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: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/6 -3/4 -2/3 -23/36 -5/9 -1/2 -11/24 -3/8 -1/3 -11/36 -3/10 -7/24 -1/4 -2/9 -5/24 -2/11 -1/6 -1/8 0/1 1/7 1/6 1/5 2/9 1/4 3/11 2/7 3/10 1/3 3/8 2/5 5/12 7/16 1/2 5/9 7/12 2/3 3/4 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -6/7 1/1 -5/6 1/0 -4/5 -1/1 -11/14 1/0 -18/23 -5/1 -7/9 -2/1 0/1 -3/4 -1/1 -11/15 -2/3 0/1 -8/11 -1/1 -13/18 -1/1 -5/7 -1/1 -2/3 -7/10 -1/2 -2/3 -1/2 -9/14 -1/2 -16/25 -5/9 -23/36 -1/2 -7/11 -1/2 -4/9 -5/8 -2/5 -8/13 -1/3 -3/5 -2/5 -1/3 -7/12 -1/3 -4/7 -1/3 -13/23 -1/3 -2/7 -9/16 -2/7 -14/25 -5/19 -5/9 -2/7 0/1 -6/11 -1/3 -1/2 -1/4 -6/13 -1/5 -11/24 0/1 -5/11 -1/4 0/1 -4/9 -1/4 -11/25 -3/13 -2/9 -7/16 -2/9 -3/7 -2/9 -1/5 -5/12 -1/5 -2/5 -1/5 -7/18 -1/6 -5/13 -2/11 -1/6 -3/8 0/1 -4/11 -1/5 -1/3 -2/11 0/1 -4/13 -1/5 -11/36 -1/5 -7/23 -1/5 -2/11 -3/10 -1/6 -5/17 -1/7 0/1 -7/24 0/1 -2/7 -1/5 -5/18 -1/6 -3/11 -1/6 0/1 -1/4 -1/6 -3/13 -1/6 -2/13 -2/9 -1/6 -5/23 -2/13 -1/7 -3/14 -1/6 -4/19 -3/19 -5/24 -2/13 -1/5 -2/13 -1/7 -2/11 -1/7 -1/6 -1/7 -1/7 -1/7 -2/15 -1/8 -2/15 0/1 -1/9 1/7 -2/21 -1/11 1/6 -1/11 1/5 -1/11 -2/23 3/14 -1/12 5/23 -1/11 -2/23 2/9 -1/12 1/4 -1/12 4/15 -1/12 3/11 -1/12 0/1 5/18 -1/12 2/7 -1/13 3/10 -1/12 1/3 -2/25 0/1 5/14 -1/12 9/25 -2/25 -1/13 13/36 -1/13 4/11 -1/13 3/8 0/1 5/13 -1/12 -2/25 2/5 -1/13 5/12 -1/13 3/7 -1/13 -2/27 10/23 -7/93 7/16 -2/27 11/25 -2/27 -3/41 4/9 -1/14 5/11 -1/14 0/1 1/2 -1/14 7/13 -1/14 -2/29 13/24 -2/29 6/11 -1/15 5/9 -2/29 0/1 14/25 -5/71 9/16 -2/29 4/7 -1/15 7/12 -1/15 3/5 -1/15 -2/31 11/18 -1/15 8/13 -1/15 5/8 -2/31 7/11 -4/63 -1/16 2/3 -1/16 9/13 -2/31 -1/16 25/36 -1/16 16/23 -7/113 7/10 -1/16 12/17 -3/49 17/24 -2/33 5/7 -2/33 -1/17 13/18 -1/17 8/11 -1/17 3/4 -1/17 10/13 -1/17 7/9 -2/35 0/1 18/23 -5/89 11/14 -1/18 15/19 -1/19 0/1 19/24 0/1 4/5 -1/17 9/11 -1/18 0/1 5/6 -1/18 6/7 -1/19 7/8 0/1 1/1 -1/19 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(95,82,-168,-145) (-1/1,-6/7) -> (-4/7,-13/23) Hyperbolic Matrix(47,40,168,143) (-6/7,-5/6) -> (5/18,2/7) Hyperbolic Matrix(47,38,-120,-97) (-5/6,-4/5) -> (-2/5,-7/18) Hyperbolic Matrix(71,56,-336,-265) (-4/5,-11/14) -> (-3/14,-4/19) Hyperbolic Matrix(385,302,552,433) (-11/14,-18/23) -> (16/23,7/10) Hyperbolic Matrix(241,188,432,337) (-18/23,-7/9) -> (5/9,14/25) Hyperbolic Matrix(71,54,-96,-73) (-7/9,-3/4) -> (-3/4,-11/15) Parabolic Matrix(145,106,264,193) (-11/15,-8/11) -> (6/11,5/9) Hyperbolic Matrix(265,192,432,313) (-8/11,-13/18) -> (11/18,8/13) Hyperbolic