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 1/0 -6/1 -3/1 -1/1 -5/1 1/0 -14/3 -1/1 -9/2 -1/1 -4/1 -1/1 1/1 -11/3 0/1 -18/5 1/1 -7/2 2/1 -3/1 1/0 -11/4 -8/1 -19/7 1/0 -27/10 -7/1 -8/3 -7/1 -13/5 -11/2 -18/7 -5/1 -5/2 -4/1 -12/5 -5/1 -3/1 -19/8 -4/1 -7/3 -4/1 -9/4 -3/1 -11/5 -5/2 -2/1 -3/1 -1/1 -11/6 -3/2 -9/5 -1/1 -7/4 0/1 -19/11 1/0 -12/7 -1/1 -5/3 0/1 -18/11 1/1 -13/8 2/1 -8/5 1/1 3/1 -3/2 1/0 -10/7 -7/1 -5/1 -27/19 -5/1 -17/12 1/0 -7/5 1/0 -18/13 -5/1 -11/8 -4/1 -4/3 -3/1 -9/7 -3/1 -14/11 -3/1 -7/3 -5/4 -2/1 -6/5 -3/1 -1/1 -7/6 1/0 -1/1 1/0 -6/7 -3/1 -5/6 1/0 -9/11 -3/1 -4/5 -3/1 -1/1 -7/9 -2/1 -3/4 1/0 -11/15 -4/1 -19/26 -4/1 -27/37 -3/1 -8/11 -5/1 -3/1 -13/18 -7/2 -18/25 -3/1 -5/7 1/0 -12/17 -5/1 -3/1 -19/27 -4/1 -7/10 -4/1 -9/13 -3/1 -2/3 -3/1 -9/14 -3/1 -7/11 -5/2 -12/19 -3/1 -7/3 -5/8 -2/1 -8/13 -3/1 -7/3 -3/5 -2/1 -10/17 -3/1 -1/1 -27/46 -1/1 -17/29 1/0 -7/12 1/0 -18/31 -3/1 -11/19 -5/2 -4/7 -3/1 -1/1 -9/16 -3/1 -5/9 -2/1 -6/11 -3/1 -7/13 -5/2 -1/2 -2/1 -6/13 -3/1 -1/1 -5/11 1/0 -4/9 -3/1 -7/16 -2/1 -3/7 -2/1 -11/26 -2/1 -19/45 -2/1 -27/64 -9/5 -8/19 -9/5 -5/3 -5/12 -3/2 -12/29 -1/1 -7/17 -3/2 -2/5 -3/1 -1/1 -7/18 -5/2 -12/31 -3/1 -7/3 -5/13 -5/2 -8/21 -11/5 -3/8 -2/1 -10/27 -9/5 -27/73 -9/5 -17/46 -2/1 -7/19 -7/4 -4/11 -5/3 -1/1 -5/14 -2/1 -6/17 -3/1 -1/1 -1/3 -2/1 0/1 -1/1 1/3 0/1 5/14 0/1 4/11 -1/1 -1/3 7/19 -1/4 3/8 0/1 5/13 1/2 7/18 1/2 2/5 -1/1 1/1 7/17 -1/2 5/12 -1/2 8/19 -1/3 -1/5 3/7 0/1 4/9 1/1 5/11 1/0 1/2 0/1 5/9 0/1 4/7 -1/1 1/1 11/19 1/2 7/12 1/0 3/5 0/1 11/18 1/2 8/13 1/3 1/1 5/8 0/1 17/27 0/1 12/19 1/3 1/1 7/11 1/2 9/14 1/1 2/3 1/1 7/10 2/1 5/7 1/0 13/18 3/2 8/11 1/1 3/1 3/4 1/0 10/13 -1/1 1/1 7/9 0/1 4/5 -1/1 1/1 9/11 1/1 5/6 1/0 11/13 1/0 6/7 1/1 1/1 1/0 8/7 -1/1 1/1 7/6 1/0 13/11 1/0 6/5 -1/1 1/1 5/4 0/1 14/11 1/3 1/1 9/7 1/1 4/3 1/1 15/11 2/1 26/19 7/3 3/1 11/8 2/1 18/13 3/1 25/18 7/2 7/5 1/0 3/2 1/0 11/7 1/0 19/12 1/0 27/17 -5/1 8/5 -5/1 -3/1 13/8 -4/1 18/11 -3/1 5/3 -2/1 17/10 -2/1 29/17 -3/2 12/7 -1/1 19/11 1/0 7/4 -2/1 16/9 -1/1 9/5 -1/1 11/6 -1/2 2/1 -1/1 1/1 11/5 1/2 9/4 1/1 7/3 2/1 26/11 1/1 3/1 45/19 1/1 19/8 2/1 31/13 1/0 12/5 1/1 3/1 5/2 2/1 18/7 3/1 31/12 13/4 13/5 7/2 21/8 4/1 29/11 9/2 37/14 4/1 