7) Hyperbolic Matrix(121,-38,86,-27) (3/10,1/3) -> (7/5,17/12) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(223,-98,66,-29) (3/7,4/9) -> (10/3,17/5) Hyperbolic Matrix(111,-50,242,-109) (4/9,5/11) -> (5/11,6/13) Parabolic Matrix(391,-216,248,-137) (6/11,5/9) -> (11/7,30/19) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(337,-208,128,-79) (8/13,5/8) -> (21/8,8/3) Hyperbolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(121,-102,70,-59) (5/6,1/1) -> (19/11,7/4) Hyperbolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(555,-788,212,-301) (17/12,10/7) -> (34/13,21/8) Hyperbolic Matrix(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(2047,-3234,626,-989) (30/19,49/31) -> (49/15,36/11) Hyperbolic Matrix(991,-1568,304,-481) (49/31,19/12) -> (13/4,49/15) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(271,-466,82,-141) (12/7,19/11) -> (23/7,10/3) Hyperbolic Matrix(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(223,-514,82,-189) (16/7,7/3) -> (19/7,30/11) Hyperbolic Matrix(67,-158,14,-33) (7/3,12/5) -> (14/3,5/1) Hyperbolic Matrix(479,-1250,146,-381) (13/5,34/13) -> (36/11,23/7) Hyperbolic Matrix(197,-538,26,-71) (30/11,11/4) -> (15/2,8/1) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic Matrix(29,-196,4,-27) (13/2,7/1) -> (7/1,15/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(61,28,-268,-123) -> Matrix(1,0,0,1) Matrix(173,78,-814,-367) -> Matrix(3,-2,2,-1) Matrix(521,234,118,53) -> Matrix(9,-8,-10,9) Matrix(59,26,-202,-89) -> Matrix(1,-2,2,-3) Matrix(835,354,526,223) -> Matrix(7,10,2,3) Matrix(109,46,-718,-303) -> Matrix(1,0,2,1) Matrix(219,92,388,163) -> Matrix(1,2,0,1) Matrix(985,408,408,169) -> Matrix(3,-2,-4,3) Matrix(271,112,-1280,-529) -> Matrix(1,0,0,1) Matrix(161,66,-866,-355) -> Matrix(1,0,-2,1) Matrix(485,198,218,89) -> Matrix(1,0,-2,1) Matrix(51,20,28,11) -> Matrix(1,-2,0,1) Matrix(49,18,-226,-83) -> Matrix(1,0,2,1) Matrix(199,72,152,55) -> Matrix(1,0,4,1) Matrix(97,34,174,61) -> Matrix(3,-2,2,-1) Matrix(185,58,118,37) -> Matrix(5,2,2,1) Matrix(183,56,232,71) -> Matrix(1,0,8,1) Matrix(47,14,-366,-109) -> Matrix(1,0,-2,1) Matrix(865,256,544,161) -> Matrix(15,-4,4,-1) Matrix(43,12,68,19) -> Matrix(1,-2,0,1) Matrix(41,10,86,21) -> Matrix(1,0,2,1) Matrix(43,10,-314,-73) -> Matrix(1,0,-2,1) Matrix(249,56,40,9) -> Matrix(1,0,0,1) Matrix(163,34,302,63) -> Matrix(1,-2,2,-3) Matrix(117,22,218,41) -> Matrix(1,0,2,1) Matrix(315,58,38,7) -> Matrix(3,2,-2,-1) Matrix(117,20,76,13) -> Matrix(1,2,0,1) Matrix(37,6,154,25) -> Matrix(1,0,2,1) Matrix(187,28,20,3) -> Matrix(1,-2,0,1) Matrix(181,22,74,9) -> Matrix(1,0,-2,1) Matrix(171,-22,70,-9) -> Matrix(1,0,-2,1) Matrix(305,-46,126,-19