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 -8/1 -6/1 -9/2 -4/1 -15/4 -27/8 -3/1 -9/4 -2/1 -3/2 -6/5 0/1 1/1 6/5 9/7 18/13 3/2 18/11 9/5 2/1 9/4 5/2 18/7 3/1 27/8 7/2 18/5 15/4 72/19 4/1 9/2 19/4 5/1 11/2 6/1 13/2 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 0/1 1/1 1/0 -7/1 0/1 1/1 -13/2 -1/1 -1/2 -6/1 0/1 -11/2 0/1 1/1 -5/1 -1/1 1/0 -9/2 0/1 -13/3 1/4 1/3 -4/1 0/1 1/1 1/0 -19/5 0/1 1/0 -15/4 -1/1 -11/3 -1/3 0/1 -18/5 0/1 -7/2 0/1 1/3 -17/5 0/1 1/2 -27/8 1/2 -10/3 1/2 2/3 1/1 -3/1 -1/1 1/1 -11/4 1/1 2/1 -30/11 2/1 -19/7 2/1 1/0 -27/10 1/0 -8/3 1/1 2/1 1/0 -29/11 1/1 2/1 -21/8 3/1 -55/21 4/1 1/0 -34/13 2/1 3/1 1/0 -13/5 5/1 1/0 -18/7 1/0 -5/2 -1/1 1/0 -12/5 1/0 -19/8 -2/1 1/0 -7/3 -1/1 0/1 -9/4 1/0 -11/5 -3/1 -2/1 -13/6 -3/2 -1/1 -2/1 -1/1 0/1 1/0 -13/7 -3/1 1/0 -11/6 -2/1 -1/1 -9/5 -1/1 -7/4 -1/1 0/1 -12/7 1/0 -17/10 -2/1 1/0 -22/13 -3/1 -5/2 -2/1 -5/3 -3/2 -1/1 -18/11 -1/1 -13/8 -1/1 -5/6 -34/21 -1/1 -3/4 -2/3 -21/13 -1/1 -5/7 -8/5 -1/1 -1/2 0/1 -3/2 -1/1 -10/7 -1/1 -3/4 -2/3 -37/26 -3/4 -2/3 -27/19 -2/3 -17/12 -2/3 -1/2 -7/5 -1/1 0/1 -18/13 -1/1 -11/8 -1/1 -4/5 -15/11 -1/1 -5/7 -34/25 -1/1 -3/4 -2/3 -53/39 -4/5 -3/4 -72/53 -3/4 -19/14 -3/4 -2/3 -4/3 -1/1 -2/3 -1/2 -17/13 -2/3 -1/2 -30/23 -2/3 -13/10 -3/5 -1/2 -9/7 -1/2 -5/4 -1/1 -1/2 -6/5 -1/2 -7/6 -1/3 0/1 -1/1 -1/2 0/1 0/1 0/1 1/1 0/1 1/2 6/5 1/2 5/4 1/2 1/1 9/7 1/2 13/10 1/2 3/5 17/13 1/2 2/3 4/3 1/2 2/3 1/1 11/8 4/5 1/1 18/13 1/1 7/5 0/1 1/1 3/2 1/1 11/7 1/1 2/1 19/12 0/1 1/0 27/17 0/1 8/5 0/1 1/2 1/1 13/8 5/6 1/1 18/11 1/1 5/3 1/1 3/2 17/10 2/1 1/0 12/7 1/0 19/11 0/1 1/0 7/4 0/1 1/1 9/5 1/1 11/6 1/1 2/1 2/1 0/1 1/1 1/0 9/4 1/0 16/7 -2/1 -1/1 1/0 23/10 -1/1 -1/2 7/3 0/1 1/1 19/8 2/1 1/0 12/5 1/0 5/2 1/1 1/0 18/7 1/0 13/5 -5/1 1/0 34/13 -3/1 -2/1 1/0 21/8 -3/1 8/3 -2/1 -1/1 1/0 27/10 1/0 19/7 -2/1 1/0 11/4 -2/1 -1/1 14/5 -1/1 -1/2 0/1 17/6 0/1 1/0 3/1 -1/1 1/1 19/6 0/1 1/0 16/5 -1/1 0/1 1/0 13/4 -3/2 -1/1 10/3 -1/1 -2/3 -1/2 27/8 -1/2 17/5 -1/2 0/1 7/2 -1/3 0/1 18/5 0/1 11/3 0/1 1/3 26/7 0/1 1/3 1/2 15/4 1/1 34/9 1/1 2/1 1/0 53/14 2/1 1/0 72/19 1/0 19/5 0/1 1/0 4/1 -1/1 0/1 1/0 9/2 0/1 14/3 0/1 1/4 1/3 19/4 0/1 1/2 5/1 1/1 1/0 16/3 -2/1 -1/1 1/0 11/2 -1/1 0/1 17/3 -1/2 0/1 6/1 0/1 19/3 0/1 1/2 13/2 1/2 1/1 7/1 -1/1 0/1 8/1 -1/1 0/1 1/0 9/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(37,306,-26,-215) (-8/1,1/0) -> (-10/7,-37/26) Hyperbolic Matrix(37,270,10,73) (-8/1,-7/1) -> (11/3,26/7) Hyperbolic Matrix(37,252,16,109) (-7/1,-13/2) -> (23/10,7/3) Hyperbolic Matrix(73,468,-56,-359) (-13/2,-6/1) -> (-30/23,-13/10) Hyperbolic Matrix(71,396,-26,-145) (-6/1,-11/2) -> (-11/4,-30/11) Hyperbolic