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): 648 Minimal number of generators: 109 Number of equivalence classes of cusps: 54 Genus: 28 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/1 -6/1 -5/1 -4/1 -7/2 -3/1 -5/2 -2/1 -8/5 -3/2 -6/5 -1/1 -6/7 -3/4 -3/5 -1/2 -3/7 -3/8 0/1 1/3 3/8 3/7 1/2 3/5 2/3 3/4 5/6 6/7 1/1 7/6 6/5 9/7 4/3 3/2 27/17 8/5 5/3 9/5 2/1 9/4 7/3 5/2 18/7 8/3 3/1 27/8 7/2 18/5 4/1 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 -7/1 -3/2 1/0 -6/1 -1/1 -5/1 -1/2 1/0 -14/3 -3/2 -9/2 -1/1 -4/1 -1/1 -1/2 0/1 -11/3 -1/2 -18/5 0/1 -7/2 -1/2 1/0 -3/1 -1/2 1/0 -11/4 -1/2 1/0 -19/7 -3/4 -1/2 -27/10 -1/1 -8/3 -1/2 -13/5 -1/2 -1/4 -18/7 0/1 -5/2 -1/2 1/0 -12/5 -1/1 -19/8 -3/4 -1/2 -7/3 -1/2 -9/4 -1/2 -11/5 -1/2 -1/4 -2/1 -1/1 -1/2 0/1 -11/6 -1/2 -9/5 -1/2 -7/4 -1/2 -1/4 -19/11 -1/2 -1/4 -12/7 -1/2 -1/4 -5/3 -1/4 -18/11 0/1 -13/8 -1/2 -1/4 -8/5 -1/4 -1/5 0/1 -3/2 0/1 -10/7 0/1 1/6 1/5 -27/19 1/5 -17/12 1/4 -7/5 1/4 1/2 -18/13 0/1 -11/8 1/4 1/2 -4/3 1/2 -9/7 1/1 -14/11 1/1 3/2 2/1 -5/4 3/2 1/0 -6/5 1/0 -7/6 1/0 -1/1 -1/2 1/0 -6/7 -1/1 -5/6 -1/2 -9/11 -1/1 -4/5 -1/1 -2/3 -1/2 -7/9 -1/2 -3/4 -1/2 -11/15 -1/2 -19/26 -1/2 -1/4 -27/37 -1/3 -8/11 -1/2 -1/3 0/1 -13/18 -1/2 -18/25 0/1 -5/7 -1/2 1/0 -12/17 -1/1 -19/27 -3/4 -7/10 -3/4 -1/2 -9/13 -1/2 -2/3 -1/2 -9/14 -1/2 -7/11 -1/2 -1/4 -12/19 -1/2 -5/8 -1/2 -1/4 -8/13 -1/1 -1/2 0/1 -3/5 -1/2 -10/17 -1/2 -2/5 -1/3 -27/46 -1/3 -17/29 -1/2 -1/4 -7/12 -1/2 -18/31 -2/5 -11/19 -1/2 -3/8 -4/7 -1/2 -2/5 -1/3 -9/16 -1/3 -5/9 -1/4 -6/11 -1/2 -1/4 -7/13 -1/2 -1/4 -1/2 -1/2 -1/4 -6/13 -1/3 -5/11 -1/2 -1/4 -4/9 -1/4 -7/16 -1/2 -1/4 -3/7 -1/2 -1/4 -11/26 -1/2 -1/4 -19/45 -1/4 -27/64 -1/3 -8/19 -1/3 -1/4 0/1 -5/12 -1/2 -12/29 -1/3 -7/17 -3/10 -1/4 -2/5 -1/3 -1/4 0/1 -7/18 -1/4 -12/31 -1/4 -5/13 -1/4 -1/6 -8/21 -1/4 -3/8 0/1 -10/27 -1/2 -27/73 -1/3 -17/46 -1/2 -1/4 -7/19 -1/2 -1/4 -4/11 -1/2 -1/3 0/1 -5/14 -3/10 -1/4 -6/17 -1/4 -1/3 -1/4 0/1 0/1 1/3 1/4 5/14 1/4 1/2 4/11 0/1 1/3 1/2 7/19 1/4 1/2 3/8 1/4 1/2 5/13 1/4 1/2 7/18 1/2 2/5 0/1 1/3 1/2 7/17 1/2 1/0 5/12 1/0 8/19 -1/1 0/1 1/0 3/7 0/1 4/9 1/6 5/11 3/14 1/4 1/2 1/4 1/2 5/9 1/2 4/7 1/3 2/5 1/2 11/19 3/8 1/2 7/12 1/2 3/5 1/2 11/18 1/2 8/13 0/1 1/2 1/1 5/8 1/4 1/2 17/27 1/4 12/19 1/3 7/11 3/8 1/2 9/14 1/2 2/3 1/2 7/10 1/2 1/0 5/7 1/2 1/0 13/18 1/4 8/11 0/1 1/3 1/2 3/4 1/2 10/13 1/2 2/3 1/1 7/9 1/2 4/5 1/2 2/3 1/1 9/11 1/1 5/6 1/0 11/13 1/2 1/0 6/7 1/2 1/0 1/1 1/2 1/0 8/7 0/1 1/2 1/1 7/6 1/2 13/11 1/2 3/4 6/5 1/1 5/4 1/2 1/0 14/11 2/3 3/4 1/1 9/7 1/1 4/3 1/0 15/11 0/1 26/19 0/1 1/4 1/3 11/8 1/2 1/0 18/13 0/1 25/18 1/2 7/5 1/2 1/0 3/2 1/2 1/0 11/7 1/2 1/0 19/12 1/0 27/17 1/1 8/5 0/1 1/1 1/0 13/8 -1/2 1/0 18/11 0/1 5/3 1/2 17/10 1/2 3/4 29/17 3/4 5/6 12/7 1/1 19/11 5/4 3/2 7/4 3/2 1/0 16/9 1/0 9/5 1/0 11/6 1/0 2/1 0/1 1/1 1/0 11/5 3/2 1/0 9/4 1/0 7/3 1/0 26/11 0/1 1/1 1/0 45/19 1/1 19/8 3/2 1/0 31/13 7/2 1/0 12/5 1/0 5/2 -1/2 1/0 18/7 0/1 31/12 1/4 13/5 1/2 1/0 21/8 1/2 1/0 29/11 1/2 1/0 37/14 3/2 1/0 45/17 1/1 8/3 1/0 3/1 0/1 10/3 1/2 27/8 1/1 44/13 0/1 1/2 1/1 17/5 1/2 1/0 7/2 1/2 1/0 18/5 0/1 11/3 1/2 4/1 0/1 1/2 1/1 13/3 1/2 9/2 1/1 14/3 3/2 5/1 3/2 1/0 11/2 3/2 1/0 28/5 2/1 5/2 3/1 45/8 3/1 17/3 1/0 6/1 1/0 7/1 -1/2 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,144,12,91) (-7/1,1/0) -> (11/7,19/12) Hyperbolic