Matrix(25,18,168,121) (-13/18,-5/7) -> (1/7,1/6) Hyperbolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(527,338,672,431) (-9/14,-16/25) -> (18/23,11/14) Hyperbolic Matrix(1201,768,1728,1105) (-16/25,-23/36) -> (25/36,16/23) Hyperbolic Matrix(599,382,864,551) (-23/36,-7/11) -> (9/13,25/36) Hyperbolic Matrix(73,46,192,121) (-7/11,-5/8) -> (3/8,5/13) Hyperbolic Matrix(71,44,192,119) (-5/8,-8/13) -> (4/11,3/8) Hyperbolic Matrix(23,14,-120,-73) (-8/13,-3/5) -> (-1/5,-2/11) Hyperbolic Matrix(71,42,120,71) (-3/5,-7/12) -> (7/12,3/5) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(337,190,768,433) (-13/23,-9/16) -> (7/16,11/25) Hyperbolic Matrix(335,188,768,431) (-9/16,-14/25) -> (10/23,7/16) Hyperbolic Matrix(337,188,432,241) (-14/25,-5/9) -> (7/9,18/23) Hyperbolic Matrix(167,92,216,119) (-5/9,-6/11) -> (10/13,7/9) Hyperbolic Matrix(23,12,-48,-25) (-6/11,-1/2) -> (-1/2,-6/13) Parabolic Matrix(313,144,576,265) (-6/13,-11/24) -> (13/24,6/11) Hyperbolic Matrix(311,142,576,263) (-11/24,-5/11) -> (7/13,13/24) Hyperbolic Matrix(71,32,264,119) (-5/11,-4/9) -> (4/15,3/11) Hyperbolic Matrix(95,42,432,191) (-4/9,-11/25) -> (5/23,2/9) Hyperbolic Matrix(191,84,216,95) (-11/25,-7/16) -> (7/8,1/1) Hyperbolic Matrix(23,10,-168,-73) (-7/16,-3/7) -> (-1/7,-1/8) Hyperbolic Matrix(71,30,168,71) (-3/7,-5/12) -> (5/12,3/7) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(119,46,432,167) (-7/18,-5/13) -> (3/11,5/18) Hyperbolic Matrix(121,46,192,73) (-5/13,-3/8) -> (5/8,7/11) Hyperbolic Matrix(119,44,192,71) (-3/8,-4/11) -> (8/13,5/8) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(313,96,864,265) (-4/13,-11/36) -> (13/36,4/11) Hyperbolic Matrix(623,190,1728,527) (-11/36,-7/23) -> (9/25,13/36) Hyperbolic Matrix(119,36,552,167) (-7/23,-3/10) -> (3/14,5/23) Hyperbolic Matrix(409,120,576,169) (-5/17,-7/24) -> (17/24,5/7) Hyperbolic Matrix(407,118,576,167) (-7/24,-2/7) -> (12/17,17/24) Hyperbolic Matrix(143,40,168,47) (-2/7,-5/18) -> (5/6,6/7) Hyperbolic Matrix(217,60,264,73) (-5/18,-3/11) -> (9/11,5/6) Hyperbolic Matrix(23,6,-96,-25) (-3/11,-1/4) -> (-1/4,-3/13) Parabolic Matrix(97,22,216,49) (-3/13,-2/9) -> (4/9,5/11) Hyperbolic Matrix(191,42,432,95) (-2/9,-5/23) -> (11/25,4/9) Hyperbolic Matrix(241,52,672,145) (-5/23,-3/14) -> (5/14,9/25) Hyperbolic Matrix(457,96,576,121) (-4/19,-5/24) -> (19/24,4/5) Hyperbolic Matrix(455,94,576,119) (-5/24,-1/5) -> (15/19,19/24) Hyperbolic Matrix(191,34,264,47) (-2/11,-1/6) -> (13/18,8/11) Hyperbolic Matrix(121,18,168,25) (-1/6,-1/7) -> (5/7,13/18) Hyperbolic