45/17 5/1 8/3 5/1 3/1 1/0 10/3 -5/1 27/8 -5/1 44/13 -5/1 -3/1 17/5 1/0 7/2 -4/1 18/5 -3/1 11/3 -2/1 4/1 -3/1 -1/1 13/3 -2/1 9/2 -1/1 14/3 -1/1 5/1 1/0 11/2 0/1 28/5 -1/1 1/1 45/8 -1/1 17/3 0/1 6/1 -1/1 1/1 7/1 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(1,2,2,5) Matrix(17,90,-24,-127) -> Matrix(1,-2,0,1) Matrix(19,90,42,199) -> Matrix(1,2,0,1) Matrix(55,252,12,55) -> Matrix(1,2,-2,-3) Matrix(17,72,-30,-127) -> Matrix(1,-2,0,1) Matrix(19,72,24,91) -> Matrix(1,0,0,1) Matrix(109,396,30,109) -> Matrix(5,-2,-2,1) Matrix(91,324,66,235) -> Matrix(5,-8,2,-3) Matrix(17,54,-6,-19) -> Matrix(1,-10,0,1) Matrix(125,342,72,197) -> Matrix(1,6,0,1) Matrix(359,972,-612,-1657) -> Matrix(1,6,0,1) Matrix(181,486,54,145) -> Matrix(1,2,0,1) Matrix(55,144,-144,-377) -> Matrix(9,52,-4,-23) Matrix(125,324,-174,-451) -> Matrix(5,28,-2,-11) Matrix(127,324,78,199) -> Matrix(7,32,-2,-9) Matrix(37,90,30,73) -> Matrix(1,4,0,1) Matrix(53,126,-114,-271) -> Matrix(1,2,0,1) Matrix(145,342,-198,-467) -> Matrix(1,0,0,1) Matrix(55,126,24,55) -> Matrix(3,10,2,7) Matrix(73,162,114,253) -> Matrix(3,8,4,11) Matrix(17,36,42,89) -> Matrix(1,2,0,1) Matrix(19,36,48,91) -> Matrix(1,2,0,1) Matrix(109,198,60,109) -> Matrix(3,4,-4,-5) Matrix(71,126,-102,-181) -> Matrix(7,4,-2,-1) Matrix(145,252,42,73) -> Matrix(1,-4,0,1) Matrix(73,126,84,145) -> Matrix(1,2,0,1) Matrix(53,90,-96,-163) -> Matrix(1,-2,0,1) Matrix(109,180,66,109) -> Matrix(5,-2,-2,1) Matrix(199,324,78,127) -> Matrix(5,-8,2,-3) Matrix(89,144,144,233) -> Matrix(1,-2,2,-3) Matrix(35,54,-24,-37) -> Matrix(1,-8,0,1) Matrix(341,486,-468,-667) -> Matrix(1,2,0,1) Matrix(685,972,432,613) -> Matrix(1,0,0,1) Matrix(89,126,12,17) -> Matrix(1,6,0,1) Matrix(233,324,-402,-559) -> Matrix(5,28,-2,-11) Matrix(235,324,66,91) -> Matrix(7,32,-2,-9) Matrix(53,72,-120,-163) -> Matrix(5,18,-2,-7) Matrix(55,72,42,55) -> Matrix(1,4,0,1) Matrix(127,162,156,199) -> Matrix(1,2,2,5) Matrix(71,90,198,251) -> Matrix(1,2,0,1) Matrix(73,90,30,37) -> Matrix(1,4,0,1) Matrix(107,126,-276,-325) -> Matrix(5,8,-2,-3) Matrix(109,126,-186,-215) -> Matrix(1,0,0,1) Matrix(145,126,84,73) -> Matrix(1,2,0,1) Matrix(107,90,-258,-217) -> Matrix(3,8,-2,-5) Matrix(109,90,132