Matrix(37,198,-20,-107) (-11/2,-5/1) -> (-13/7,-11/6) Hyperbolic Matrix(35,162,-8,-37) (-5/1,-9/2) -> (-9/2,-13/3) Parabolic Matrix(71,306,-42,-181) (-13/3,-4/1) -> (-22/13,-5/3) Hyperbolic Matrix(37,144,28,109) (-4/1,-19/5) -> (17/13,4/3) Hyperbolic Matrix(325,1224,-124,-467) (-19/5,-15/4) -> (-21/8,-55/21) Hyperbolic Matrix(179,666,-68,-253) (-15/4,-11/3) -> (-29/11,-21/8) 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(73,252,42,145) (-7/2,-17/5) -> (19/11,7/4) Hyperbolic Matrix(287,972,106,359) (-17/5,-27/8) -> (27/10,19/7) Hyperbolic Matrix(145,486,54,181) (-27/8,-10/3) -> (8/3,27/10) Hyperbolic Matrix(71,234,-44,-145) (-10/3,-3/1) -> (-21/13,-8/5) Hyperbolic Matrix(71,198,-52,-145) (-3/1,-11/4) -> (-11/8,-15/11) Hyperbolic Matrix(397,1080,-304,-827) (-30/11,-19/7) -> (-17/13,-30/23) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(109,288,14,37) (-8/3,-29/11) -> (7/1,8/1) Hyperbolic Matrix(1403,3672,-1032,-2701) (-55/21,-34/13) -> (-34/25,-53/39) Hyperbolic Matrix(145,378,28,73) (-34/13,-13/5) -> (5/1,16/3) 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(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(181,432,106,253) (-12/5,-19/8) -> (17/10,12/7) Hyperbolic Matrix(145,342,92,217) (-19/8,-7/3) -> (11/7,19/12) Hyperbolic Matrix(71,162,-32,-73) (-7/3,-9/4) -> (-9/4,-11/5) Parabolic Matrix(107,234,16,35) (-11/5,-13/6) -> (13/2,7/1) Hyperbolic Matrix(109,234,34,73) (-13/6,-2/1) -> (16/5,13/4) Hyperbolic Matrix(251,468,96,179) (-2/1,-13/7) -> (13/5,34/13) Hyperbolic Matrix(109,198,60,109) (-11/6,-9/5) -> (9/5,11/6) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(73,126,-62,-107) (-7/4,-12/7) -> (-6/5,-7/6) Hyperbolic Matrix(253,432,106,181) (-12/7,-17/10) -> (19/8,12/5) Hyperbolic Matrix(361,612,128,217) (-17/10,-22/13) -> (14/5,17/6) 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(611,990,266,431) (-13/8,-34/21) -> (16/7,23/10) Hyperbolic Matrix(757,1224,-556,-899) (-34/21,-21/13) -> (-15/11,-34/25) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(253,360,26,37) (-37/26,-27/19) -> (9/1,1/0) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(179,252,76,107) (-17/12,-7/5) -> (7/3,19/8) 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(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(253,342,54,73) (-19/14,-4/3) -> (14/3,19/4) Hyperbolic Matrix(109,144,28,37) (-4/3,-17/13) -> (19/5,4/1) Hyperbolic Matrix(181,234,140,181) (-13/10,-9/7) -> (9/7,13/10) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(109,126,32,37) (-7/6,-1/1) -> (17/5,7/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(359,-468,56,-73) (13/10,17/13) -> (19/3,13/2) Hyperbolic