Matrix(121,14,216,25) (-1/8,0/1) -> (14/25,9/16) Hyperbolic Matrix(73,-10,168,-23) (0/1,1/7) -> (3/7,10/23) Hyperbolic Matrix(73,-14,120,-23) (1/6,1/5) -> (3/5,11/18) Hyperbolic Matrix(265,-56,336,-71) (1/5,3/14) -> (11/14,15/19) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(145,-82,168,-95) (9/16,4/7) -> (6/7,7/8) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-20,1) Matrix(95,82,-168,-145) -> Matrix(3,-2,-10,7) Matrix(47,40,168,143) -> Matrix(1,-2,-12,25) Matrix(47,38,-120,-97) -> Matrix(1,2,-6,-11) Matrix(71,56,-336,-265) -> Matrix(1,-2,-6,13) Matrix(385,302,552,433) -> Matrix(1,-2,-16,33) Matrix(241,188,432,337) -> Matrix(1,0,-14,1) Matrix(71,54,-96,-73) -> Matrix(1,2,-2,-3) Matrix(145,106,264,193) -> Matrix(3,2,-44,-29) Matrix(265,192,432,313) -> Matrix(1,2,-16,-31) Matrix(25,18,168,121) -> Matrix(5,4,-54,-43) Matrix(71,50,-240,-169) -> Matrix(3,2,-20,-13) Matrix(47,32,-72,-49) -> Matrix(3,2,-8,-5) Matrix(527,338,672,431) -> Matrix(1,0,-16,1) Matrix(1201,768,1728,1105) -> Matrix(23,12,-370,-193) Matrix(599,382,864,551) -> Matrix(13,6,-206,-95) Matrix(73,46,192,121) -> Matrix(5,2,-58,-23) Matrix(71,44,192,119) -> Matrix(5,2,-68,-27) Matrix(23,14,-120,-73) -> Matrix(1,0,-4,1) Matrix(71,42,120,71) -> Matrix(11,4,-168,-61) Matrix(97,56,168,97) -> Matrix(19,6,-282,-89) Matrix(337,190,768,433) -> Matrix(27,8,-368,-109) Matrix(335,188,768,431) -> Matrix(29,8,-388,-107) Matrix(337,188,432,241) -> Matrix(1,0,-14,1) Matrix(167,92,216,119) -> Matrix(1,0,-14,1) Matrix(23,12,-48,-25) -> Matrix(7,2,-32,-9) Matrix(313,144,576,265) -> Matrix(11,2,-160,-29) Matrix(311,142,576,263) -> Matrix(7,2,-102,-29) Matrix(71,32,264,119) -> Matrix(1,0,-8,1) Matrix(95,42,432,191) -> Matrix(17,4,-200,-47) Matrix(191,84,216,95) -> Matrix(9,2,-158,-35) Matrix(23,10,-168,-73) -> Matrix(19,4,-138,-29) Matrix(71,30,168,71) -> Matrix(19,4,-252,-53) Matrix(49,20,120,49) -> Matrix(11,2,-138,-25) Matrix(119,46,432,167) -> Matrix(11,2,-138,-25) Matrix(121,46,192,73) -> Matrix(13,2,-202,-31) Matrix(119,44,192,71) -> Matrix(9,2,-140,-31) Matrix(23,8,-72,-25) -> Matrix(1,0,0,1) Matrix(313,96,864,265) -> Matrix(11,2,-138,-25) Matrix(623,190,1728,527) -> Matrix(21,4,-268,-51) Matrix(119,36,552,167) -> Matrix(1,0,-6,1) Matrix(409,120,576,169) -> Matrix(15,2,-248,-33) Matrix(407,118,576,167) -> Matrix(7,2,-116,-33) Matrix(143,40,168,47) -> Matrix(11,2,-204,-37) Matrix(217,60,264,73) -> Matrix(1,0,-12,1) Matrix(23,6,-96,-25) -> Matrix(11,2,-72,-13) Matrix(97,22,216,49) -> Matrix(13,2,-176,-27) Matrix(191,42,432,95) -> Matrix(25,4,-344,-55) Matrix(241,52,672,145) -> Matrix(1,0,-6,1) Matrix(457,96,576,121) -> Matrix(13,2,-202,-31) Matrix(455,94,576,119) -> Matrix(13,2,-254,-39) Matrix(191,34,264,47) -> Matrix(13,2,-228,-35) Matrix(121,18,168,25) -> Matrix(29,4,-486,-67) Matrix(121,14,216,25) -> Matrix(31,4,-442,-57) Matrix(73,-10,168,-23) -> Matrix(43,4,-570,-53) Matrix(73,-14,120,-23) -> Matrix(1,0,-4,1) Matrix(265,-56,336,-71) -> Matrix(23,2,-426,-37) Matrix(25,-6,96,-23) -> Matrix(23,2,-288,-25) Matrix(169,-50,240,-71) -> Matrix(23,2,-380,-33) Matrix(25,-8,72,-23) -> Matrix(1,0,0,1) Matrix(97,-38,120,-47) -> Matrix(25,2,-438,-35) Matrix(25,-12,48,-23) -> Matrix(27,2,-392,-29) Matrix(145,-82,168,-95) -> Matrix(29,2,-566,-39) Matrix(49,-32,72,-47) -> Matrix(31,2,-512,-33) Matrix(73,-54,96,-71) -> Matrix(33,2,-578,-35) 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: ((1,4,16,49,57,59,62,60,50,17,5,2)(3,10,39,11)(6,22,32,31,51,61,35,9,34,14,13,23)(7,28,29,8)(12,26,25,58,46,15,38,53,20,19,33,41)(18,52,44,43)(21,54,64,45,27,37,36,56,63,42,30,40)(24,55,48,47); (1,2,8,32,58,52,64,54,55,33,9,3)(4,14,45,15)(5,20,21,6)(7,26,12,11,40,16,48,61,51,18,17,27)(10,37,50,24,23,13,44,49,30,29,53,38)(19,43,63,56,47,46,22,39,62,59,28,34)(25,36,35,57)(31,42,41,60); (2,6,24,56,25,7)(3,12,42,43,13,4)(5,18,19)(8,30,31)(9,36,10)(11,22,21)(14,28,27)(15,47,16)(17,37)(20,29,59,35,48,54)(32,46)(33,34)(38,45,52,51,60,39)(40,49)(41,55,50)(44,58,57)) ----------------------------------------------------------------------- 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 0/1 -1/9 1 12 1/7 (-2/21,-1/11) 0 12 1/6 -1/11 3 2 1/5 (-1/11,-2/23) 0 12 3/14 -1/12 1 6 5/23 (-1/11,-2/23) 0 12 2/9 -1/12 1 4 1/4 -1/12 1 3 4/15 -1/12 1 4 3/11 (-1/12,0/1) 0 12 5/18 -1/12 1 2 2/7 -1/13 1 12 3/10 -1/12 1 6 1/3 0 4 5/14 -1/12 1 6 9/25 (-2/25,-1/13) 0 12 13/36 -1/13 1 1 4/11 -1/13 1 12 3/8 0/1 1 3 5/13 (-1/12,-2/25) 0 12 2/5 -1/13 1 12 5/12 -1/13 3 1 3/7 (-1/13,-2/27) 0 12 10/23 -7/93 1 12 7/16 -2/27 2 3 11/25 (-2/27,-3/41) 0 12 4/9 -1/14 1 4 5/11 (-1/14,0/1) 0 12 1/2 -1/14 1 6 7/13 (-1/14,-2/29) 0 12 13/24 -2/29 1 1 6/11 -1/15 1 12 5/9 0 4 14/25 -5/71 1 12 9/16 -2/29 2 3 4/7 -1/15 1 12 7/12 -1/15 5 1 3/5 (-1/15,-2/31) 0 12 11/18 -1/15 3 2 8/13 -1/15 1 12 5/8 -2/31 1 3 7/11 (-4/63,-1/16) 0 12 2/3 -1/16 1 4 9/13 (-2/31,-1/16) 0 12 25/36 -1/16 9 1 16/23 -7/113 1 12 7/10 -1/16 1 6 12/17 -3/49 1 12 17/24 -2/33 2 1 5/7 (-2/33,-1/17) 0 12 13/18 -1/17 3 2 8/11 -1/17 1 12 3/4 -1/17 1 3 10/13 -1/17 1 12 7/9 0 4 18/23 -5/89 1 12 11/14 -1/18 1 6 