,109) -> Matrix(1,4,0,1) Matrix(199,162,156,127) -> Matrix(1,2,2,5) Matrix(91,72,24,19) -> Matrix(1,0,0,1) Matrix(71,54,-96,-73) -> Matrix(1,-2,0,1) Matrix(1331,972,-3600,-2629) -> Matrix(7,30,-4,-17) Matrix(199,144,474,343) -> Matrix(1,4,-4,-15) Matrix(1099,792,426,307) -> Matrix(19,60,6,19) Matrix(613,432,972,685) -> Matrix(1,4,2,9) Matrix(487,342,-1152,-809) -> Matrix(7,30,-4,-17) Matrix(235,162,132,91) -> Matrix(3,8,-2,-5) Matrix(55,36,84,55) -> Matrix(1,2,2,5) Matrix(253,162,114,73) -> Matrix(3,8,4,11) Matrix(199,126,30,19) -> Matrix(1,2,2,5) Matrix(143,90,-402,-253) -> Matrix(3,8,-2,-5) Matrix(233,144,144,89) -> Matrix(7,18,-2,-5) Matrix(89,54,-150,-91) -> Matrix(3,8,-2,-5) Matrix(827,486,-1962,-1153) -> Matrix(7,16,-4,-9) Matrix(991,576,714,415) -> Matrix(7,24,2,7) Matrix(125,72,342,197) -> Matrix(1,2,-2,-3) Matrix(289,162,66,37) -> Matrix(3,8,-2,-5) Matrix(199,108,234,127) -> Matrix(1,2,2,5) Matrix(235,126,-636,-341) -> Matrix(11,24,-6,-13) Matrix(235,108,198,91) -> Matrix(1,2,0,1) Matrix(199,90,42,19) -> Matrix(1,2,0,1) Matrix(125,54,-294,-127) -> Matrix(15,32,-8,-17) Matrix(3113,1314,552,233) -> Matrix(1,2,-6,-11) Matrix(343,144,474,199) -> Matrix(7,12,4,7) Matrix(1045,432,612,253) -> Matrix(1,0,0,1) Matrix(89,36,42,17) -> Matrix(1,2,0,1) Matrix(91,36,48,19) -> Matrix(1,2,0,1) Matrix(1117,432,468,181) -> Matrix(5,12,2,5) Matrix(143,54,-384,-145) -> Matrix(19,40,-10,-21) Matrix(1799,666,678,251) -> Matrix(15,26,4,7) Matrix(197,72,342,125) -> Matrix(1,2,-2,-3) Matrix(251,90,198,71) -> Matrix(1,2,0,1) Matrix(307,108,54,19) -> Matrix(1,2,0,1) Matrix(1,0,6,1) -> Matrix(1,2,-2,-3) Matrix(253,-90,402,-143) -> Matrix(1,0,2,1) Matrix(631,-234,240,-89) -> Matrix(25,4,6,1) Matrix(377,-144,144,-55) -> Matrix(15,-4,4,-1) Matrix(325,-126,276,-107) -> Matrix(1,0,-2,1) Matrix(217,-90,258,-107) -> Matrix(1,0,2,1) Matrix(467,-198,342,-145) -> Matrix(13,2,6,1) Matrix(163,-72,120,-53) -> Matrix(3,-2,2,-1) Matrix(271,-126,114,-53) -> Matrix(1,2,0,1) Matrix(163,-90,96,-53) -> Matrix(1,-2,0,1) Matrix(127,-72,30,-17) -> Matrix(1,-2,0,1) Matrix(559,-324,402,-233) -> Matrix(7,-4,2,-1) Matrix(91,-54,150,-89) -> Matrix(1,0,2,1) Matrix(235,-144,204,-125) -> Matrix