Matrix(145,-198,52,-71) (4/3,11/8) -> (11/4,14/5) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(181,-288,22,-35) (27/17,8/5) -> (8/1,9/1) Hyperbolic Matrix(145,-234,44,-71) (8/5,13/8) -> (13/4,10/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(107,-198,20,-37) (11/6,2/1) -> (16/3,11/2) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(467,-1224,124,-325) (34/13,21/8) -> (15/4,34/9) Hyperbolic Matrix(253,-666,68,-179) (21/8,8/3) -> (26/7,15/4) Hyperbolic Matrix(145,-396,26,-71) (19/7,11/4) -> (11/2,17/3) Hyperbolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(469,-1494,124,-395) (19/6,16/5) -> (34/9,53/14) Hyperbolic Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic Matrix(37,-216,6,-35) (17/3,6/1) -> (6/1,19/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(37,306,-26,-215) -> Matrix(3,-2,-4,3) Matrix(37,270,10,73) -> Matrix(1,0,2,1) Matrix(37,252,16,109) -> Matrix(1,0,0,1) Matrix(73,468,-56,-359) -> Matrix(5,2,-8,-3) Matrix(71,396,-26,-145) -> Matrix(3,-2,2,-1) Matrix(37,198,-20,-107) -> Matrix(1,-2,0,1) Matrix(35,162,-8,-37) -> Matrix(1,0,4,1) Matrix(71,306,-42,-181) -> Matrix(5,-2,-2,1) Matrix(37,144,28,109) -> Matrix(1,-2,2,-3) Matrix(325,1224,-124,-467) -> Matrix(1,4,0,1) Matrix(179,666,-68,-253) -> Matrix(5,2,2,1) Matrix(109,396,30,109) -> Matrix(1,0,6,1) Matrix(71,252,20,71) -> Matrix(1,0,-6,1) Matrix(73,252,42,145) -> Matrix(1,0,-2,1) Matrix(287,972,106,359) -> Matrix(5,-2,-2,1) Matrix(145,486,54,181) -> Matrix(1,0,-2,1) Matrix(71,234,-44,-145) -> Matrix(3,-2,-4,3) Matrix(71,198,-52,-145) -> Matrix(3,-2,-4,3) Matrix(397,1080,-304,-827) -> Matrix(1,0,-2,1) Matrix(359,972,106,287) -> Matrix(1,-2,-2,5) Matrix(181,486,54,145) -> Matrix(1,0,-2,1) Matrix(109,288,14,37) -> Matrix(1,-2,0,1) Matrix(1403,3672,-1032,-2701) -> Matrix(3,-8,-4,11) Matrix(145,378,28,73) -> Matrix(1,-4,0,1) Matrix(181,468,70,181) -> Matrix(1,-10,0,1) Matrix(71,180,28,71) -> Matrix(1,2,0,1) Matrix(37,90,30,73) -> Matrix(1,0,2,1) Matrix(181,432,106,253) -> Matrix(1,4,0,1) Matrix(145,342,92,217) -> Matrix(1,2,0,1) Matrix(71,162,-32,-73) -> Matrix(1,-2,0,1) Matrix(107,234,16,35) -> Matrix(1,2,0,1) Matrix(109,234,34,73) -> Matrix(1,0,0,1) Matrix(251,468,96,179) -> Matrix(1,-2,0,1) Matrix(109,198,60,109) -> Matrix(3,4,2,3) Matrix(71,126,40,71) -> Matrix(1,0,2,1) Matrix(73,126,-62,-107) -> Matrix(1,0,-2,1) Matrix(253,432,106,181) -> Matrix(1,4,0,1) Matrix(361,612,128,217) -> Matrix(1,2,0,1) Matrix(109,180,66,109) -> Matrix(5,6,4,5) Matrix(287,468,176,287) -> Matrix(