15/19 (-1/19,0/1) 0 12 19/24 0/1 2 1 4/5 -1/17 1 12 9/11 (-1/18,0/1) 0 12 5/6 -1/18 1 2 6/7 -1/19 1 12 7/8 0/1 2 3 1/1 (-1/19,0/1) 0 12 1/0 0/1 10 1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(73,-10,168,-23) (0/1,1/7) -> (3/7,10/23) Hyperbolic Matrix(121,-18,168,-25) (1/7,1/6) -> (5/7,13/18) Glide Reflection Matrix(73,-14,120,-23) (1/6,1/5) -> (3/5,11/18) Hyperbolic Matrix(265,-56,336,-71) (1/5,3/14) -> (11/14,15/19) Hyperbolic Matrix(241,-52,672,-145) (3/14,5/23) -> (5/14,9/25) Glide Reflection Matrix(191,-42,432,-95) (5/23,2/9) -> (11/25,4/9) Glide Reflection Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(119,-32,264,-71) (4/15,3/11) -> (4/9,5/11) Glide Reflection Matrix(217,-60,264,-73) (3/11,5/18) -> (9/11,5/6) Glide Reflection Matrix(143,-40,168,-47) (5/18,2/7) -> (5/6,6/7) Glide Reflection Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(649,-234,1800,-649) (9/25,13/36) -> (9/25,13/36) Reflection Matrix(287,-104,792,-287) (13/36,4/11) -> (13/36,4/11) Reflection Matrix(119,-44,192,-71) (4/11,3/8) -> (8/13,5/8) Glide Reflection Matrix(121,-46,192,-73) (3/8,5/13) -> (5/8,7/11) Glide Reflection Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(431,-188,768,-335) (10/23,7/16) -> (14/25,9/16) Glide Reflection Matrix(191,-84,216,-95) (7/16,11/25) -> (7/8,1/1) Glide Reflection Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(337,-182,624,-337) (7/13,13/24) -> (7/13,13/24) Reflection Matrix(287,-156,528,-287) (13/24,6/11) -> (13/24,6/11) Reflection Matrix(167,-92,216,-119) (6/11,5/9) -> (10/13,7/9) Glide Reflection Matrix(337,-188,432,-241) (5/9,14/25) -> (7/9,18/23) Glide Reflection Matrix(145,-82,168,-95) (9/16,4/7) -> (6/7,7/8) Hyperbolic Matrix(97,-56,168,-97) (4/7,7/12) -> (4/7,7/12) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(313,-192,432,-265) (11/18,8/13) -> (13/18,8/11) Glide Reflection Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(649,-450,936,-649) (9/13,25/36) -> (9/13,25/36) Reflection Matrix(1151,-800,1656,-1151) (25/36,16/23) -> (25/36,16/23) Reflection Matrix(433,-302,552,-385) (16/23,7/10) -> (18/23,11/14) Glide Reflection Matrix(577,-408,816,-577) (12/17,17/24) -> (12/17,17/24) Reflection Matrix(239,-170,336,-239) (17/24,5/7) -> (17/24,5/7) Reflection Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(721,-570,912,-721) (15/19,19/24) -> (15/19,19/24) Reflection Matrix(191,-152,240,-191) (19/24,4/5) -> (19/24,4/5) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,18,1) (0/1,1/0) -> (-1/9,0/1) Matrix(73,-10,168,-23) -> Matrix(43,4,-570,-53) Matrix(121,-18,168,-25) -> Matrix(43,4,-720,-67) Matrix(73,-14,120,-23) -> Matrix(1,0,-4,1) 