(1,0,-2,1) Matrix(181,-126,102,-71) -> Matrix(3,-4,-2,3) Matrix(127,-90,24,-17) -> Matrix(1,-2,0,1) Matrix(451,-324,174,-125) -> Matrix(7,-4,2,-1) Matrix(73,-54,96,-71) -> Matrix(1,-2,0,1) Matrix(325,-252,138,-107) -> Matrix(1,2,0,1) Matrix(163,-180,48,-53) -> Matrix(1,-4,0,1) Matrix(433,-594,78,-107) -> Matrix(1,-2,-2,5) Matrix(37,-54,24,-35) -> Matrix(1,-8,0,1) Matrix(667,-1062,282,-449) -> Matrix(1,6,0,1) Matrix(919,-1566,348,-593) -> Matrix(1,6,0,1) Matrix(1063,-2520,402,-953) -> Matrix(9,-14,2,-3) Matrix(19,-54,6,-17) -> Matrix(1,-10,0,1) Matrix(809,-2736,144,-487) -> 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): 32 Degree of the the map X: 32 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 1/0 -6/1 -3/1 -1/1 -5/1 1/0 -9/2 -1/1 -4/1 -1/1 1/1 -11/3 0/1 -18/5 1/1 -7/2 2/1 -3/1 1/0 -11/4 -8/1 -19/7 1/0 -27/10 -7/1 -8/3 -7/1 -5/2 -4/1 -12/5 -5/1 -3/1 -7/3 -4/1 -9/4 -3/1 -2/1 -3/1 -1/1 -9/5 -1/1 -7/4 0/1 -19/11 1/0 -12/7 -1/1 -5/3 0/1 -18/11 1/1 -13/8 2/1 -8/5 1/1 3/1 -3/2 1/0 -10/7 -7/1 -5/1 -27/19 -5/1 -17/12 1/0 -7/5 1/0 -4/3 -3/1 -9/7 -3/1 -5/4 -2/1 -6/5 -3/1 -1/1 -1/1 1/0 0/1 -1/1 1/2 0/1 5/9 0/1 4/7 -1/1 1/1 7/12 1/0 3/5 0/1 11/18 1/2 8/13 1/3 1/1 5/8 0/1 12/19 1/3 1/1 7/11 1/2 9/14 1/1 2/3 1/1 7/10 2/1 5/7 1/0 8/11 1/1 3/1 3/4 1/0 10/13 -1/1 1/1 7/9 0/1 4/5 -1/1 1/1 9/11 1/1 5/6 1/0 6/7 1/1 1/1 1/0 8/7 -1/1 1/1 7/6 1/0 6/5 -1/1 1/1 5/4 0/1 9/7 1/1 4/3 1/1 11/8 2/1 18/13 3/1 7/5 1/0 3/2 1/0 11/7 1/0 19/12 1/0 27/17 -5/1 8/5 -5/1 -3/1 13/8 -4/1 18/11 -3/1 5/3 -2/1 17/10 -2/1 12/7 -1/1 19/11 1/0 7/4 -2/1 16/9 -1/1 9/5 -1/1 2/1 -1/1 1/1 9/4 1/1 7/3 2/1 26/11 1/1 3/1 45/19 1/1 19/8 2/1 12/5 1/1 3/1 5/2 2/1 18/7 3/1 13/5 7/2 8/3 5/1 3/1 1/0 10/3 -5/1 27/8 -5/1 44/13 -5/1 -3/1 17/5 1/0 7/2 -4/1 18/5 -3/1 11/3 -2/1 4/1 -3/1 -1/1 13/3 -2/1 9/2 -1/1 5/1 1/0 11/2 0/1 6/1 -1/1 1/1 7/1 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(1,2,2,5) Matrix(17,90,27,143) -> Matrix(0,-1,1,0) Matrix(19,90,-15,-71) -> Matrix(2,5,-1,-2) Matrix(17,72,21,89) -> Matrix(0,-1,1,0) Matrix(19,72,24,91) -> Matrix(1,0,0,1) Matrix(109,396,30,109) -> Matrix(5,-2,-2,1) Matrix(91,324,66,235) -> Matrix(5,-8,2,-3) Matrix(17,54,-6,-19) -> Matrix(1,-10,0,1) Matrix(125,342,72,197) -> Matrix(1,6,0,1) Matrix(505,1368,213,577) -> Matrix(2,15,1,8) Matrix(181,486,54,145) -> Matrix(1,2,0,1) Matrix(55,144,21,55) -> Matrix(4,23,1,6) Matrix(37,90,30,73) -> Matrix(1,4,0,1) Matrix(53,126,45,107) -> Matrix(0,-1,1,4) Matrix(55,126,24,55) -> Matrix(3,10,2,7) Matrix(17,36,-9,-19) -> Matrix(2,5,-1,-2) Matrix(71,126,111,197) -> Matrix(2,1,3,2) Matrix(145,252,42,73) -> Matrix(1,-4,0,1) Matrix(73,126,84,145) -> Matrix(1,2,0,1) Matrix(53,90,63,107) -> Matrix(0,-1,1,0) Matrix(109,180,66,109) -> Matrix(5,-2,-2,1) Matrix(199,324,78,127) -> Matrix(5,-8,2,-3) Matrix(89,144,144,233) -> Matrix(1,-2,2,-3) Matrix(35,54,-24,-37) -> Matrix(1,-8,0,1) Matrix(1025,1458,303,431) -> Matrix(4,25,-1,-6) Matrix(685,972,432,613) -> Matrix(1,0,0,1) Matrix(89,126,12,17) -> Matrix(1,6,0,1) Matrix(53,72,39,53) -> Matrix(2,7,1,4) Matrix(55,72,42,55) -> Matrix(1,4,0,1) Matrix(73,90,30,37) -> Matrix(1,4,0,1) Matrix(107,126,45,53) -> Matrix(2,3,1,2) Matrix(1,0,3,1) -> Matrix(0,-1,1,2) Matrix(163,-90,96,-53) -> Matrix(1,-2,0,1) Matrix(127,-72,30,-17) -> Matrix(1,-2,0,1) Matrix(125,-72,33,-19) -> Matrix(2,1,-1,0) Matrix(91,-54,150,-89) -> Matrix(1,0,2,1) Matrix(235,-144,204,-125) -> Matrix(1,0,-2,1) Matrix(251,-162,141,-91) -> Matrix(2,-3,-1,2) Matrix(181,-126,102,-71) -> Matrix(3,-4,-2,3) Matrix(127,-90,24,-17) -> Matrix(1,-2,0,1) Matrix(199,-144,123,-89) -> Matrix(4,-7,-1,2) Matrix(73,-54,96,-71) -> Matrix(1,-2,0,1) Matrix(325,-252,138,-107) -> Matrix(1,2,0,1) Matrix(197,-162,45,-37) -> Matrix(2,-3,-1,2) Matrix(163,-180,48,-53) -> Matrix(1,-4,0,1) Matrix(71,-90,15,-19) -> Matrix(0,-1,1,0) Matrix(181,-252,51,-71) -> Matrix(4,-15,-1,4) Matrix(37,-54,24,-35) -> Matrix(1,-8,0,1) Matrix(667,-1062,282,-449) -> Matrix(1,6,0,1) Matrix(287,-468,111,-181) -> Matrix(10,33,3,10) Matrix(253,-432,147,-251) -> Matrix(0,-1,1,2) Matrix(19,-36,9,-17) -> Matrix(0,-1,1,0) Matrix(19,-54,6,-17) -> Matrix(1,-10,0,1) Matrix(19,-108,3,-17) -> Matrix(0,-1,1,0) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 ----------------------------------------------------------------------- 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 -1/1 1 3 2/3 1/1 1 9 7/10 2/1 1 9 5/7 1/0 1 9 8/11 (1/1,3/1).