Matrix(19,126,30,199) (-7/1,-6/1) -> (12/19,7/11) Hyperbolic Matrix(17,90,-24,-127) (-6/1,-5/1) -> (-5/7,-12/17) Hyperbolic Matrix(19,90,42,199) (-5/1,-14/3) -> (4/9,5/11) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(17,72,-30,-127) (-9/2,-4/1) -> (-4/7,-9/16) Hyperbolic Matrix(19,72,24,91) (-4/1,-11/3) -> (7/9,4/5) Hyperbolic Matrix(109,396,30,109) (-11/3,-18/5) -> (18/5,11/3) Hyperbolic Matrix(91,324,66,235) (-18/5,-7/2) -> (11/8,18/13) 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,-612,-1657) (-19/7,-27/10) -> (-27/46,-17/29) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(55,144,-144,-377) (-8/3,-13/5) -> (-5/13,-8/21) Hyperbolic Matrix(125,324,-174,-451) (-13/5,-18/7) -> (-18/25,-5/7) Hyperbolic Matrix(127,324,78,199) (-18/7,-5/2) -> (13/8,18/11) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(53,126,-114,-271) (-12/5,-19/8) -> (-1/2,-6/13) Hyperbolic Matrix(145,342,-198,-467) (-19/8,-7/3) -> (-11/15,-19/26) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(73,162,114,253) (-9/4,-11/5) -> (7/11,9/14) Hyperbolic Matrix(17,36,42,89) (-11/5,-2/1) -> (2/5,7/17) Hyperbolic Matrix(19,36,48,91) (-2/1,-11/6) -> (7/18,2/5) Hyperbolic Matrix(109,198,60,109) (-11/6,-9/5) -> (9/5,11/6) Hyperbolic Matrix(71,126,-102,-181) (-9/5,-7/4) -> (-7/10,-9/13) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(73,126,84,145) (-19/11,-12/7) -> (6/7,1/1) Hyperbolic Matrix(53,90,-96,-163) (-12/7,-5/3) -> (-5/9,-6/11) Hyperbolic Matrix(109,180,66,109) (-5/3,-18/11) -> (18/11,5/3) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(89,144,144,233) (-13/8,-8/5) -> (8/13,5/8) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(341,486,-468,-667) (-10/7,-27/19) -> (-27/37,-8/11) 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(233,324,-402,-559) (-7/5,-18/13) -> (-18/31,-11/19) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(53,72,-120,-163) (-11/8,-4/3) -> (-4/9,-7/16) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(127,162,156,199) (-9/7,-14/11) -> (4/5,9/11) Hyperbolic Matrix(71,90,198,251) (-14/11,-5/4) -> (5/14,4/11) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(107,126,-276,-325) (-6/5,-7/6) -> (-7/18,-12/31) Hyperbolic Matrix(109,126,-186,-215) (-7/6,-1/1) -> (-17/29,-7/12) Hyperbolic Matrix(145,126,84,73) (-1/1,-6/7) -> (12/7,19/11) Hyperbolic Matrix(107,90,-258,-217) (-6/7,-5/6) -> (-5/12,-12/29) Hyperbolic Matrix(109,90,132,109) (-5/6,-9/11) -> (9/11,5/6) Hyperbolic Matrix(199,162,156,127) (-9/11,-4/5) -> (14/11,9/7) Hyperbolic Matrix(91,72,24,19) (-4/5,-7/9) -> (11/3,4/1) Hyperbolic Matrix(71,54,-96,-73) (-7/9,-3/4) -> (-3/4,-11/15) Parabolic Matrix(1331,972,-3600,-2629) (-19/26,-27/37) -> (-27/73,-17/46) Hyperbolic Matrix(199,144,474,343) (-8/11,-13/18) -> (5/12,8/19) Hyperbolic Matrix(1099,792,426,307) (-13/18,-18/25) -> (18/7,31/12) Hyperbolic Matrix(613,432,972,685) (-12/17,-19/27) -> (17/27,12/19) Hyperbolic Matrix(487,342,-1152,-809) (-19/27,-7/10) -> (-11/26,-19/45) Hyperbolic Matrix(235,162,132,91) (-9/13,-2/3) -> (16/9,9/5) Hyperbolic Matrix(55,36,84,55) (-2/3,-9/14) -> (9/14,2/3) Hyperbolic Matrix(253,162,114,73) (-9/14,-7/11) -> (11/5,9/4) Hyperbolic Matrix(199,126,30,19) (-7/11,-12/19) -> (6/1,7/1) Hyperbolic