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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/7 -5/13 -7/19 -1/3 -3/11 -3/13 -1/5 -1/7 0/1 1/7 1/3 7/17 1/2 3/5 5/7 1/1 9/7 7/5 3/2 5/3 9/5 31/17 13/7 2/1 11/5 7/3 45/19 17/7 5/2 3/1 25/7 11/3 4/1 13/3 5/1 27/5 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/3 -6/13 1/4 1/3 -5/11 1/4 -4/9 1/3 2/5 -3/7 0/1 -8/19 1/5 2/9 -5/12 1/4 3/11 -12/29 2/7 1/3 -7/17 3/10 -2/5 1/3 2/5 -5/13 1/2 -8/21 0/1 1/1 -3/8 0/1 1/1 -13/35 1/4 -10/27 0/1 1/3 -7/19 0/1 -11/30 0/1 1/3 -15/41 1/4 -4/11 1/4 1/3 -1/3 1/2 -4/13 3/4 1/1 -3/10 1/1 1/0 -8/27 0/1 1/1 -5/17 0/1 -7/24 0/1 1/1 -2/7 0/1 1/3 -3/11 1/2 -4/15 2/3 1/1 -1/4 2/3 1/1 -4/17 1/1 8/7 -3/13 3/2 -11/48 9/5 2/1 -8/35 2/1 3/1 -5/22 1/1 2/1 -2/9 3/1 1/0 -1/5 0/1 -2/11 3/7 1/2 -1/6 2/3 1/1 -2/13 1/1 4/3 -1/7 1/0 -3/22 -1/1 0/1 -2/15 0/1 1/1 -1/8 1/1 1/0 0/1 0/1 1/1 1/7 0/1 2/13 0/1 1/3 1/6 1/2 1/1 1/5 1/2 1/4 1/1 2/1 1/3 0/1 3/8 1/3 2/5 8/21 1/3 2/5 5/13 1/2 2/5 1/2 3/5 7/17 2/3 12/29 3/4 1/1 5/12 2/3 1/1 3/7 3/4 1/2 1/1 2/1 3/5 1/0 5/8 -1/1 0/1 17/27 0/1 12/19 0/1 1/1 7/11 1/0 9/14 -5/1 1/0 2/3 -1/1 0/1 9/13 1/2 7/10 0/1 1/1 5/7 0/1 13/18 0/1 1/3 21/29 1/2 8/11 0/1 1/1 11/15 1/2 3/4 1/2 1/1 1/1 1/0 5/4 -3/2 -1/1 9/7 -1/1 13/10 -1/1 -5/6 4/3 -1/1 -1/2 11/8 -1/1 0/1 7/5 0/1 17/12 0/1 1/1 10/7 -1/1 0/1 3/2 0/1 1/1 5/3 1/0 7/4 -2/1 -1/1 16/9 3/1 1/0 25/14 -7/1 1/0 9/5 1/0 20/11 -4/1 -3/1 31/17 -3/1 11/6 -3/1 -2/1 13/7 -2/1 2/1 -2/1 -1/1 11/5 -1/1 20/9 -1/1 -18/19 9/4 -1/1 -8/9 7/3 -3/4 26/11 -9/13 -2/3 45/19 -2/3 64/27 -2/3 -15/23 19/8 -2/3 -3/5 12/5 -1/1 -2/3 17/7 -2/3 22/9 -2/3 -3/5 5/2 -3/5 -1/2 3/1 0/1 7/2 3/1 1/0 25/7 1/0 43/12 -17/1 1/0 18/5 -7/1 1/0 11/3 1/0 37/10 -4/1 -3/1 63/17 -3/1 26/7 -3/1 -2/1 15/4 -2/1 -1/1 4/1 -2/1 -1/1 13/3 -1/1 22/5 -1/1 -10/11 9/2 -1/1 -4/5 5/1 -1/2 16/3 -1/3 0/1 27/5 0/1 38/7 0/1 1/1 11/2 -1/1 0/1 6/1 -1/1 -1/2 7/1 0/1 8/1 1/1 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(89,42,214,101) (-1/2,-6/13) -> (12/29,5/12) Hyperbolic Matrix(239,110,-654,-301) (-6/13,-5/11) -> (-15/41,-4/11) Hyperbolic Matrix(31,14,-206,-93) (-5/11,-4/9) -> (-2/13,-1/7) Hyperbolic Matrix(83,36,-196,-85) (-4/9,-3/7) -> (-3/7,-8/19) Parabolic