(2/1,1/0) 0 9 3/4 1/0 1 3 10/13 (-1/1,1/1).(0/1,1/0) 0 9 7/9 0/1 2 9 4/5 (-1/1,1/1).(0/1,1/0) 0 9 5/6 1/0 2 9 1/1 1/0 1 9 6/5 (-1/1,1/1).(0/1,1/0) 0 3 5/4 0/1 1 9 9/7 1/1 2 3 4/3 1/1 1 9 18/13 3/1 5 3 7/5 1/0 1 9 3/2 1/0 4 3 11/7 1/0 1 9 19/12 1/0 2 9 27/17 -5/1 1 3 8/5 (-5/1,-3/1).(-4/1,1/0) 0 9 13/8 -4/1 1 9 18/11 -3/1 7 3 5/3 -2/1 2 9 17/10 -2/1 1 9 12/7 -1/1 1 3 19/11 1/0 1 9 7/4 -2/1 1 9 16/9 -1/1 1 9 9/5 -1/1 4 3 2/1 (-1/1,1/1).(0/1,1/0) 0 9 9/4 1/1 4 3 7/3 2/1 2 9 26/11 (1/1,3/1).(2/1,1/0) 0 9 19/8 2/1 1 9 12/5 (1/1,3/1).(2/1,1/0) 0 3 5/2 2/1 1 9 18/7 3/1 7 3 8/3 5/1 1 9 3/1 1/0 5 3 10/3 -5/1 1 9 27/8 -5/1 1 3 44/13 (-5/1,-3/1).(-4/1,1/0) 0 9 17/5 1/0 1 9 7/2 -4/1 1 9 18/5 -3/1 5 3 11/3 -2/1 2 9 4/1 (-3/1,-1/1).(-2/1,1/0) 0 9 13/3 -2/1 2 9 9/2 -1/1 2 3 5/1 1/0 1 9 11/2 0/1 1 9 6/1 (-1/1,1/1).(0/1,1/0) 0 3 7/1 1/0 1 9 1/0 1/0 2 9 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(1,0,3,-1) (0/1,2/3) -> (0/1,2/3) Reflection 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(91,-72,24,-19) (7/9,4/5) -> (11/3,4/1) Glide Reflection Matrix(89,-72,21,-17) (4/5,5/6) -> (4/1,13/3) Glide Reflection Matrix(107,-90,63,-53) (5/6,1/1) -> (5/3,17/10) Glide Reflection Matrix(107,-126,45,-53) (1/1,6/5) -> (19/8,12/5) Glide Reflection Matrix(73,-90,30,-37) (6/5,5/4) -> (12/5,5/2) Glide Reflection Matrix(71,-90,15,-19) (5/4,9/7) -> (9/2,5/1) Hyperbolic Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(53,-72,39,-53) (4/3,18/13) -> (4/3,18/13) Reflection 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(91,-144,12,-19) (11/7,19/12) -> (7/1,1/0) Glide Reflection Matrix(647,-1026,408,-647) (19/12,27/17) -> (19/12,27/17) Reflection Matrix(883,-1404,261,-415) (27/17,8/5) -> (27/8,44/13) Glide Reflection Matrix(199,-324,78,-127) (13/8,18/11) -> (5/2,18/7) Glide Reflection Matrix(109,-180,66,-109) (18/11,5/3) -> (18/11,5/3) Reflection Matrix(253,-432,147,-251) (17/10,12/7) -> (12/7,19/11) Parabolic Matrix(145,-252,42,-73) (19/11,7/4) -> (17/5,7/2) Glide Reflection Matrix(161,-288,90,-161) (16/9,9/5) -> (16/9,9/5) Reflection Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(539,-1278,159,-377) (26/11,19/8) -> (44/13,17/5) Glide