Matrix(143,90,-402,-253) (-12/19,-5/8) -> (-5/14,-6/17) Hyperbolic Matrix(233,144,144,89) (-5/8,-8/13) -> (8/5,13/8) Hyperbolic Matrix(89,54,-150,-91) (-8/13,-3/5) -> (-3/5,-10/17) Parabolic Matrix(827,486,-1962,-1153) (-10/17,-27/46) -> (-27/64,-8/19) Hyperbolic Matrix(991,576,714,415) (-7/12,-18/31) -> (18/13,25/18) Hyperbolic Matrix(125,72,342,197) (-11/19,-4/7) -> (4/11,7/19) Hyperbolic Matrix(289,162,66,37) (-9/16,-5/9) -> (13/3,9/2) Hyperbolic Matrix(199,108,234,127) (-6/11,-7/13) -> (11/13,6/7) Hyperbolic Matrix(235,126,-636,-341) (-7/13,-1/2) -> (-17/46,-7/19) Hyperbolic Matrix(235,108,198,91) (-6/13,-5/11) -> (13/11,6/5) Hyperbolic Matrix(199,90,42,19) (-5/11,-4/9) -> (14/3,5/1) Hyperbolic Matrix(125,54,-294,-127) (-7/16,-3/7) -> (-3/7,-11/26) Parabolic Matrix(3113,1314,552,233) (-19/45,-27/64) -> (45/8,17/3) Hyperbolic Matrix(343,144,474,199) (-8/19,-5/12) -> (13/18,8/11) Hyperbolic Matrix(1045,432,612,253) (-12/29,-7/17) -> (29/17,12/7) Hyperbolic Matrix(89,36,42,17) (-7/17,-2/5) -> (2/1,11/5) Hyperbolic Matrix(91,36,48,19) (-2/5,-7/18) -> (11/6,2/1) Hyperbolic Matrix(1117,432,468,181) (-12/31,-5/13) -> (31/13,12/5) Hyperbolic Matrix(143,54,-384,-145) (-8/21,-3/8) -> (-3/8,-10/27) Parabolic Matrix(1799,666,678,251) (-10/27,-27/73) -> (45/17,8/3) Hyperbolic Matrix(197,72,342,125) (-7/19,-4/11) -> (4/7,11/19) Hyperbolic Matrix(251,90,198,71) (-4/11,-5/14) -> (5/4,14/11) Hyperbolic Matrix(307,108,54,19) (-6/17,-1/3) -> (17/3,6/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(253,-90,402,-143) (1/3,5/14) -> (5/8,17/27) Hyperbolic Matrix(631,-234,240,-89) (7/19,3/8) -> (21/8,29/11) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(325,-126,276,-107) (5/13,7/18) -> (7/6,13/11) Hyperbolic Matrix(217,-90,258,-107) (7/17,5/12) -> (5/6,11/13) Hyperbolic Matrix(467,-198,342,-145) (8/19,3/7) -> (15/11,26/19) Hyperbolic Matrix(163,-72,120,-53) (3/7,4/9) -> (4/3,15/11) Hyperbolic Matrix(271,-126,114,-53) (5/11,1/2) -> (19/8,31/13) Hyperbolic Matrix(163,-90,96,-53) (1/2,5/9) -> (5/3,17/10) Hyperbolic Matrix(127,-72,30,-17) (5/9,4/7) -> (4/1,13/3) Hyperbolic Matrix(559,-324,402,-233) (11/19,7/12) -> (25/18,7/5) Hyperbolic Matrix(91,-54,150,-89) (7/12,3/5) -> (3/5,11/18) Parabolic Matrix(235,-144,204,-125) (11/18,8/13) -> (8/7,7/6) Hyperbolic Matrix(181,-126,102,-71) (2/3,7/10) -> (7/4,16/9) Hyperbolic Matrix(127,-90,24,-17) (7/10,5/7) -> (5/1,11/2) Hyperbolic Matrix(451,-324,174,-125) (5/7,13/18) -> (31/12,13/5) Hyperbolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(325,-252,138,-107) (10/13,7/9) -> (7/3,26/11) Hyperbolic Matrix(163,-180,48,-53) (1/1,8/7) -> (44/13,17/5) Hyperbolic Matrix(433,-594,78,-107) (26/19,11/8) -> (11/2,28/5) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(667,-1062,282,-449) (27/17,8/5) -> (26/11,45/19) Hyperbolic Matrix(919,-1566,348,-593) (17/10,29/17) -> (29/11,37/14) Hyperbolic