0/1 Matrix(265,-56,336,-71) -> Matrix(23,2,-426,-37) Matrix(241,-52,672,-145) -> Matrix(-1,0,24,1) *** -> (-1/12,0/1) Matrix(191,-42,432,-95) -> Matrix(47,4,-646,-55) Matrix(25,-6,96,-23) -> Matrix(23,2,-288,-25) -1/12 Matrix(119,-32,264,-71) -> Matrix(-1,0,26,1) *** -> (-1/13,0/1) Matrix(217,-60,264,-73) -> Matrix(-1,0,30,1) *** -> (-1/15,0/1) Matrix(143,-40,168,-47) -> Matrix(25,2,-462,-37) Matrix(169,-50,240,-71) -> Matrix(23,2,-380,-33) Matrix(25,-8,72,-23) -> Matrix(1,0,0,1) Matrix(649,-234,1800,-649) -> Matrix(51,4,-650,-51) (9/25,13/36) -> (-2/25,-1/13) Matrix(287,-104,792,-287) -> Matrix(25,2,-312,-25) (13/36,4/11) -> (-1/12,-1/13) Matrix(119,-44,192,-71) -> Matrix(27,2,-418,-31) Matrix(121,-46,192,-73) -> Matrix(23,2,-356,-31) Matrix(97,-38,120,-47) -> Matrix(25,2,-438,-35) Matrix(49,-20,120,-49) -> Matrix(25,2,-312,-25) (2/5,5/12) -> (-1/12,-1/13) Matrix(71,-30,168,-71) -> Matrix(53,4,-702,-53) (5/12,3/7) -> (-1/13,-2/27) Matrix(431,-188,768,-335) -> Matrix(107,8,-1538,-115) Matrix(191,-84,216,-95) -> Matrix(27,2,-472,-35) Matrix(25,-12,48,-23) -> Matrix(27,2,-392,-29) -1/14 Matrix(337,-182,624,-337) -> Matrix(57,4,-812,-57) (7/13,13/24) -> (-1/14,-2/29) Matrix(287,-156,528,-287) -> Matrix(59,4,-870,-59) (13/24,6/11) -> (-2/29,-1/15) Matrix(167,-92,216,-119) -> Matrix(-1,0,32,1) *** -> (-1/16,0/1) Matrix(337,-188,432,-241) -> Matrix(-1,0,32,1) *** -> (-1/16,0/1) Matrix(145,-82,168,-95) -> Matrix(29,2,-566,-39) Matrix(97,-56,168,-97) -> Matrix(89,6,-1320,-89) (4/7,7/12) -> (-3/44,-1/15) Matrix(71,-42,120,-71) -> Matrix(61,4,-930,-61) (7/12,3/5) -> (-1/15,-2/31) Matrix(313,-192,432,-265) -> Matrix(31,2,-542,-35) Matrix(49,-32,72,-47) -> Matrix(31,2,-512,-33) -1/16 Matrix(649,-450,936,-649) -> Matrix(63,4,-992,-63) (9/13,25/36) -> (-2/31,-1/16) Matrix(1151,-800,1656,-1151) -> Matrix(225,14,-3616,-225) (25/36,16/23) -> (-1/16,-7/113) Matrix(433,-302,552,-385) -> Matrix(33,2,-610,-37) Matrix(577,-408,816,-577) -> Matrix(197,12,-3234,-197) (12/17,17/24) -> (-3/49,-2/33) Matrix(239,-170,336,-239) -> Matrix(67,4,-1122,-67) (17/24,5/7) -> (-2/33,-1/17) Matrix(73,-54,96,-71) -> Matrix(33,2,-578,-35) -1/17 Matrix(721,-570,912,-721) -> Matrix(-1,0,38,1) (15/19,19/24) -> (-1/19,0/1) Matrix(191,-152,240,-191) -> Matrix(-1,0,34,1) (19/24,4/5) -> (-1/17,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,38,1) (1/1,1/0) -> (-1/19,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.