Reflection Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(53,-234,12,-53) (13/3,9/2) -> (13/3,9/2) Reflection Matrix(19,-108,3,-17) (11/2,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,2,0,-1) (0/1,1/0) -> (-1/1,1/0) Matrix(1,0,3,-1) -> Matrix(0,1,1,0) (0/1,2/3) -> (-1/1,1/1) Matrix(181,-126,102,-71) -> Matrix(3,-4,-2,3) Matrix(127,-90,24,-17) -> Matrix(1,-2,0,1) 1/0 Matrix(199,-144,123,-89) -> Matrix(4,-7,-1,2) Matrix(73,-54,96,-71) -> Matrix(1,-2,0,1) 1/0 Matrix(325,-252,138,-107) -> Matrix(1,2,0,1) 1/0 Matrix(91,-72,24,-19) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(89,-72,21,-17) -> Matrix(-2,1,1,0) Matrix(107,-90,63,-53) -> Matrix(-2,1,1,0) Matrix(107,-126,45,-53) -> Matrix(2,1,1,0) Matrix(73,-90,30,-37) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(71,-90,15,-19) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(55,-72,42,-55) -> Matrix(-1,2,0,1) (9/7,4/3) -> (1/1,1/0) Matrix(53,-72,39,-53) -> Matrix(2,-3,1,-2) (4/3,18/13) -> (1/1,3/1) Matrix(181,-252,51,-71) -> Matrix(4,-15,-1,4) Matrix(37,-54,24,-35) -> Matrix(1,-8,0,1) 1/0 Matrix(91,-144,12,-19) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(647,-1026,408,-647) -> Matrix(1,10,0,-1) (19/12,27/17) -> (-5/1,1/0) Matrix(883,-1404,261,-415) -> Matrix(4,15,-1,-4) *** -> (-5/1,-3/1) Matrix(199,-324,78,-127) -> Matrix(5,18,2,7) Matrix(109,-180,66,-109) -> Matrix(5,12,-2,-5) (18/11,5/3) -> (-3/1,-2/1) Matrix(253,-432,147,-251) -> Matrix(0,-1,1,2) -1/1 Matrix(145,-252,42,-73) -> Matrix(1,6,0,-1) *** -> (-3/1,1/0) Matrix(161,-288,90,-161) -> Matrix(5,6,-4,-5) (16/9,9/5) -> (-3/2,-1/1) Matrix(19,-36,9,-17) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(55,-126,24,-55) -> Matrix(3,-4,2,-3) (9/4,7/3) -> (1/1,2/1) Matrix(539,-1278,159,-377) -> Matrix(4,-9,-1,2) Matrix(55,-144,21,-55) -> Matrix(4,-15,1,-4) (18/7,8/3) -> (3/1,5/1) Matrix(19,-54,6,-17) -> Matrix(1,-10,0,1) 1/0 Matrix(161,-540,48,-161) -> Matrix(1,10,0,-1) (10/3,27/8) -> (-5/1,1/0) Matrix(109,-396,30,-109) -> Matrix(5,12,-2,-5) (18/5,11/3) -> (-3/1,-2/1) Matrix(53,-234,12,-53) -> Matrix(3,4,-2,-3) (13/3,9/2) -> (-2/1,-1/1) Matrix(19,-108,3,-17) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.