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): 432 Minimal number of generators: 73 Number of equivalence classes of cusps: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -3/1 -8/3 -2/1 -9/5 -27/17 -3/2 -15/11 -9/7 -6/5 -9/8 0/1 1/1 6/5 9/7 15/11 18/13 3/2 27/17 18/11 9/5 2/1 9/4 5/2 18/7 8/3 3/1 17/5 7/2 18/5 11/3 72/19 4/1 13/3 9/2 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/0 -5/1 1/1 1/0 -9/2 1/0 -13/3 -2/1 1/0 -17/4 -1/1 -4/1 -1/1 1/1 -11/3 2/1 1/0 -18/5 1/0 -7/2 -1/1 -3/1 0/1 1/1 1/0 -11/4 -1/1 -19/7 -1/1 0/1 -27/10 0/1 -8/3 -1/1 1/1 -13/5 -1/2 0/1 -18/7 0/1 -5/2 1/1 -17/7 1/1 2/1 -12/5 1/0 -19/8 -1/1 -7/3 0/1 1/0 -9/4 0/1 -11/5 0/1 1/2 -2/1 -1/1 1/1 -9/5 1/0 -16/9 -3/1 -1/1 -23/13 0/1 1/0 -7/4 1/1 -19/11 0/1 1/1 -12/7 1/0 -5/3 1/1 1/0 -18/11 1/0 -13/8 -5/1 -34/21 -11/3 -3/1 -21/13 -3/1 -2/1 1/0 -8/5 -3/1 -1/1 -27/17 -1/1 -19/12 -1/1 -11/7 0/1 1/0 -14/9 -3/1 -1/1 -17/11 -4/3 -1/1 -3/2 -1/1 1/1 -19/13 -4/3 -1/1 -16/11 -1/1 -3/5 -13/9 -1/2 0/1 -10/7 -1/1 1/1 -27/19 -1/1 -17/12 -1/1 -7/5 -1/2 0/1 -18/13 0/1 -11/8 1/3 -26/19 1/3 1/1 -15/11 0/1 1/2 1/1 -34/25 5/7 1/1 -53/39 8/9 1/1 -72/53 1/1 -19/14 1/1 -4/3 -1/1 1/1 -9/7 0/1 -14/11 1/3 1/1 -19/15 1/1 2/1 -5/4 1/1 -16/13 -1/1 -1/3 -11/9 0/1 1/0 -17/14 -1/1 -6/5 0/1 -19/16 1/1 -13/11 0/1 1/0 -7/6 -1/1 -8/7 -1/1 1/1 -9/8 0/1 -1/1 0/1 1/0 0/1 0/1 1/1 0/1 1/2 8/7 1/3 1/1 7/6 1/3 13/11 0/1 1/2 6/5 0/1 11/9 0/1 1/2 5/4 1/1 9/7 0/1 13/10 1/3 4/3 1/3 1/1 19/14 1/1 15/11 0/1 1/1 1/0 11/8 -1/1 18/13 0/1 7/5 0/1 1/4 17/12 1/3 27/19 1/3 10/7 1/3 1/1 3/2 1/3 1/1 11/7 0/1 1/2 30/19 0/1 19/12 1/3 27/17 1/3 8/5 1/3 3/7 29/18 1/3 21/13 2/5 3/7 1/2 55/34 3/7 34/21 3/7 11/25 13/8 5/11 18/11 1/2 5/3 1/2 1/1 12/7 1/2 19/11 0/1 1/1 7/4 1/1 9/5 1/2 11/6 3/5 13/7 1/2 2/3 2/1 1/3 1/1 13/6 -1/1 11/5 0/1 1/0 9/4 0/1 7/3 0/1 1/2 12/5 1/2 17/7 2/3 1/1 22/9 -1/1 1/1 5/2 1/1 18/7 0/1 13/5 0/1 1/4 34/13 3/11 1/3 21/8 1/3 1/1 8/3 1/3 1/1 3/1 0/1 1/2 1/1 10/3 1/3 1/1 37/11 0/1 1/2 27/8 0/1 17/5 0/1 1/3 7/2 1/3 18/5 1/2 11/3 1/2 2/3 15/4 3/5 1/1 34/9 7/9 1/1 53/14 1/1 72/19 1/1 19/5 1/1 2/1 4/1 1/3 1/1 17/4 1/3 30/7 2/5 13/3 2/5 1/2 9/2 1/2 5/1 1/2 1/1 6/1 1/2 7/1 1/2 2/3 1/0 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,126,-8,-53) (-6/1,1/0) -> (-12/5,-19/8) Hyperbolic Matrix(17,90,10,53) (-6/1,-5/1) -> (5/3,12/7) Hyperbolic Matrix(19,90,4,19) (-5/1,-9/2) -> (9/2,5/1) Hyperbolic Matrix(53,234,12,53) (-9/2,-13/3) -> (13/3,9/2) Hyperbolic Matrix(109,468,-92,-395) (-13/3,-17/4) -> (-19/16,-13/11) Hyperbolic Matrix(35,144,26,107) (-17/4,-4/1) -> (4/3,19/14) Hyperbolic Matrix(53,198,-34,-127) (-4/1,-11/3) -> (-11/7,-14/9) Hyperbolic Matrix(109,396,30,109) (-11/3,-18/5) -> (18/5,11/3) Hyperbolic