) -> Matrix(1,2,-2,-3) Matrix(137,-24,40,-7) -> Matrix(1,-2,0,1) Matrix(211,-46,78,-17) -> Matrix(1,0,2,1) Matrix(79,-18,338,-77) -> Matrix(1,0,6,1) Matrix(51,-14,62,-17) -> Matrix(1,0,2,1) Matrix(223,-66,98,-29) -> Matrix(1,0,-2,1) Matrix(121,-38,86,-27) -> Matrix(1,-2,2,-3) Matrix(47,-18,34,-13) -> Matrix(1,0,2,1) Matrix(67,-28,12,-5) -> Matrix(1,0,0,1) Matrix(223,-98,66,-29) -> Matrix(3,2,-2,-1) Matrix(111,-50,242,-109) -> Matrix(1,0,10,1) Matrix(391,-216,248,-137) -> Matrix(11,-14,4,-5) Matrix(61,-36,100,-59) -> Matrix(1,-6,0,1) Matrix(337,-208,128,-79) -> Matrix(1,4,0,1) Matrix(37,-26,10,-7) -> Matrix(3,2,-2,-1) Matrix(127,-98,162,-125) -> Matrix(1,0,14,1) Matrix(121,-102,70,-59) -> Matrix(9,-4,-2,1) Matrix(57,-70,22,-27) -> Matrix(1,0,2,1) Matrix(127,-162,98,-125) -> Matrix(1,0,10,1) Matrix(555,-788,212,-301) -> Matrix(5,-4,4,-3) Matrix(91,-132,20,-29) -> Matrix(3,-2,-4,3) Matrix(2047,-3234,626,-989) -> Matrix(53,-156,-18,53) Matrix(991,-1568,304,-481) -> Matrix(25,-78,-8,25) Matrix(61,-100,36,-59) -> Matrix(1,-10,0,1) Matrix(271,-466,82,-141) -> Matrix(5,28,-2,-11) Matrix(111,-242,50,-109) -> Matrix(1,2,-2,-3) Matrix(223,-514,82,-189) -> Matrix(1,0,2,1) Matrix(67,-158,14,-33) -> Matrix(1,2,-2,-3) Matrix(479,-1250,146,-381) -> Matrix(17,-14,-6,5) Matrix(197,-538,26,-71) -> Matrix(5,-2,-2,1) Matrix(25,-72,8,-23) -> Matrix(1,-4,0,1) Matrix(79,-338,18,-77) -> Matrix(13,14,-14,-15) Matrix(29,-196,4,-27) -> Matrix(1,-4,0,1) 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): 21 Degree of the the map X: 21 Degree of the the map Y: 64 Permutation triple for Y: ((1,6,22,49,50,23,7,2)(3,12,38,57,27,8,13,4)(5,10,9,31,60,28,25,18)(11,34,48,51,59,45,19,35)(14,39,33,32,47,55,56,40)(15,20,29,54,24,26,43,16)(17,21,37,62,61,52,41,44)(30,36,58,53,46,64,63,42); (1,4,16,35,63,39,17,5)(3,10,33,11)(6,15,42,21)(7,25,41,40,46,45,26,8)(9,20,32,22,34,37,12,30)(13,14)(18,19)(23,52,53,24)(27,59,56,28)(29,62)(36,49)(50,55,54,60,58,57,61,51); (1,2,8,28,54,62,34,33,63,64,40,59,61,29,9,3)(4,14,41,23,51,48,22,21,39,13,26,53,60,31,30,15)(5,19,46,52,57,38,37,42,35,18,7,24,55,47,20,6)(10,17,44,25,56,50,36,12,11,16,43,45,27,58,49,32)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 1 1 0/1 (0/1,1/0) 0 16 1/7 1/0 1 2 1/6 (-1/1,0/1) 0 16 1/5 1/0 1 8 2/9 (-1/2,0/1) 0 16 3/13 0/1 3 1 1/4 (0/1,1/1) 0 16 2/7 (0/1,1/2) 0 16 3/10 (0/1,1/1) 0 16 1/3 1/0 1 4 2/5 (0/1,1/0) 0 16 3/7 -1/2 1 8 4/9 (-1/4,0/1) 0 16 5/11 0/1 5 1 1/2 (0/1,1/1) 0 16 6/11 (3/4,1/1) 0 16 5/9 3/2 1 8 4/7 (2/1,1/0) 0 16 3/5 1/0 3 2 2/3 (-1/1,1/0) 0 16 3/4 (-1/3,0/1) 0 16 7/9 0/1 7 1 4/5 (0/1,1/4) 0 16 5/6 (0/1,1/3) 0 16 1/1 1/0 1 8 5/4 (-1/3,0/1) 0 16 9/7 0/1 5 1 4/3 (0/1,1/2) 0 