11,10,12,11) Matrix(611,990,266,431) -> Matrix(5,4,-4,-3) Matrix(757,1224,-556,-899) -> Matrix(1,0,0,1) Matrix(35,54,-24,-37) -> Matrix(1,2,-2,-3) Matrix(253,360,26,37) -> Matrix(3,2,4,3) Matrix(685,972,432,613) -> Matrix(3,2,-2,-1) Matrix(179,252,76,107) -> Matrix(1,0,2,1) Matrix(181,252,130,181) -> Matrix(1,0,2,1) Matrix(287,396,208,287) -> Matrix(9,8,10,9) Matrix(3815,5184,1006,1367) -> Matrix(5,4,-4,-3) Matrix(3817,5184,1008,1369) -> Matrix(11,8,4,3) Matrix(253,342,54,73) -> Matrix(3,2,10,7) Matrix(109,144,28,37) -> Matrix(3,2,-2,-1) Matrix(181,234,140,181) -> Matrix(11,6,20,11) Matrix(71,90,56,71) -> Matrix(3,2,4,3) Matrix(73,90,30,37) -> Matrix(1,0,2,1) Matrix(109,126,32,37) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(107,-126,62,-73) -> Matrix(1,0,-2,1) Matrix(359,-468,56,-73) -> Matrix(3,-2,8,-5) Matrix(145,-198,52,-71) -> Matrix(3,-2,-4,3) Matrix(37,-54,24,-35) -> Matrix(3,-2,2,-1) Matrix(181,-288,22,-35) -> Matrix(1,0,-2,1) Matrix(145,-234,44,-71) -> Matrix(3,-2,-4,3) Matrix(107,-180,22,-37) -> Matrix(1,-2,2,-3) Matrix(107,-198,20,-37) -> Matrix(1,-2,0,1) Matrix(73,-162,32,-71) -> Matrix(1,-2,0,1) Matrix(467,-1224,124,-325) -> Matrix(1,4,0,1) Matrix(253,-666,68,-179) -> Matrix(1,2,2,5) Matrix(145,-396,26,-71) -> Matrix(1,2,-2,-3) Matrix(37,-108,12,-35) -> Matrix(1,0,0,1) Matrix(469,-1494,124,-395) -> Matrix(1,2,0,1) Matrix(37,-162,8,-35) -> Matrix(1,0,4,1) Matrix(37,-216,6,-35) -> 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): 18 Degree of the the map X: 18 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 2 1 1/1 (0/1,1/2) 0 18 6/5 1/2 2 3 5/4 (1/2,1/1) 0 18 9/7 1/2 2 2 13/10 (1/2,3/5) 0 18 17/13 (1/2,2/3) 0 18 4/3 0 9 11/8 (4/5,1/1) 0 18 18/13 1/1 4 1 7/5 (0/1,1/1) 0 18 3/2 1/1 2 6 11/7 (1/1,2/1) 0 18 19/12 (0/1,1/0) 0 18 27/17 0/1 2 2 8/5 0 9 13/8 (5/6,1/1) 0 18 18/11 1/1 8 1 5/3 (1/1,3/2) 0 18 17/10 (2/1,1/0) 0 18 12/7 1/0 2 3 19/11 (0/1,1/0) 0 18 7/4 (0/1,1/1) 0 18 9/5 1/1 2 2 11/6 (1/1,2/1) 0 18 2/1 0 9 9/4 1/0 2 2 16/7 0 9 23/10 (-1/1,-1/2) 0 18 7/3 (0/1,1/1) 0 18 19/8 (2/1,1/0) 0 18 12/5 1/0 2 3 5/2 (1/1,1/0) 0 18 18/7 1/0 6 1 13/5 (-5/1,1/0) 0 18 34/13 0 9 21/8 -3/1 2 6 8/3 0 9 27/10 1/0 2 2 19/7 (-2/1,1/0) 0 18 11/4 (-2/1,-1/1) 0 18 14/5 0 9 17/6 (0/1,1/0) 0 18 3/1 0 6 19/6 (0/1,1/0) 0 18 16/5 0 9 13/4 (-3/2,-1/1) 0 18 10/3 0 9 27/8 -1/2 2 2 17/5 (-1/2,0/1) 0 18 7/2 (-1/3,0/1) 0 18 18/5 0/1 6 1 11/3 (0/1,1/3) 0 18 26/7 0 9 15/4 1/1 2 6 34/9 0 9 53/14 (2/1,1/0) 0 18 72/19 1/0 2 1 19/5 (0/1,1/0) 0 18 4/1 0 9 9/2 0/1 4 2 14/3 0 9 19/4 (0/1,1/2) 0 18 5/1 (1/1,1/0) 0 18 16/3 0 9 11/2 (-1/1,0/1) 0 18 17/3 (-1/2,0/1) 0 18 6/1 0/1 2 3 19/3 (0/1,1/2) 0 18 13/2 (1/2,1/1) 0 18 7/1 (-1/1,0/1) 0 18 8/1 0 9 9/1 0/1 2 2 1/0 (0/1,1/0) 0 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(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(73,-90,30,-37) (6/5,5/4) -> (12/5,5/2) Glide Reflection Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(181,-234,140,-181) (9/7,13/10) -> (9/7,13/10) Reflection Matrix(359,-468,56,-73) (13/10,17/13) -> (19/3,13/2) Hyperbolic Matrix(109,-144,28,-37) (17/13,4/3) -> (19/5,4/1) Glide Reflection Matrix(145,-198,52,-71) (4/3,11/8) -> (11/4,14/5) 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(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(217,-342,92,-145) (11/7,19/12) -> (7/3,19/8) Glide Reflection Matrix(647,-1026,408,-647) (19/12,27/17) -> (19/12,27/17) Reflection Matrix(181,-288,22,-35) (27/17,8/5) -> (8/1,9/1) Hyperbolic Matrix(145,-234,44,-71) (8/5,13/8) -> (13/4,10/3) Hyperbolic 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(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(253,-432,106,-181) (17/10,12/7) -> (19/8,12/5) Glide Reflection Matrix(145,-252,42,-73) (19/11,7/4) -> (17/5,7/2) Glide Reflection Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(109,-198,60,-109) (9/5,11/6) -> (9/5,11/6) Reflection Matrix(107,-198,20,-37) (11/6,2/1) -> (16/3,11/2) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(251,-576,78,-179) (16/7,23/10) -> (16/5,13/4) Glide Reflection Matrix(109,-252,16,-37) (23/10,7/3) -> (13/2,7/1) 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(145,-378,28,-73) (13/5,34/13) -> (5/1,16/3) Glide Reflection Matrix(467,-1224,124,-325) (34/13,21/8) -> (15/4,34/9) Hyperbolic Matrix(253,-666,68,-179) (21/8,8/3) -> (26/7,15/4) Hyperbolic Matrix(181,-486,54,-145) (8/3,27/10) -> (10/3,27/8) Glide Reflection Matrix(359,-972,106,-287) (27/10,19/7) -> (27/8,17/5) Glide Reflection Matrix(145,-396,26,-71) (19/7,11/4) -> (11/2,17/3) Hyperbolic Matrix(179,-504,38,-107) (14/5,17/6) -> (14/3,19/4) Glide Reflection Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(469,-1494,124,-395) (19/6,16/5) -> (34/9,53/14) Hyperbolic 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(73,-270,10,-37) (11/3,26/7) -> (7/1,8/1) Glide 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(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic Matrix(37,-216,6,-35) (17/3,6/1) -> (6/1,19/3) Parabolic Matrix(-1,18,0,1) (9/1,1/0) -> (9/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(107,-126,62,-73) -> Matrix(1,0,-2,1) 0/1 