Matrix(71,252,20,71) (-18/5,-7/2) -> (7/2,18/5) Hyperbolic Matrix(17,54,-6,-19) (-7/2,-3/1) -> (-3/1,-11/4) Parabolic Matrix(125,342,72,197) (-11/4,-19/7) -> (19/11,7/4) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(107,288,-94,-253) (-27/10,-8/3) -> (-8/7,-9/8) Hyperbolic Matrix(89,234,-62,-163) (-8/3,-13/5) -> (-13/9,-10/7) Hyperbolic Matrix(181,468,70,181) (-13/5,-18/7) -> (18/7,13/5) Hyperbolic Matrix(71,180,28,71) (-18/7,-5/2) -> (5/2,18/7) Hyperbolic Matrix(73,180,-58,-143) (-5/2,-17/7) -> (-19/15,-5/4) Hyperbolic Matrix(179,432,104,251) (-17/7,-12/5) -> (12/7,19/11) Hyperbolic Matrix(107,252,76,179) (-19/8,-7/3) -> (7/5,17/12) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(89,198,40,89) (-9/4,-11/5) -> (11/5,9/4) Hyperbolic Matrix(91,198,-74,-161) (-11/5,-2/1) -> (-16/13,-11/9) Hyperbolic Matrix(89,162,-50,-91) (-2/1,-9/5) -> (-9/5,-16/9) Parabolic Matrix(559,990,214,379) (-16/9,-23/13) -> (13/5,34/13) Hyperbolic Matrix(143,252,122,215) (-23/13,-7/4) -> (7/6,13/11) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(251,432,104,179) (-19/11,-12/7) -> (12/5,17/7) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(109,180,66,109) (-5/3,-18/11) -> (18/11,5/3) Hyperbolic Matrix(287,468,176,287) (-18/11,-13/8) -> (13/8,18/11) Hyperbolic Matrix(289,468,134,217) (-13/8,-34/21) -> (2/1,13/6) Hyperbolic Matrix(757,1224,-556,-899) (-34/21,-21/13) -> (-15/11,-34/25) Hyperbolic Matrix(413,666,-302,-487) (-21/13,-8/5) -> (-26/19,-15/11) Hyperbolic Matrix(305,486,214,341) (-8/5,-27/17) -> (27/19,10/7) Hyperbolic Matrix(613,972,432,685) (-27/17,-19/12) -> (17/12,27/19) Hyperbolic Matrix(251,396,-206,-325) (-19/12,-11/7) -> (-11/9,-17/14) Hyperbolic Matrix(395,612,162,251) (-14/9,-17/11) -> (17/7,22/9) Hyperbolic Matrix(71,108,-48,-73) (-17/11,-3/2) -> (-3/2,-19/13) Parabolic Matrix(1025,1494,-754,-1099) (-19/13,-16/11) -> (-34/25,-53/39) Hyperbolic Matrix(161,234,86,125) (-16/11,-13/9) -> (13/7,2/1) Hyperbolic Matrix(341,486,214,305) (-10/7,-27/19) -> (27/17,8/5) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(89,126,12,17) (-17/12,-7/5) -> (7/1,1/0) Hyperbolic Matrix(181,252,130,181) (-7/5,-18/13) -> (18/13,7/5) Hyperbolic Matrix(287,396,208,287) (-18/13,-11/8) -> (11/8,18/13) Hyperbolic Matrix(197,270,170,233) (-11/8,-26/19) -> (8/7,7/6) Hyperbolic Matrix(3815,5184,1006,1367) (-53/39,-72/53) -> (72/19,19/5) Hyperbolic Matrix(3817,5184,1008,1369) (-72/53,-19/14) -> (53/14,72/19) Hyperbolic Matrix(107,144,26,35) (-19/14,-4/3) -> (4/1,17/4) Hyperbolic Matrix(125,162,-98,-127) (-4/3,-9/7) -> (-9/7,-14/11) Parabolic Matrix(269,342,70,89) (-14/11,-19/15) -> (19/5,4/1) Hyperbolic Matrix(305,378,188,233) (-5/4,-16/13) -> (34/21,13/8) Hyperbolic Matrix(179,216,-150,-181) (-17/14,-6/5) -> (-6/5,-19/16) Parabolic