Matrix(1063,-2520,402,-953) (45/19,19/8) -> (37/14,45/17) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(809,-2736,144,-487) (27/8,44/13) -> (28/5,45/8) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,144,12,91) -> Matrix(1,2,0,1) Matrix(19,126,30,199) -> Matrix(3,4,8,11) Matrix(17,90,-24,-127) -> Matrix(1,0,0,1) Matrix(19,90,42,199) -> Matrix(1,2,4,9) Matrix(55,252,12,55) -> Matrix(5,6,4,5) Matrix(17,72,-30,-127) -> Matrix(3,2,-8,-5) Matrix(19,72,24,91) -> Matrix(3,2,4,3) Matrix(109,396,30,109) -> Matrix(1,0,4,1) Matrix(91,324,66,235) -> Matrix(1,0,2,1) Matrix(17,54,-6,-19) -> Matrix(1,0,0,1) Matrix(125,342,72,197) -> Matrix(1,2,0,1) Matrix(359,972,-612,-1657) -> Matrix(3,2,-8,-5) Matrix(181,486,54,145) -> Matrix(3,2,4,3) Matrix(55,144,-144,-377) -> Matrix(1,0,-2,1) Matrix(125,324,-174,-451) -> Matrix(1,0,2,1) Matrix(127,324,78,199) -> Matrix(1,0,0,1) Matrix(37,90,30,73) -> Matrix(1,0,2,1) Matrix(53,126,-114,-271) -> Matrix(3,2,-8,-5) Matrix(145,342,-198,-467) -> Matrix(3,2,-8,-5) Matrix(55,126,24,55) -> Matrix(1,0,2,1) Matrix(73,162,114,253) -> Matrix(5,2,12,5) Matrix(17,36,42,89) -> Matrix(1,0,4,1) Matrix(19,36,48,91) -> Matrix(1,0,4,1) Matrix(109,198,60,109) -> Matrix(1,0,2,1) Matrix(71,126,-102,-181) -> Matrix(5,2,-8,-3) Matrix(145,252,42,73) -> Matrix(1,0,4,1) Matrix(73,126,84,145) -> Matrix(1,0,4,1) Matrix(53,90,-96,-163) -> Matrix(1,0,0,1) Matrix(109,180,66,109) -> Matrix(1,0,6,1) Matrix(199,324,78,127) -> Matrix(1,0,2,1) Matrix(89,144,144,233) -> Matrix(1,0,6,1) Matrix(35,54,-24,-37) -> Matrix(1,0,10,1) Matrix(341,486,-468,-667) -> Matrix(1,0,-8,1) Matrix(685,972,432,613) -> Matrix(1,0,-4,1) Matrix(89,126,12,17) -> Matrix(1,0,-4,1) Matrix(233,324,-402,-559) -> Matrix(5,-2,-12,5) Matrix(235,324,66,91) -> Matrix(1,0,-2,1) Matrix(53,72,-120,-163) -> Matrix(1,0,-6,1) Matrix(55,72,42,55) -> Matrix(3,-2,2,-1) Matrix(127,162,156,199) -> Matrix(3,-4,4,-5) Matrix(71,90,198,251) -> Matrix(1,-2,4,-7) Matrix(73,90,30,37) -> Matrix(1,-2,0,1) Matrix(107,126,-276,-325) -> Matrix(1,2,-4,-7) Matrix(109,126,-186,-215) -> Matrix(1,0,-2,1) Matrix(145,126,84,73) -> Matrix(3,4,2,3) Matrix(107,90,-258,-217) -> Matrix(3,2,-8,-5) Matrix(109,90,132,109) -> Matrix(1,0,2,1) Matrix(199,162,156,127) -> Matrix(5,4,6,5) Matrix(91,72,24,19) -> Matrix(3,2,4,3) Matrix(71,54,-96,-73) -> Matrix(3,2,-8,-5) Matrix(1331,972,-3600,-2629) -> Matrix(1,0,0,1) Matrix(199,144,474,343) -> Matrix(1,0,2,1) Matrix(1099,792,426,307) -> Matrix(1,0,6,1) Matrix(613,432,972,685) -> Matrix(5,4,16,13) Matrix(487,342,-1152,-809) -> Matrix(3,2,-8,-5) Matrix(235,162,132,91) -> Matrix(1,0,2,1) Matrix(55,36,84,55) -> Matrix(5,2,12,5) Matrix(253,162,114,73) -> Matrix(5,2,2,1) Matrix(199,126,30,19) -> Matrix(1,0,2,1) Matrix(143,90,-402,-253) -> Matrix(5,2,-18,-7) Matrix(233,144,144,89) -> Matrix(1,0,2,1) Matrix(89,54,-150,-91) -> Matrix(3,2,-8,-5) Matrix(827,486,-1962,-1153) -> Matrix(5,2,-18,-7) Matrix(991,576,714,415) -> Matrix(5,2,12,5) Matrix(125,72,342,197) -> Matrix(5,2,12,5) Matrix(289,162,66,37) -> Matrix(7,2,10,3) Matrix(199,108,234,127) -> Matrix(1,0,4,1) Matrix(235,126,-636,-341) -> Matrix(1,0,0,1) Matrix(235,108,198,91) -> Matrix(7,2,10,3) Matrix(199,90,42,19) -> Matrix(5,2,2,1) Matrix(125,54,-294,-127) -> Matrix(1,0,0,1) Matrix(3113,1314,552,233) -> Matrix(9,2,4,1) Matrix(343,144,474,199) -> Matrix(1,0,6,1) Matrix(1045,432,612,253) -> Matrix(19,6,22,7) Matrix(89,36,42,17) -> Matrix(1,0,4,1) Matrix(91,36,48,19) -> Matrix(1,0,4,1) Matrix(1117,432,468,181) -> Matrix(17,4,4,1) Matrix(143,54,-384,-145) -> Matrix(1,0,2,1) Matrix(1799,666,678,251) -> Matrix(