Matrix(73,-90,30,-37) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(71,-90,56,-71) -> Matrix(3,-2,4,-3) (5/4,9/7) -> (1/2,1/1) Matrix(181,-234,140,-181) -> Matrix(11,-6,20,-11) (9/7,13/10) -> (1/2,3/5) Matrix(359,-468,56,-73) -> Matrix(3,-2,8,-5) 1/2 Matrix(109,-144,28,-37) -> Matrix(3,-2,-2,1) Matrix(145,-198,52,-71) -> Matrix(3,-2,-4,3) Matrix(287,-396,208,-287) -> Matrix(9,-8,10,-9) (11/8,18/13) -> (4/5,1/1) Matrix(181,-252,130,-181) -> Matrix(1,0,2,-1) (18/13,7/5) -> (0/1,1/1) Matrix(37,-54,24,-35) -> Matrix(3,-2,2,-1) 1/1 Matrix(217,-342,92,-145) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(647,-1026,408,-647) -> Matrix(1,0,0,-1) (19/12,27/17) -> (0/1,1/0) Matrix(181,-288,22,-35) -> Matrix(1,0,-2,1) 0/1 Matrix(145,-234,44,-71) -> Matrix(3,-2,-4,3) Matrix(287,-468,176,-287) -> Matrix(11,-10,12,-11) (13/8,18/11) -> (5/6,1/1) Matrix(109,-180,66,-109) -> Matrix(5,-6,4,-5) (18/11,5/3) -> (1/1,3/2) Matrix(107,-180,22,-37) -> Matrix(1,-2,2,-3) 1/1 Matrix(253,-432,106,-181) -> Matrix(-1,4,0,1) *** -> (2/1,1/0) Matrix(145,-252,42,-73) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(71,-126,40,-71) -> Matrix(1,0,2,-1) (7/4,9/5) -> (0/1,1/1) Matrix(109,-198,60,-109) -> Matrix(3,-4,2,-3) (9/5,11/6) -> (1/1,2/1) Matrix(107,-198,20,-37) -> Matrix(1,-2,0,1) 1/0 Matrix(73,-162,32,-71) -> Matrix(1,-2,0,1) 1/0 Matrix(251,-576,78,-179) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(109,-252,16,-37) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(71,-180,28,-71) -> Matrix(-1,2,0,1) (5/2,18/7) -> (1/1,1/0) Matrix(181,-468,70,-181) -> Matrix(1,10,0,-1) (18/7,13/5) -> (-5/1,1/0) Matrix(145,-378,28,-73) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(467,-1224,124,-325) -> Matrix(1,4,0,1) 1/0 Matrix(253,-666,68,-179) -> Matrix(1,2,2,5) Matrix(181,-486,54,-145) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(359,-972,106,-287) -> Matrix(1,2,-2,-5) Matrix(145,-396,26,-71) -> Matrix(1,2,-2,-3) -1/1 Matrix(179,-504,38,-107) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(37,-108,12,-35) -> Matrix(1,0,0,1) Matrix(469,-1494,124,-395) -> Matrix(1,2,0,1) 1/0 Matrix(71,-252,20,-71) -> Matrix(-1,0,6,1) (7/2,18/5) -> (-1/3,0/1) Matrix(109,-396,30,-109) -> Matrix(1,0,6,-1) (18/5,11/3) -> (0/1,1/3) Matrix(73,-270,10,-37) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(2015,-7632,532,-2015) -> Matrix(-1,4,0,1) (53/14,72/19) -> (2/1,1/0) Matrix(721,-2736,190,-721) -> Matrix(1,0,0,-1) (72/19,19/5) -> (0/1,1/0) Matrix(37,-162,8,-35) -> Matrix(1,0,4,1) 0/1 Matrix(37,-216,6,-35) -> Matrix(1,0,4,1) 0/1 Matrix(-1,18,0,1) -> Matrix(1,0,0,-1) (9/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.