Matrix(199,234,108,127) (-13/11,-7/6) -> (11/6,13/7) Hyperbolic Matrix(251,288,156,179) (-7/6,-8/7) -> (8/5,29/18) Hyperbolic Matrix(323,360,96,107) (-9/8,-1/1) -> (37/11,27/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(269,-306,80,-91) (1/1,8/7) -> (10/3,37/11) Hyperbolic Matrix(395,-468,92,-109) (13/11,6/5) -> (30/7,13/3) Hyperbolic Matrix(325,-396,206,-251) (6/5,11/9) -> (11/7,30/19) Hyperbolic Matrix(161,-198,74,-91) (11/9,5/4) -> (13/6,11/5) Hyperbolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(235,-306,96,-125) (13/10,4/3) -> (22/9,5/2) Hyperbolic Matrix(899,-1224,556,-757) (19/14,15/11) -> (21/13,55/34) Hyperbolic Matrix(487,-666,302,-413) (15/11,11/8) -> (29/18,21/13) Hyperbolic Matrix(163,-234,62,-89) (10/7,3/2) -> (21/8,8/3) Hyperbolic Matrix(127,-198,34,-53) (3/2,11/7) -> (11/3,15/4) Hyperbolic Matrix(683,-1080,160,-253) (30/19,19/12) -> (17/4,30/7) Hyperbolic Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(467,-1224,124,-325) (34/13,21/8) -> (15/4,34/9) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,126,-8,-53) -> Matrix(1,-2,0,1) Matrix(17,90,10,53) -> Matrix(1,-2,2,-3) Matrix(19,90,4,19) -> Matrix(1,-2,2,-3) Matrix(53,234,12,53) -> Matrix(1,4,2,9) Matrix(109,468,-92,-395) -> Matrix(1,2,0,1) Matrix(35,144,26,107) -> Matrix(1,0,2,1) Matrix(53,198,-34,-127) -> Matrix(1,-2,0,1) Matrix(109,396,30,109) -> Matrix(1,-4,2,-7) Matrix(71,252,20,71) -> Matrix(1,2,2,5) Matrix(17,54,-6,-19) -> Matrix(1,0,0,1) Matrix(125,342,72,197) -> Matrix(1,0,2,1) Matrix(359,972,106,287) -> Matrix(1,0,4,1) Matrix(107,288,-94,-253) -> Matrix(1,0,0,1) Matrix(89,234,-62,-163) -> Matrix(1,0,0,1) Matrix(181,468,70,181) -> Matrix(1,0,6,1) Matrix(71,180,28,71) -> Matrix(1,0,0,1) Matrix(73,180,-58,-143) -> Matrix(1,0,0,1) Matrix(179,432,104,251) -> Matrix(1,-2,2,-3) Matrix(107,252,76,179) -> Matrix(1,0,4,1) Matrix(55,126,24,55) -> Matrix(1,0,2,1) Matrix(89,198,40,89) -> Matrix(1,0,-2,1) Matrix(91,198,-74,-161) -> Matrix(1,0,-2,1) Matrix(89,162,-50,-91) -> Matrix(1,-2,0,1) Matrix(559,990,214,379) -> Matrix(1,0,4,1) Matrix(143,252,122,215) -> Matrix(1,0,2,1) Matrix(145,252,42,73) -> Matrix(1,0,2,1) Matrix(251,432,104,179) -> Matrix(1,-2,2,-3) Matrix(53,90,10,17) -> Matrix(1,-2,2,-3) Matrix(109,180,66,109) -> Matrix(1,-2,2,-3) Matrix(287,468,176,287) -> Matrix(1,10,2,21) Matrix(289,468,134,217) -> Matrix(1,4,0,1) Matrix(757,1224,-556,-899) -> Matrix(1,2,2,5) Matrix(413,666,-302,-487) -> Matrix(1,2,2,5) Matrix(305,486,214,341) -> Matrix(1,2,2,5) Matrix(613,972,432,685) -> Matrix(1,0,4,1) Matrix(251,396,-206,-325) -> Matrix(1,0,0,1) Matrix(395,612,162,251) -> Matrix(1,2,0,1) Matrix(71,108,-48,-73) -> Matrix(1,0,0,1) Matrix(1025,1494,-754,-1099) -> Matrix(5,4,6,5) Matrix(161,234,86,125) -> Matrix(3,2,4,3) Matrix(341,486,214,305) -> Matrix(1,2,2,5) Matrix(685,972,432,613) -> Matrix(1,0,4,1) Matrix(89,126,12,17) -> Matrix(3,2,4,3) Matrix(181,252,130,181) -> Matrix(1,0,6,1) Matrix(287,396,208,287) -> Matrix(1,0,-4,1) Matrix(197,270,170,233) -> Matrix(1,0,0,1) Matrix(3815,5184,1006,1367) -> Matrix(11,-10,10,-9) Matrix(3817,5184,1008,1369) -> Matrix(11,-12,12,-13) Matrix(107,144,26,35) -> Matrix(1,0,2,1) Matrix(125,162,-98,-127) -> Matrix(1,0,2,1) Matrix(269,342,70,89) -> Matrix(1,0,0,1) Matrix(305,378,188,233) -> Matrix(1,4,2,9) Matrix(179,216,-150,-181) -> Matrix(1,0,2,1) Matrix(199,234,108,127) -> Matrix(1,-2,2,-3) Matrix(251,288,156,179) -> Matrix(1,2,2,5) Matrix(323,360,96,107) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(269,-306,80,-91) -> Matrix(1,0,0,1) Matrix(395,-468,92,-109) -> Matrix(3,-2,8,-5) Matrix(325,-396,206,-251) -> Matrix(1,0,0,1) Matrix(161,-198,74,-91) -> Matrix(1,0,-2,1) Matrix(127,-162,98,-125) -> Matrix(1,0,2,1) Matrix(235,-306,96,-125) -> Matrix(1,0,-2,1) Matrix(899,-1224,556,-757) -> Matrix(1,2,2,5) Matrix(487,-666,302,-413) -> Matrix(1,2,2,5) Matrix(163,-234,62,-89) -> Matrix(1,0,0,1) Matrix(127,-198,34,-53) -> Matrix(5,-2,8,-3) Matrix(683,-1080,160,-253) -> Matrix(7,-2,18,-5) Matrix(2269,-3672,600,-971) -> Matrix(37,-16,44,-19) Matrix(91,-162,50,-89) -> Matrix(5,-2,8,-3) Matrix(53,-126,8,-19) -> Matrix(5,-2,8,-3) Matrix(467,-1224,124,-325) -> Matrix(5,-2,8,-3) Matrix(19,-54,6,-17) -> Matrix(1,0,0,1) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 72 ----------------------------------------------------------------------- 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 0/1 1 1 1/1 (0/1,1/2) 0 18 8/7 0 9 7/6 1/3 1 18 13/11 (0/1,1/2) 0 18 6/5 0/1 1 3 11/9 (0/1,1/2) 0 18 5/4 1/1 1 18 9/7 0/1 2 2 13/10 1/3 1 18 4/3 0 9 19/14 1/1 1 18 15/11 0 6 11/8 -1/1 1 18 18/13 0/1 5 1 7/5 (0/1,1/4) 0 18 17/12 1/3 1 18 27/19 1/3 2 2 10/7 0 9 3/2 0 6 11/7 (0/1,1/2) 0 18 30/19 0/1 1 3 19/12 1/3 1 18 27/17 1/3 2 2 8/5 0 9 29/18 1/3 1 18 21/13 0 6 55/34 3/7 1 18 34/21 0 9 13/8 5/11 1 18 18/11 1/2 6 1 5/3 (1/2,1/1) 0 18 12/7 1/2 1 3 19/11 (0/1,1/1) 0 18 7/4 1/1 1 18 9/5 1/2 2 2 11/6 3/5 1 18 13/7 (1/2,2/3) 0 18 2/1 0 9 13/6 -1/1 1 18 11/5 (0/1,1/0) 0 18 9/4 0/1 2 2 7/3 (0/1,1/2) 0 18 12/5 1/2 1 3 17/7 (2/3,1/1) 0 18 22/9 0 9 5/2 1/1 1 18 18/7 0/1 3 1 13/5 (0/1,1/4) 0 18 34/13 0 9 21/8 0 6 8/3 0 9 3/1 0 6 10/3 0 9 37/11 (0/1,1/2) 0 18 27/8 0/1 1 2 17/5 (0/1,1/3) 0 18 7/2 1/3 1 18 18/5 1/2 3 1 11/3 (1/2,2/3) 0 18 15/4 0 6 34/9 0 9 53/14 1/1 1 18 72/19 1/1 11 1 19/5 (1/1,2/1) 0 18 4/1 0 9 17/4 1/3 1 18 30/7 2/5 1 3 13/3 (2/5,1/2) 0 18 9/2 1/2 3 2 5/1 (1/2,1/1) 0 18 6/1 1/2 1 3 7/1 (1/2,2/3) 0 18 1/0 