5,2,2,1) Matrix(197,72,342,125) -> Matrix(5,2,12,5) Matrix(251,90,198,71) -> Matrix(7,2,10,3) Matrix(307,108,54,19) -> Matrix(9,2,4,1) Matrix(1,0,6,1) -> Matrix(1,0,8,1) Matrix(253,-90,402,-143) -> Matrix(1,0,0,1) Matrix(631,-234,240,-89) -> Matrix(1,0,-2,1) Matrix(377,-144,144,-55) -> Matrix(1,0,-2,1) Matrix(325,-126,276,-107) -> Matrix(5,-2,8,-3) Matrix(217,-90,258,-107) -> Matrix(1,0,0,1) Matrix(467,-198,342,-145) -> Matrix(1,0,4,1) Matrix(163,-72,120,-53) -> Matrix(1,0,-6,1) Matrix(271,-126,114,-53) -> Matrix(7,-2,4,-1) Matrix(163,-90,96,-53) -> Matrix(5,-2,8,-3) Matrix(127,-72,30,-17) -> Matrix(5,-2,8,-3) Matrix(559,-324,402,-233) -> Matrix(5,-2,8,-3) Matrix(91,-54,150,-89) -> Matrix(5,-2,8,-3) Matrix(235,-144,204,-125) -> Matrix(1,0,0,1) Matrix(181,-126,102,-71) -> Matrix(3,-2,2,-1) Matrix(127,-90,24,-17) -> Matrix(3,-2,2,-1) Matrix(451,-324,174,-125) -> Matrix(1,0,0,1) Matrix(73,-54,96,-71) -> Matrix(5,-2,8,-3) Matrix(325,-252,138,-107) -> Matrix(3,-2,2,-1) Matrix(163,-180,48,-53) -> Matrix(1,0,0,1) Matrix(433,-594,78,-107) -> Matrix(3,-2,2,-1) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(667,-1062,282,-449) -> Matrix(1,0,0,1) Matrix(919,-1566,348,-593) -> Matrix(5,-4,4,-3) Matrix(1063,-2520,402,-953) -> Matrix(1,0,0,1) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) Matrix(809,-2736,144,-487) -> Matrix(1,2,0,1) 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): 24 Degree of the the map X: 24 Degree of the the map Y: 108 ----------------------------------------------------------------------- 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): 324 Minimal number of generators: 55 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: 36 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/1 -6/1 -5/1 -4/1 -3/1 -2/1 -8/5 -3/2 -6/5 -1/1 0/1 3/5 2/3 3/4 6/7 1/1 6/5 4/3 3/2 8/5 5/3 12/7 2/1 9/4 7/3 18/7 8/3 3/1 27/8 18/5 4/1 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 -7/1 -3/2 1/0 -6/1 -1/1 -5/1 -1/2 1/0 -9/2 -1/1 -4/1 -1/1 -1/2 0/1 -11/3 -1/2 -18/5 0/1 -7/2 -1/2 1/0 -3/1 -1/2 1/0 -11/4 -1/2 1/0 -19/7 -3/4 -1/2 -27/10 -1/1 -8/3 -1/2 -5/2 -1/2 1/0 -12/5 -1/1 -7/3 -1/2 -9/4 -1/2 -2/1 -1/1 -1/2 0/1 -9/5 -1/2 -7/4 -1/2 -1/4 -19/11 -1/2 -1/4 -12/7 -1/2 -1/4 -5/3 -1/4 -18/11 0/1 -13/8 -1/2 -1/4 -8/5 -1/4 -1/5 0/1 -3/2 0/1 -10/7 0/1 1/6 1/5 -27/19 1/5 -17/12 1/4 -7/5 1/4 1/2 -4/3 1/2 -9/7 1/1 -5/4 3/2 1/0 -6/5 1/0 -1/1 -1/2 1/0 0/1 0/1 1/2 1/4 1/2 5/9 1/2 4/7 1/3 2/5 1/2 7/12 1/2 3/5 1/2 11/18 1/2 8/13 0/1 1/2 1/1 5/8 1/4 1/2 12/19 1/3 7/11 3/8 1/2 9/14 1/2 2/3 1/2 7/10 1/2 1/0 5/7 1/2 1/0 8/11 0/1 1/3 1/2 3/4 1/2 10/13 1/2 2/3 1/1 7/9 1/2 4/5 1/2 2/3 1/1 9/11 1/1 5/6 1/0 6/7 1/2 1/0 1/1 1/2 1/0 8/7 0/1 1/2 1/1 7/6 1/2 6/5 1/1 5/4 1/2 1/0 9/7 1/1 4/3 1/0 11/8 1/2 1/0 18/13 0/1 7/5 1/2 1/0 3/2 1/2 1/0 11/7 1/2 1/0 19/12 1/0 27/17 1/1 8/5 0/1 1/1 1/0 13/8 -1/2 1/0 18/11 0/1 5/3 1/2 17/10 1/2 3/4 12/7 1/1 19/11 5/4 3/2 7/4 3/2 1/0 16/9 1/0 9/5 1/0 2/1 0/1 1/1 1/0 9/4 1/0 7/3 1/0 26/11 0/1 1/1 1/0 45/19 1/1 19/8 3/2 1/0 12/5 1/0 5/2 -1/2 1/0 18/7 0/1 13/5 1/2 1/0 8/3 1/0 3/1 0/1 10/3 1/2 27/8 1/1 44/13 0/1 1/2 1/1 17/5 1/2 1/0 7/2 1/2 1/0 18/5 0/1 11/3 1/2 4/1 0/1 1/2 1/1 13/3 1/2 9/2 1/1 5/1 3/2 1/0 11/2 3/2 1/0 6/1 1/0 7/1 -1/2 