16 7/5 1/2 1 4 17/12 (2/3,1/1) 0 16 10/7 (1/1,1/0) 0 16 3/2 (1/1,2/1) 0 16 11/7 5/2 1 8 30/19 (23/8,3/1) 0 16 49/31 3/1 13 1 19/12 (3/1,16/5) 0 16 8/5 (4/1,1/0) 0 16 5/3 1/0 5 2 2/1 (-1/1,1/0) 0 16 11/5 -1/1 1 1 9/4 (-1/1,0/1) 0 16 16/7 (0/1,1/0) 0 16 7/3 1/0 1 8 12/5 (-1/1,-1/2) 0 16 17/7 -1/2 1 2 5/2 (-1/1,0/1) 0 16 13/5 1/2 1 8 8/3 (0/1,1/0) 0 16 19/7 1/2 1 8 30/11 (0/1,1/2) 0 16 11/4 (2/3,1/1) 0 16 3/1 1/0 2 4 13/4 (-10/3,-3/1) 0 16 10/3 (-5/2,-2/1) 0 16 17/5 1/0 1 8 7/2 (-3/1,-2/1) 0 16 4/1 (-3/2,-1/1) 0 16 13/3 -1/1 7 1 9/2 (-1/1,-4/5) 0 16 14/3 (-1/1,-3/4) 0 16 5/1 -1/2 1 8 6/1 (0/1,1/0) 0 16 7/1 1/0 2 2 8/1 (-2/1,1/0) 0 16 1/0 (-1/1,0/1) 0 16 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,14,-1) (0/1,1/7) -> (0/1,1/7) Reflection Matrix(137,-22,56,-9) (1/7,1/6) -> (17/7,5/2) Glide Reflection Matrix(137,-24,40,-7) (1/6,1/5) -> (17/5,7/2) Hyperbolic Matrix(211,-46,78,-17) (1/5,2/9) -> (8/3,19/7) Hyperbolic Matrix(53,-12,234,-53) (2/9,3/13) -> (2/9,3/13) Reflection Matrix(25,-6,104,-25) (3/13,1/4) -> (3/13,1/4) Reflection Matrix(51,-14,62,-17) (1/4,2/7) -> (4/5,5/6) Hyperbolic Matrix(223,-66,98,-29) (2/7,3/10) -> (9/4,16/7) Hyperbolic Matrix(121,-38,86,-27) (3/10,1/3) -> (7/5,17/12) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(223,-98,66,-29) (3/7,4/9) -> (10/3,17/5) Hyperbolic Matrix(89,-40,198,-89) (4/9,5/11) -> (4/9,5/11) Reflection Matrix(21,-10,44,-21) (5/11,1/2) -> (5/11,1/2) Reflection Matrix(207,-112,146,-79) (1/2,6/11) -> (17/12,10/7) Glide Reflection Matrix(391,-216,248,-137) (6/11,5/9) -> (11/7,30/19) Hyperbolic Matrix(163,-92,62,-35) (5/9,4/7) -> (13/5,8/3) Glide Reflection Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(55,-42,72,-55) (3/4,7/9) -> (3/4,7/9) Reflection Matrix(71,-56,90,-71) (7/9,4/5) -> (7/9,4/5) Reflection Matrix(69,-58,44,-37) (5/6,1/1) -> (3/2,11/7) Glide Reflection Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(1861,-2940,1178,-1861) (30/19,49/31) -> (30/19,49/31) Reflection Matrix(1177,-1862,744,-1177) (49/31,19/12) -> (49/31,19/12) Reflection Matrix(185,-294,56,-89) (19/12,8/5) -> (13/4,10/3) Glide Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(21,-44,10,-21) (2/1,11/5) -> (2/1,11/5) Reflection Matrix(89,-198,40,-89) (11/5,9/4) -> (11/5,9/4) Reflection Matrix(223,-514,82,-189) (16/7,7/3) -> (19/7,30/11) Hyperbolic Matrix(67,-158,14,-33) (7/3,12/5) -> (14/3,5/1) Hyperbolic Matrix(169,-408,70,-169) (12/5,17/7) -> (12/5,17/7) Reflection Matrix(43,-118,4,-11) (30/11,11/4) -> (8/1,1/0) Glide