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 -1/1 1/1 -7/1 -1/1 -13/2 -1/2 0/1 -6/1 0/1 -11/2 -1/1 0/1 -5/1 -1/1 -9/2 -1/1 -1/3 -13/3 -1/1 -4/1 -1/1 -1/3 -19/5 -3/5 -15/4 -1/2 -2/5 -1/3 -11/3 -1/3 -18/5 -1/3 -7/2 -1/3 -1/4 -17/5 -1/1 -27/8 -1/3 -10/3 -1/3 -3/11 -3/1 -1/3 -1/5 -11/4 -1/5 0/1 -30/11 0/1 -19/7 -3/13 -27/10 -1/5 -8/3 -1/5 -1/7 -29/11 -1/5 -21/8 -1/5 -1/6 0/1 -55/21 -5/29 -34/13 -3/19 -1/7 -13/5 -1/7 -18/7 0/1 -5/2 -1/4 0/1 -12/5 -1/4 -19/8 -1/4 -2/9 -7/3 -1/5 -9/4 -1/5 -11/5 -1/5 -13/6 -2/11 -1/6 -2/1 -1/5 -1/7 -13/7 -1/7 -11/6 -2/13 -1/7 -9/5 -1/7 -7/4 -1/7 -1/8 -12/7 -1/8 -17/10 -2/15 -1/8 -22/13 -1/7 -1/9 -5/3 -1/7 -18/11 -1/8 -13/8 -1/8 -2/17 -34/21 -1/7 -1/9 -21/13 -3/25 -1/9 -8/5 -1/7 -1/9 -3/2 -1/8 -1/9 0/1 -10/7 -1/7 -1/9 -37/26 -1/8 0/1 -27/19 -1/7 -1/9 -17/12 -1/8 0/1 -7/5 -1/9 -18/13 -1/9 -11/8 -1/9 -2/19 -15/11 -1/9 -3/29 -34/25 -3/29 -5/49 -53/39 -11/109 -72/53 -1/10 -19/14 -1/10 -2/21 -4/3 -1/9 -1/11 -17/13 -3/31 -30/23 0/1 -13/10 -1/10 0/1 -9/7 -1/9 -1/11 -5/4 -1/10 0/1 -6/5 -1/10 -7/6 -1/10 -1/11 -1/1 -1/13 0/1 0/1 1/1 1/13 6/5 1/10 5/4 0/1 1/10 9/7 1/11 1/9 13/10 0/1 1/10 17/13 3/31 4/3 1/11 1/9 11/8 2/19 1/9 18/13 1/9 7/5 1/9 3/2 0/1 1/9 1/8 11/7 1/9 19/12 0/1 1/8 27/17 1/9 1/7 8/5 1/9 1/7 13/8 2/17 1/8 18/11 1/8 5/3 1/7 17/10 1/8 2/15 12/7 1/8 19/11 5/37 7/4 1/8 1/7 9/5 1/7 11/6 1/7 2/13 2/1 1/7 1/5 9/4 1/5 16/7 1/5 3/13 23/10 0/1 1/4 7/3 1/5 19/8 2/9 1/4 12/5 1/4 5/2 0/1 1/4 18/7 0/1 13/5 1/7 34/13 1/7 3/19 21/8 0/1 1/6 1/5 8/3 1/7 1/5 27/10 1/5 19/7 3/13 11/4 0/1 1/5 14/5 1/5 3/13 17/6 4/17 1/4 3/1 1/5 1/3 19/6 4/17 1/4 16/5 3/11 1/3 13/4 0/1 1/4 10/3 3/11 1/3 27/8 1/3 17/5 1/1 7/2 1/4 1/3 18/5 1/3 11/3 1/3 26/7 1/3 3/7 15/4 1/3 2/5 1/2 34/9 3/7 5/11 53/14 10/21 1/2 72/19 1/2 19/5 3/5 4/1 1/3 1/1 9/2 1/3 1/1 14/3 1/3 1/1 19/4 1/2 2/3 5/1 1/1 16/3 3/5 1/1 11/2 0/1 1/1 17/3 3/1 6/1 0/1 19/3 3/7 13/2 0/1 1/2 7/1 1/1 8/1 -1/1 1/1 9/1 -1/1 1/1 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(1,0,-8,1) Matrix(37,270,10,73) -> Matrix(1,2,2,5) Matrix(37,252,16,109) -> Matrix(1,0,6,1) Matrix(73,468,-56,-359) -> Matrix(1,0,-8,1) Matrix(71,396,-26,-145) -> Matrix(1,0,-4,1) Matrix(37,198,-20,-107) -> Matrix(3,2,-20,-13) Matrix(35,162,-8,-37) -> Matrix(1,0,0,1) Matrix(71,306,-42,-181) -> Matrix(1,0,-6,1) Matrix(37,144,28,109) -> Matrix(1,0,12,1) Matrix(325,1224,-124,-467) -> Matrix(5,2,-28,-11) Matrix(179,666,-68,-253) -> Matrix(5,2,-28,-11) Matrix(109,396,30,109) -> Matrix(11,4,30,11) Matrix(71,252,20,71) -> Matrix(7,2,24,7) Matrix(73,252,42,145) -> Matrix(7,2,52,15) Matrix(287,972,106,359) -> Matrix(5,2,22,9) Matrix(145,486,54,181) -> Matrix(7,2,38,11) Matrix(71,234,-44,-145) -> Matrix(7,2,-60,-17) Matrix(71,198,-52,-145) -> Matrix(9,2,-86,-19) Matrix(397,1080,-304,-827) -> Matrix(1,0,-6,1) Matrix(359,972,106,287) -> Matrix(9,2,22,5) Matrix(181,486,54,145) -> Matrix(11,2,38,7) Matrix(109,288,14,37) -> Matrix(1,0,6,1) Matrix(1403,3672,-1032,-2701) -> Matrix(37,6,-364,-59) Matrix(145,378,28