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,144,12,91) (-7/1,1/0) -> (11/7,19/12) Hyperbolic Matrix(19,126,30,199) (-7/1,-6/1) -> (12/19,7/11) Hyperbolic Matrix(17,90,27,143) (-6/1,-5/1) -> (5/8,12/19) Hyperbolic Matrix(19,90,-15,-71) (-5/1,-9/2) -> (-9/7,-5/4) Hyperbolic Matrix(17,72,21,89) (-9/2,-4/1) -> (4/5,9/11) Hyperbolic Matrix(19,72,24,91) (-4/1,-11/3) -> (7/9,4/5) Hyperbolic Matrix(109,396,30,109) (-11/3,-18/5) -> (18/5,11/3) Hyperbolic Matrix(91,324,66,235) (-18/5,-7/2) -> (11/8,18/13) 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(505,1368,213,577) (-19/7,-27/10) -> (45/19,19/8) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(55,144,21,55) (-8/3,-5/2) -> (13/5,8/3) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(53,126,45,107) (-12/5,-7/3) -> (7/6,6/5) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(17,36,-9,-19) (-9/4,-2/1) -> (-2/1,-9/5) Parabolic Matrix(71,126,111,197) (-9/5,-7/4) -> (7/11,9/14) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(73,126,84,145) (-19/11,-12/7) -> (6/7,1/1) Hyperbolic Matrix(53,90,63,107) (-12/7,-5/3) -> (5/6,6/7) Hyperbolic Matrix(109,180,66,109) (-5/3,-18/11) -> (18/11,5/3) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(89,144,144,233) (-13/8,-8/5) -> (8/13,5/8) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(1025,1458,303,431) (-10/7,-27/19) -> (27/8,44/13) 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(53,72,39,53) (-7/5,-4/3) -> (4/3,11/8) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(107,126,45,53) (-6/5,-1/1) -> (19/8,12/5) Hyperbolic Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(163,-90,96,-53) (1/2,5/9) -> (5/3,17/10) Hyperbolic Matrix(127,-72,30,-17) (5/9,4/7) -> (4/1,13/3) Hyperbolic Matrix(125,-72,33,-19) (4/7,7/12) -> (11/3,4/1) Hyperbolic Matrix(91,-54,150,-89) (7/12,3/5) -> (3/5,11/18) Parabolic Matrix(235,-144,204,-125) (11/18,8/13) -> (8/7,7/6) Hyperbolic Matrix(251,-162,141,-91) (9/14,2/3) -> (16/9,9/5) Hyperbolic Matrix(181,-126,102,-71) (2/3,7/10) -> (7/4,16/9) Hyperbolic Matrix(127,-90,24,-17) (7/10,5/7) -> (5/1,11/2) Hyperbolic Matrix(199,-144,123,-89) (5/7,8/11) -> (8/5,13/8) Hyperbolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(325,-252,138,-107) (10/13,7/9) -> (7/3,26/11) Hyperbolic Matrix(197,-162,45,-37) (9/11,5/6) -> (13/3,9/2) Hyperbolic Matrix(163,-180,48,-53) (1/1,8/7) -> (44/13,17/5) Hyperbolic Matrix(71,-90,15,-19) (5/4,9/7) -> (9/2,5/1) Hyperbolic Matrix(181,-252,51,-71) (18/13,7/5) -> (7/2,18/5) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(667,-1062,282,-449) (27/17,8/5) -> (26/11,45/19) Hyperbolic Matrix(287,-468,111,-181) (13/8,18/11) -> (18/7,13/5) Hyperbolic Matrix(253,-432,147,-251) (17/10,12/7) -> (12/7,19/11) Parabolic Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(19,-108,3,-17) (11/2,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,144,12,91) -> Matrix(1,2,0,1) Matrix(19,126,30,199) -> Matrix(3,4,8,11) Matrix(17,90,27,143) -> Matrix(1,0,4,1) Matrix(19,90,-15,-71) -> Matrix(1,2,0,1) Matrix(17,72,21,89) -> Matrix(3,2,4,3) Matrix(19,72,24,91) -> Matrix(3,2,4,3) Matrix(109,396,30,109) -> Matrix(1,0,4,1) Matrix(91,324,66,235) -> Matrix(1,0,2,1) Matrix(17,54,-6,-19) -> Matrix(1,0,0,1) Matrix(125,342,72,197) -> Matrix(1,2,0,1) Matrix(505