1/1 1 18 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(269,-306,80,-91) (1/1,8/7) -> (10/3,37/11) Hyperbolic Matrix(251,-288,156,-179) (8/7,7/6) -> (8/5,29/18) Glide Reflection Matrix(199,-234,108,-127) (7/6,13/11) -> (11/6,13/7) Glide Reflection Matrix(395,-468,92,-109) (13/11,6/5) -> (30/7,13/3) Hyperbolic Matrix(325,-396,206,-251) (6/5,11/9) -> (11/7,30/19) Hyperbolic Matrix(161,-198,74,-91) (11/9,5/4) -> (13/6,11/5) Hyperbolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(235,-306,96,-125) (13/10,4/3) -> (22/9,5/2) Hyperbolic Matrix(107,-144,26,-35) (4/3,19/14) -> (4/1,17/4) Glide Reflection Matrix(899,-1224,556,-757) (19/14,15/11) -> (21/13,55/34) Hyperbolic Matrix(487,-666,302,-413) (15/11,11/8) -> (29/18,21/13) Hyperbolic Matrix(287,-396,208,-287) (11/8,18/13) -> (11/8,18/13) Reflection Matrix(181,-252,130,-181) (18/13,7/5) -> (18/13,7/5) Reflection Matrix(89,-126,12,-17) (7/5,17/12) -> (7/1,1/0) Glide Reflection Matrix(685,-972,432,-613) (17/12,27/19) -> (19/12,27/17) Glide Reflection Matrix(341,-486,214,-305) (27/19,10/7) -> (27/17,8/5) Glide Reflection Matrix(163,-234,62,-89) (10/7,3/2) -> (21/8,8/3) Hyperbolic Matrix(127,-198,34,-53) (3/2,11/7) -> (11/3,15/4) Hyperbolic Matrix(683,-1080,160,-253) (30/19,19/12) -> (17/4,30/7) Hyperbolic Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(289,-468,134,-217) (34/21,13/8) -> (2/1,13/6) Glide Reflection Matrix(287,-468,176,-287) (13/8,18/11) -> (13/8,18/11) Reflection Matrix(109,-180,66,-109) (18/11,5/3) -> (18/11,5/3) Reflection Matrix(53,-90,10,-17) (5/3,12/7) -> (5/1,6/1) Glide Reflection Matrix(251,-432,104,-179) (12/7,19/11) -> (12/5,17/7) Glide Reflection Matrix(145,-252,42,-73) (19/11,7/4) -> (17/5,7/2) Glide Reflection Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(251,-468,96,-179) (13/7,2/1) -> (13/5,34/13) Glide Reflection Matrix(89,-198,40,-89) (11/5,9/4) -> (11/5,9/4) Reflection Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(199,-486,52,-127) (17/7,22/9) -> (19/5,4/1) Glide Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(181,-468,70,-181) (18/7,13/5) -> (18/7,13/5) Reflection Matrix(467,-1224,124,-325) (34/13,21/8) -> (15/4,34/9) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(593,-1998,176,-593) (37/11,27/8) -> (37/11,27/8) Reflection Matrix(271,-918,80,-271) (27/8,17/5) -> (27/8,17/5) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(2015,-7632,532,-2015) (53/14,72/19) -> (53/14,72/19) Reflection Matrix(721,-2736,190,-721) (72/19,19/5) -> (72/19,19/5) Reflection