,73) -> Matrix(1,0,8,1) Matrix(181,468,70,181) -> Matrix(1,0,14,1) Matrix(71,180,28,71) -> Matrix(1,0,8,1) Matrix(37,90,30,73) -> Matrix(1,0,14,1) Matrix(181,432,106,253) -> Matrix(1,0,12,1) Matrix(145,342,92,217) -> Matrix(9,2,76,17) Matrix(71,162,-32,-73) -> Matrix(9,2,-50,-11) Matrix(107,234,16,35) -> Matrix(11,2,16,3) Matrix(109,234,34,73) -> Matrix(11,2,38,7) Matrix(251,468,96,179) -> Matrix(13,2,84,13) Matrix(109,198,60,109) -> Matrix(27,4,182,27) Matrix(71,126,40,71) -> Matrix(15,2,112,15) Matrix(73,126,-62,-107) -> Matrix(15,2,-158,-21) Matrix(253,432,106,181) -> Matrix(1,0,12,1) Matrix(361,612,128,217) -> Matrix(17,2,76,9) Matrix(109,180,66,109) -> Matrix(15,2,112,15) Matrix(287,468,176,287) -> Matrix(33,4,272,33) Matrix(611,990,266,431) -> Matrix(17,2,76,9) Matrix(757,1224,-556,-899) -> Matrix(33,4,-322,-39) Matrix(35,54,-24,-37) -> Matrix(1,0,0,1) Matrix(253,360,26,37) -> Matrix(1,0,8,1) Matrix(685,972,432,613) -> Matrix(1,0,16,1) Matrix(179,252,76,107) -> Matrix(17,2,76,9) Matrix(181,252,130,181) -> Matrix(17,2,144,17) Matrix(287,396,208,287) -> Matrix(37,4,342,37) Matrix(3815,5184,1006,1367) -> Matrix(139,14,268,27) Matrix(3817,5184,1008,1369) -> Matrix(121,12,252,25) Matrix(253,342,54,73) -> Matrix(1,0,12,1) Matrix(109,144,28,37) -> Matrix(1,0,12,1) Matrix(181,234,140,181) -> Matrix(1,0,20,1) Matrix(71,90,56,71) -> Matrix(1,0,20,1) Matrix(73,90,30,37) -> Matrix(1,0,14,1) Matrix(109,126,32,37) -> Matrix(1,0,14,1) Matrix(1,0,2,1) -> Matrix(1,0,26,1) Matrix(107,-126,62,-73) -> Matrix(21,-2,158,-15) Matrix(359,-468,56,-73) -> Matrix(1,0,-8,1) Matrix(145,-198,52,-71) -> Matrix(19,-2,86,-9) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(181,-288,22,-35) -> Matrix(1,0,-8,1) Matrix(145,-234,44,-71) -> Matrix(17,-2,60,-7) Matrix(107,-180,22,-37) -> Matrix(1,0,-6,1) Matrix(107,-198,20,-37) -> Matrix(13,-2,20,-3) Matrix(73,-162,32,-71) -> Matrix(11,-2,50,-9) Matrix(467,-1224,124,-325) -> Matrix(11,-2,28,-5) Matrix(253,-666,68,-179) -> Matrix(11,-2,28,-5) Matrix(145,-396,26,-71) -> Matrix(1,0,-4,1) Matrix(37,-108,12,-35) -> Matrix(1,0,0,1) Matrix(469,-1494,124,-395) -> Matrix(23,-6,50,-13) Matrix(37,-162,8,-35) -> Matrix(1,0,0,1) Matrix(37,-216,6,-35) -> Matrix(1,0,2,1) 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. ----------------------------------------------------------------------- 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): 216 Minimal number of generators: 37 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: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -3/1 -3/2 0/1 1/1 9/7 3/2 27/17 9/5 2/1 9/4 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 -6/1 0/1 -5/1 -1/1 -9/2 -1/1 -1/3 -4/1 -1/1 -1/3 -7/2 -1/3 -1/4 -3/1 -1/3 -1/5 -11/4 -1/5 0/1 -19/7 -3/13 -27/10 -1/5 -8/3 -1/5 -1/7 -5/2 -1/4 0/1 -12/5 -1/4 -19/8 -1/4 -2/9 -7/3 -1/5 -9/4 -1/5 -2/1 -1/5 -1/7 -9/5 -1/7 -7/4 -1/7 -1/8 -12/7 -1/8 -5/3 -1/7 -18/11 -1/8 -13/8 -1/8 -2/17 -8/5 -1/7 -1/9 -3/2 -1/8 -1/9 0/1 -10/7 -1/7 -1/9 -27/19 -1/7 -1/9 -17/12 -1/8 0/1 -7/5 -1/9 -18/13 -1/9 -11/8 -1/9 -2/19 -4/3 -1/9 -1/11 -9/7 -1/9 -1/11 -5/4 -1/10 0/1 -6/5 -1/10 -7/6 -1/10 -1/11 -1/1 -1/13 0/1 0/1 1/1 1/13 6/5 1/10 5/4 0/1 1/10 9/7 1/11 1/9 4/3 1/11 1/9 7/5 1/9 3/2 0/1 1/9 1/8 11/7 1/9 19/12 0/1 1/8 27/17 1/9 1/7 8/5 1/9 1/7 5/3 1/7 12/7 1/8 19/11 5/37 7/4 1/8 1/7 9/5 1/7 2/1 1/7 1/5 9/4 1/5 7/3 1/5 12/5 1/4 5/2 0/1 1/4 18/7 0/1 13/5 1/7 8/3 1/7 1/5 3/1 1/5 1/3 10/3 3/11 1/3 27/8 1/3 17/5 1/1 7/2 1/4 1/3 18/5 1/3 11/3 1/3 4/1 1/3 1/1 9/2 1/3 1/1 5/1 1/1 6/1 0/1 7/1 1/1 1/0 0/1 1/0 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(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) 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(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(145,342,92,217) (-19/8,-7/3) -> (11/7,19/12) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) 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(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic 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(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) 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(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) 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(2,1,-9,-4) Matrix(17,90,10,53) -> Matrix(2,1,15,8) Matrix(19,90,4,19) -> Matrix(2,1,3,2) Matrix(17,72,4,17) -> Matrix(2,1,3,2) Matrix(19,72,-14,-53) -> Matrix(2,1,-21,-10) Matrix(17,54,-6,-19) -> Matrix(4,1,-17,-4) Matrix(125,342,72,197) -> Matrix(6,1,47,8) Matrix(359,972,106,287) -> Matrix(9,2,22,5) Matrix(181,486,54,145) -> Matrix(11,2,38,7) Matrix(55,144,-34,-89) -> Matrix(6,1,-49,-8) Matrix(37,90,30,73) -> Matrix(1,0,14,1) Matrix(145,342,92,217) -> Matrix(9,2,76,17) Matrix(55,126,24,55) -> Matrix(4,1,15,4) Matrix(17,36,8,17) -> Matrix(6,1,35,6) Matrix(19,36,10,19) -> Matrix(6,1,35,6) Matrix(71,126,40,71) -> Matrix(15,2,112,15) Matrix(73,126,-62,-107) -> Matrix(15,2,-158,-21) Matrix(53,90,10,17) -> Matrix(8,1,15,2) Matrix(197,324,76,125) -> Matrix(8,1,63,8) Matrix(199,324,78,127) -> Matrix(8,1,15,2) Matrix(35,54,-24,-37) -> Matrix(1,0,0,1) Matrix(341,486,214,305) -> Matrix(8,1,63,8) Matrix(685,972,432,613) -> Matrix(1,0,16,1) Matrix(89,126,12,17) -> Matrix(8,1,-1,0) Matrix(233,324,64,89) -> Matrix(26,3,69,8) Matrix(235,324,66,91) -> Matrix(28,3,93,10) Matrix(55,72,42,55) -> Matrix(10,1,99,10) Matrix(71,90,56,71) -> Matrix(1,0,20,1) Matrix(73,90,30,37) -> Matrix(1,0,14,1) Matrix(109,126,32,37) -> Matrix(1,0,14,1) Matrix(1,0,2,1) -> Matrix(1,0,26,1) Matrix(107,-126,62,-73) -> Matrix(21,-2,158,-15) Matrix(53,-72,14,-19) -> Matrix(10,-1,21,-2) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(89,-144,34,-55) -> Matrix(8,-1,49,-6) Matrix(53,-126,8,-19) -> Matrix(4,-1,9,-2) Matrix(19,-54,6,-17) -> Matrix(4,-1,17,-4) 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): 18 ----------------------------------------------------------------------- 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 13 1 1/1 1/13 1 18 6/5 1/10 1 3 5/4 (0/1,1/10) 0 18 9/7 (0/1,1/10).