,1368,213,577) -> Matrix(1,0,2,1) Matrix(181,486,54,145) -> Matrix(3,2,4,3) Matrix(55,144,21,55) -> Matrix(1,0,2,1) Matrix(37,90,30,73) -> Matrix(1,0,2,1) Matrix(53,126,45,107) -> Matrix(3,2,4,3) Matrix(55,126,24,55) -> Matrix(1,0,2,1) Matrix(17,36,-9,-19) -> Matrix(1,0,0,1) Matrix(71,126,111,197) -> Matrix(5,2,12,5) Matrix(145,252,42,73) -> Matrix(1,0,4,1) Matrix(73,126,84,145) -> Matrix(1,0,4,1) Matrix(53,90,63,107) -> Matrix(1,0,4,1) Matrix(109,180,66,109) -> Matrix(1,0,6,1) Matrix(199,324,78,127) -> Matrix(1,0,2,1) Matrix(89,144,144,233) -> Matrix(1,0,6,1) Matrix(35,54,-24,-37) -> Matrix(1,0,10,1) Matrix(1025,1458,303,431) -> Matrix(1,0,-4,1) Matrix(685,972,432,613) -> Matrix(1,0,-4,1) Matrix(89,126,12,17) -> Matrix(1,0,-4,1) Matrix(53,72,39,53) -> Matrix(1,0,-2,1) Matrix(55,72,42,55) -> Matrix(3,-2,2,-1) Matrix(73,90,30,37) -> Matrix(1,-2,0,1) Matrix(107,126,45,53) -> Matrix(1,2,0,1) Matrix(1,0,3,1) -> Matrix(1,0,4,1) Matrix(163,-90,96,-53) -> Matrix(5,-2,8,-3) Matrix(127,-72,30,-17) -> Matrix(5,-2,8,-3) Matrix(125,-72,33,-19) -> Matrix(5,-2,8,-3) Matrix(91,-54,150,-89) -> Matrix(5,-2,8,-3) Matrix(235,-144,204,-125) -> Matrix(1,0,0,1) Matrix(251,-162,141,-91) -> Matrix(1,0,-2,1) Matrix(181,-126,102,-71) -> Matrix(3,-2,2,-1) Matrix(127,-90,24,-17) -> Matrix(3,-2,2,-1) Matrix(199,-144,123,-89) -> Matrix(1,0,-2,1) Matrix(73,-54,96,-71) -> Matrix(5,-2,8,-3) Matrix(325,-252,138,-107) -> Matrix(3,-2,2,-1) Matrix(197,-162,45,-37) -> Matrix(1,-2,2,-3) Matrix(163,-180,48,-53) -> Matrix(1,0,0,1) Matrix(71,-90,15,-19) -> Matrix(3,-2,2,-1) Matrix(181,-252,51,-71) -> Matrix(1,0,0,1) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(667,-1062,282,-449) -> Matrix(1,0,0,1) Matrix(287,-468,111,-181) -> Matrix(1,0,2,1) Matrix(253,-432,147,-251) -> Matrix(7,-6,6,-5) Matrix(19,-36,9,-17) -> Matrix(1,0,0,1) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) Matrix(19,-108,3,-17) -> Matrix(1,-2,0,1) 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 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): 12 ----------------------------------------------------------------------- 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 -3/1 (-1/2,1/0) 0 3 -8/3 -1/2 2 9 -5/2 0 9 -12/5 -1/1 2 3 -7/3 -1/2 1 9 -2/1 0 9 -7/4 0 9 -12/7 (-1/2,-1/4) 0 3 -5/3 -1/4 1 9 -8/5 0 9 -3/2 0/1 5 3 -7/5 0 9 -4/3 1/2 2 9 -6/5 1/0 4 3 -1/1 0 9 0/1 0/1 4 3 1/2 0 9 5/9 1/2 1 9 4/7 0 9 7/12 1/2 1 9 3/5 1/2 1 3 2/3 1/2 2 9 5/7 0 9 3/4 1/2 1 3 7/9 1/2 1 9 4/5 0 9 1/1 0 9 6/5 1/1 2 3 5/4 0 9 4/3 1/0 2 9 3/2 (1/2,1/0) 0 3 5/3 1/2 1 9 17/10 0 9 12/7 1/1 6 3 7/4 0 9 9/5 1/0 1 3 2/1 0 9 9/4 1/0 1 3 7/3 1/0 1 9 12/5 1/0 4 3 5/2 0 9 18/7 0/1 6 3 13/5 0 9 8/3 1/0 2 9 3/1 0/1 1 3 10/3 1/2 2 9 7/2 0 9 18/5 0/1 2 3 11/3 1/2 1 9 4/1 0 9 9/2 1/1 5 3 5/1 0 9 6/1 1/0 2 3 1/0 1/0 1 9 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,-1) (-3/1,1/0) -> (-3/1,1/0) Reflection Matrix(17,48,-6,-17) (-3/1,-8/3) -> (-3/1,-8/3) Reflection Matrix(55,144,21,55) (-8/3,-5/2) -> (13/5,8/3) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(109,258,30,71) (-12/5,-7/3) -> (18/5,11/3) Glide Reflection Matrix(19,42,24,53) (-7/3,-2/1) -> (7/9,4/5) Glide Reflection Matrix(17,30,21,37) (-2/1,-7/4) -> (4/5,1/1) Glide