Reflection Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(53,-234,12,-53) (13/3,9/2) -> (13/3,9/2) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(15,-112,2,-15) (7/1,8/1) -> (7/1,8/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,14,-1) -> Matrix(1,0,0,-1) (0/1,1/7) -> (0/1,1/0) Matrix(137,-22,56,-9) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(137,-24,40,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(211,-46,78,-17) -> Matrix(1,0,2,1) 0/1 Matrix(53,-12,234,-53) -> Matrix(-1,0,4,1) (2/9,3/13) -> (-1/2,0/1) Matrix(25,-6,104,-25) -> Matrix(1,0,2,-1) (3/13,1/4) -> (0/1,1/1) Matrix(51,-14,62,-17) -> Matrix(1,0,2,1) 0/1 Matrix(223,-66,98,-29) -> Matrix(1,0,-2,1) 0/1 Matrix(121,-38,86,-27) -> Matrix(1,-2,2,-3) 1/1 Matrix(47,-18,34,-13) -> Matrix(1,0,2,1) 0/1 Matrix(67,-28,12,-5) -> Matrix(1,0,0,1) Matrix(223,-98,66,-29) -> Matrix(3,2,-2,-1) -1/1 Matrix(89,-40,198,-89) -> Matrix(-1,0,8,1) (4/9,5/11) -> (-1/4,0/1) Matrix(21,-10,44,-21) -> Matrix(1,0,2,-1) (5/11,1/2) -> (0/1,1/1) Matrix(207,-112,146,-79) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(391,-216,248,-137) -> Matrix(11,-14,4,-5) Matrix(163,-92,62,-35) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(41,-24,70,-41) -> Matrix(-1,4,0,1) (4/7,3/5) -> (2/1,1/0) Matrix(19,-12,30,-19) -> Matrix(1,2,0,-1) (3/5,2/3) -> (-1/1,1/0) Matrix(37,-26,10,-7) -> Matrix(3,2,-2,-1) -1/1 Matrix(55,-42,72,-55) -> Matrix(-1,0,6,1) (3/4,7/9) -> (-1/3,0/1) Matrix(71,-56,90,-71) -> Matrix(1,0,8,-1) (7/9,4/5) -> (0/1,1/4) Matrix(69,-58,44,-37) -> Matrix(5,-2,2,-1) Matrix(57,-70,22,-27) -> Matrix(1,0,2,1) 0/1 Matrix(71,-90,56,-71) -> Matrix(-1,0,6,1) (5/4,9/7) -> (-1/3,0/1) Matrix(55,-72,42,-55) -> Matrix(1,0,4,-1) (9/7,4/3) -> (0/1,1/2) Matrix(91,-132,20,-29) -> Matrix(3,-2,-4,3) Matrix(1861,-2940,1178,-1861) -> Matrix(47,-138,16,-47) (30/19,49/31) -> (23/8,3/1) Matrix(1177,-1862,744,-1177) -> Matrix(31,-96,10,-31) (49/31,19/12) -> (3/1,16/5) Matrix(185,-294,56,-89) -> Matrix(5,-18,-2,7) Matrix(49,-80,30,-49) -> Matrix(-1,8,0,1) (8/5,5/3) -> (4/1,1/0) Matrix(11,-20,6,-11) -> Matrix(1,2,0,-1) (5/3,2/1) -> (-1/1,1/0) Matrix(21,-44,10,-21) -> Matrix(1,2,0,-1) (2/1,11/5) -> (-1/1,1/0) Matrix(89,-198,40,-89) -> Matrix(-1,0,2,1) (11/5,9/4) -> (-1/1,0/1) Matrix(223,-514,82,-189) -> Matrix(1,0,2,1) 0/1 Matrix(67,-158,14,-33) -> Matrix(1,2,-2,-3) -1/1 Matrix(169,-408,70,-169) -> Matrix(3,2,-4,-3) (12/5,17/7) -> (-1/1,-1/2) Matrix(43,-118,4,-11) -> Matrix(3,-2,-2,1) Matrix(25,-72,8,-23) -> Matrix(1,-4,0,1) 1/0 Matrix(25,-104,6,-25) -> Matrix(5,6,-4,-5) (4/1,13/3) -> (-3/2,-1/1) Matrix(53,-234,12,-53) -> Matrix(9,8,-10,-9) (13/3,9/2) -> (-1/1,-4/5) Matrix(13,-84,2,-13) -> Matrix(1,0,0,-1) (6/1,7/1) -> (0/1,1/0) Matrix(15,-112,2,-15) -> Matrix(1,4,0,-1) (7/1,8/1) -> (-2/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.