(1/11,1/9) 0 2 4/3 (1/11,1/9) 0 9 7/5 1/9 1 18 3/2 0 6 11/7 1/9 1 18 19/12 (0/1,1/8) 0 18 27/17 (0/1,1/8).(1/9,1/7) 0 2 8/5 (1/9,1/7) 0 9 5/3 1/7 1 18 12/7 1/8 1 3 19/11 5/37 1 18 7/4 (1/8,1/7) 0 18 9/5 1/7 3 2 2/1 (1/7,1/5) 0 9 9/4 1/5 2 2 7/3 1/5 1 18 12/5 1/4 1 3 5/2 (0/1,1/4) 0 18 18/7 0/1 3 1 13/5 1/7 1 18 8/3 (1/7,1/5) 0 9 3/1 (0/1,1/4).(1/5,1/3) 0 6 10/3 (3/11,1/3) 0 9 27/8 1/3 4 2 17/5 1/1 1 18 7/2 (1/4,1/3) 0 18 18/5 1/3 3 1 11/3 1/3 1 18 4/1 (1/3,1/1) 0 9 9/2 (1/3,1/1) 0 2 5/1 1/1 1 18 6/1 0/1 1 3 7/1 1/1 1 18 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(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) 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(271,-432,170,-271) (27/17,8/5) -> (27/17,8/5) Reflection Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(53,-90,10,-17) (5/3,12/7) -> (5/1,6/1) 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(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) 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(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(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(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(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,26,-1) (0/1,1/1) -> (0/1,1/13) Matrix(107,-126,62,-73) -> Matrix(21,-2,158,-15) Matrix(73,-90,30,-37) -> Matrix(1,0,14,-1) *** -> (0/1,1/7) Matrix(71,-90,56,-71) -> Matrix(1,0,20,-1) (5/4,9/7) -> (0/1,1/10) Matrix(55,-72,42,-55) -> Matrix(10,-1,99,-10) (9/7,4/3) -> (1/11,1/9) Matrix(53,-72,14,-19) -> Matrix(10,-1,21,-2) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(91,-144,12,-19) -> Matrix(-8,1,1,0) Matrix(647,-1026,408,-647) -> Matrix(1,0,16,-1) (19/12,27/17) -> (0/1,1/8) Matrix(271,-432,170,-271) -> Matrix(8,-1,63,-8) (27/17,8/5) -> (1/9,1/7) Matrix(89,-144,34,-55) -> Matrix(8,-1,49,-6) 1/7 Matrix(53,-90,10,-17) -> Matrix(8,-1,15,-2) Matrix(145,-252,42,-73) -> Matrix(15,-2,52,-7) Matrix(71,-126,40,-71) -> Matrix(15,-2,112,-15) (7/4,9/5) -> (1/8,1/7) Matrix(19,-36,10,-19) -> Matrix(6,-1,35,-6) (9/5,2/1) -> (1/7,1/5) Matrix(17,-36,8,-17) -> Matrix(6,-1,35,-6) (2/1,9/4) -> (1/7,1/5) Matrix(55,-126,24,-55) -> Matrix(4,-1,15,-4) (9/4,7/3) -> (1/5,1/3) Matrix(53,-126,8,-19) -> Matrix(4,-1,9,-2) 1/3 Matrix(71,-180,28,-71) -> Matrix(1,0,8,-1) (5/2,18/7) -> (0/1,1/4) Matrix(181,-468,70,-181) -> Matrix(1,0,14,-1) (18/7,13/5) -> (0/1,1/7) Matrix(19,-54,6,-17) -> Matrix(4,-1,17,-4) (0/1,1/4).(1/5,1/3) Matrix(161,-540,48,-161) -> Matrix(10,-3,33,-10) (10/3,27/8) -> (3/11,1/3) Matrix(271,-918,80,-271) -> Matrix(2,-1,3,-2) (27/8,17/5) -> (1/3,1/1) Matrix(71,-252,20,-71) -> Matrix(7,-2,24,-7) (7/2,18/5) -> (1/4,1/3) Matrix(109,-396,30,-109) -> Matrix(11,-4,30,-11) (18/5,11/3) -> (1/3,2/5) Matrix(17,-72,4,-17) -> Matrix(2,-1,3,-2) (4/1,9/2) -> (1/3,1/1) Matrix(19,-90,4,-19) -> Matrix(2,-1,3,-2) (9/2,5/1) -> (1/3,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.