Reflection Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(181,312,-105,-181) (-26/15,-12/7) -> (-26/15,-12/7) Reflection Matrix(71,120,-42,-71) (-12/7,-5/3) -> (-12/7,-5/3) Reflection Matrix(37,60,66,107) (-5/3,-8/5) -> (5/9,4/7) Glide Reflection Matrix(53,84,12,19) (-8/5,-3/2) -> (4/1,9/2) Glide Reflection Matrix(55,78,12,17) (-3/2,-7/5) -> (9/2,5/1) Glide Reflection Matrix(35,48,27,37) (-7/5,-4/3) -> (5/4,4/3) Glide Reflection Matrix(19,24,-15,-19) (-4/3,-6/5) -> (-4/3,-6/5) Reflection Matrix(37,42,15,17) (-6/5,-1/1) -> (12/5,5/2) Glide Reflection Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(163,-90,96,-53) (1/2,5/9) -> (5/3,17/10) Hyperbolic Matrix(125,-72,33,-19) (4/7,7/12) -> (11/3,4/1) Hyperbolic Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(35,-24,51,-35) (2/3,12/17) -> (2/3,12/17) Reflection Matrix(127,-90,24,-17) (7/10,5/7) -> (5/1,11/2) Hyperbolic Matrix(91,-66,51,-37) (5/7,3/4) -> (7/4,9/5) Glide Reflection Matrix(109,-84,48,-37) (3/4,7/9) -> (9/4,7/3) Glide Reflection Matrix(53,-60,15,-17) (1/1,6/5) -> (7/2,18/5) Glide Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(215,-366,84,-143) (17/10,12/7) -> (5/2,18/7) Glide Reflection Matrix(163,-282,63,-109) (12/7,7/4) -> (18/7,13/5) Glide Reflection Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,6,0,-1) -> Matrix(1,1,0,-1) (-3/1,1/0) -> (-1/2,1/0) Matrix(17,48,-6,-17) -> Matrix(1,1,0,-1) (-3/1,-8/3) -> (-1/2,1/0) Matrix(55,144,21,55) -> Matrix(1,0,2,1) 0/1 Matrix(37,90,30,73) -> Matrix(1,0,2,1) 0/1 Matrix(109,258,30,71) -> Matrix(1,1,4,3) Matrix(19,42,24,53) -> Matrix(3,1,4,1) Matrix(17,30,21,37) -> Matrix(3,1,4,1) Matrix(145,252,42,73) -> Matrix(1,0,4,1) 0/1 Matrix(181,312,-105,-181) -> Matrix(3,1,-8,-3) (-26/15,-12/7) -> (-1/2,-1/4) Matrix(71,120,-42,-71) -> Matrix(3,1,-8,-3) (-12/7,-5/3) -> (-1/2,-1/4) Matrix(37,60,66,107) -> Matrix(3,1,10,3) Matrix(53,84,12,19) -> Matrix(5,1,6,1) Matrix(55,78,12,17) -> Matrix(5,-1,4,-1) Matrix(35,48,27,37) -> Matrix(3,-1,2,-1) Matrix(19,24,-15,-19) -> Matrix(-1,1,0,1) (-4/3,-6/5) -> (1/2,1/0) Matrix(37,42,15,17) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(1,0,3,1) -> Matrix(1,0,4,1) 0/1 Matrix(163,-90,96,-53) -> Matrix(5,-2,8,-3) 1/2 Matrix(125,-72,33,-19) -> Matrix(5,-2,8,-3) 1/2 Matrix(71,-42,120,-71) -> Matrix(7,-3,16,-7) (7/12,3/5) -> (3/8,1/2) Matrix(19,-12,30,-19) -> Matrix(3,-1,8,-3) (3/5,2/3) -> (1/4,1/2) Matrix(35,-24,51,-35) -> Matrix(-1,1,0,1) (2/3,12/17) -> (1/2,1/0) Matrix(127,-90,24,-17) -> Matrix(3,-2,2,-1) 1/1 Matrix(91,-66,51,-37) -> Matrix(3,-1,2,-1) Matrix(109,-84,48,-37) -> Matrix(1,-1,-2,1) Matrix(53,-60,15,-17) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(17,-24,12,-17) -> Matrix(-1,1,0,1) (4/3,3/2) -> (1/2,1/0) Matrix(19,-30,12,-19) -> Matrix(-1,1,0,1) (3/2,5/3) -> (1/2,1/0) Matrix(215,-366,84,-143) -> Matrix(1,-1,-4,3) Matrix(163,-282,63,-109) -> Matrix(1,-1,2,-3) Matrix(19,-36,9,-17) -> Matrix(1,0,0,1) Matrix(71,-168,30,-71) -> Matrix(-1,1,0,1) (7/3,12/5) -> (1/2,1/0) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) 0/1 Matrix(71,-240,21,-71) -> Matrix(-1,1,0,1) (10/3,24/7) -> (1/2,1/0) Matrix(17,-96,3,-17) -> Matrix(-1,3,0,1) (16/3,6/1) -> (3/2,1/0) Matrix(-1,12,0,1) -> Matrix(-1,1,0,1) (6/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.