Matrix(53,-234,12,-53) (13/3,9/2) -> (13/3,9/2) Reflection Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,2,-1) (0/1,1/0) -> (0/1,1/1) Matrix(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(269,-306,80,-91) -> Matrix(1,0,0,1) Matrix(251,-288,156,-179) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(199,-234,108,-127) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(395,-468,92,-109) -> Matrix(3,-2,8,-5) 1/2 Matrix(325,-396,206,-251) -> Matrix(1,0,0,1) Matrix(161,-198,74,-91) -> Matrix(1,0,-2,1) 0/1 Matrix(127,-162,98,-125) -> Matrix(1,0,2,1) 0/1 Matrix(235,-306,96,-125) -> Matrix(1,0,-2,1) 0/1 Matrix(107,-144,26,-35) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(899,-1224,556,-757) -> Matrix(1,2,2,5) Matrix(487,-666,302,-413) -> Matrix(1,2,2,5) Matrix(287,-396,208,-287) -> Matrix(-1,0,2,1) (11/8,18/13) -> (-1/1,0/1) Matrix(181,-252,130,-181) -> Matrix(1,0,8,-1) (18/13,7/5) -> (0/1,1/4) Matrix(89,-126,12,-17) -> Matrix(7,-2,10,-3) Matrix(685,-972,432,-613) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(341,-486,214,-305) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(163,-234,62,-89) -> Matrix(1,0,0,1) Matrix(127,-198,34,-53) -> Matrix(5,-2,8,-3) 1/2 Matrix(683,-1080,160,-253) -> Matrix(7,-2,18,-5) 1/3 Matrix(2269,-3672,600,-971) -> Matrix(37,-16,44,-19) Matrix(289,-468,134,-217) -> Matrix(9,-4,2,-1) Matrix(287,-468,176,-287) -> Matrix(21,-10,44,-21) (13/8,18/11) -> (5/11,1/2) Matrix(109,-180,66,-109) -> Matrix(3,-2,4,-3) (18/11,5/3) -> (1/2,1/1) Matrix(53,-90,10,-17) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(251,-432,104,-179) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(145,-252,42,-73) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(91,-162,50,-89) -> Matrix(5,-2,8,-3) 1/2 Matrix(251,-468,96,-179) -> Matrix(3,-2,10,-7) Matrix(89,-198,40,-89) -> Matrix(1,0,0,-1) (11/5,9/4) -> (0/1,1/0) Matrix(55,-126,24,-55) -> Matrix(1,0,4,-1) (9/4,7/3) -> (0/1,1/2) Matrix(53,-126,8,-19) -> Matrix(5,-2,8,-3) 1/2 Matrix(199,-486,52,-127) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(71,-180,28,-71) -> Matrix(1,0,2,-1) (5/2,18/7) -> (0/1,1/1) Matrix(181,-468,70,-181) -> Matrix(1,0,8,-1) (18/7,13/5) -> (0/1,1/4) Matrix(467,-1224,124,-325) -> Matrix(5,-2,8,-3) 1/2 Matrix(19,-54,6,-17) -> Matrix(1,0,0,1) Matrix(593,-1998,176,-593) -> Matrix(1,0,4,-1) (37/11,27/8) -> (0/1,1/2) Matrix(271,-918,80,-271) -> Matrix(1,0,6,-1) (27/8,17/5) -> (0/1,1/3) Matrix(71,-252,20,-71) -> Matrix(5,-2,12,-5) (7/2,18/5) -> (1/3,1/2) Matrix(109,-396,30,-109) -> Matrix(7,-4,12,-7) (18/5,11/3) -> (1/2,2/3) Matrix(2015,-7632,532,-2015) -> Matrix(19,-18,20,-19) (53/14,72/19) -> (9/10,1/1) Matrix(721,-2736,190,-721) -> Matrix(3,-4,2,-3) (72/19,19/5) -> (1/1,2/1) Matrix(53,-234,12,-53) -> Matrix(9,-4,20,-9) (13/3,9/2) -> (2/5,1/2) Matrix(19,-90,4,-19) -> Matrix(3,-2,4,-3) (9/2,5/1) -> (1/2,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.