Matrix(195,82,302,127) (-8/19,-5/12) -> (9/14,2/3) Hyperbolic Matrix(53,22,330,137) (-5/12,-12/29) -> (2/13,1/6) Hyperbolic Matrix(547,226,-1474,-609) (-12/29,-7/17) -> (-13/35,-10/27) Hyperbolic Matrix(83,34,-354,-145) (-7/17,-2/5) -> (-4/17,-3/13) Hyperbolic Matrix(129,50,-338,-131) (-2/5,-5/13) -> (-5/13,-8/21) Parabolic Matrix(127,48,336,127) (-8/21,-3/8) -> (3/8,8/21) Hyperbolic Matrix(409,152,-1784,-663) (-3/8,-13/35) -> (-3/13,-11/48) Hyperbolic Matrix(687,254,1090,403) (-10/27,-7/19) -> (17/27,12/19) Hyperbolic Matrix(605,222,962,353) (-7/19,-11/30) -> (5/8,17/27) Hyperbolic Matrix(153,56,-1112,-407) (-11/30,-15/41) -> (-1/7,-3/22) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(209,64,160,49) (-4/13,-3/10) -> (13/10,4/3) Hyperbolic Matrix(47,14,-366,-109) (-3/10,-8/27) -> (-2/15,-1/8) Hyperbolic Matrix(385,114,206,61) (-8/27,-5/17) -> (13/7,2/1) Hyperbolic Matrix(499,146,270,79) (-5/17,-7/24) -> (11/6,13/7) Hyperbolic Matrix(227,66,-994,-289) (-7/24,-2/7) -> (-8/35,-5/22) Hyperbolic Matrix(65,18,-242,-67) (-2/7,-3/11) -> (-3/11,-4/15) Parabolic Matrix(241,64,64,17) (-4/15,-1/4) -> (15/4,4/1) Hyperbolic Matrix(233,56,104,25) (-1/4,-4/17) -> (20/9,9/4) Hyperbolic Matrix(3737,856,1576,361) (-11/48,-8/35) -> (64/27,19/8) Hyperbolic Matrix(185,42,22,5) (-5/22,-2/9) -> (8/1,1/0) Hyperbolic Matrix(19,4,-100,-21) (-2/9,-1/5) -> (-1/5,-2/11) Parabolic Matrix(173,30,98,17) (-2/11,-1/6) -> (7/4,16/9) Hyperbolic Matrix(249,40,56,9) (-1/6,-2/13) -> (22/5,9/2) Hyperbolic Matrix(1001,136,184,25) (-3/22,-2/15) -> (38/7,11/2) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(15,-2,98,-13) (0/1,1/7) -> (1/7,2/13) Parabolic Matrix(81,-14,110,-19) (1/6,1/5) -> (11/15,3/4) Hyperbolic Matrix(65,-14,14,-3) (1/5,1/4) -> (9/2,5/1) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(545,-208,752,-287) (8/21,5/13) -> (21/29,8/11) Hyperbolic Matrix(289,-112,80,-31) (5/13,2/5) -> (18/5,11/3) Hyperbolic Matrix(239,-98,578,-237) (2/5,7/17) -> (7/17,12/29) Parabolic Matrix(157,-66,226,-95) (5/12,3/7) -> (9/13,7/10) Hyperbolic Matrix(77,-34,34,-15) (3/7,1/2) -> (9/4,7/3) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(391,-248,216,-137) (12/19,7/11) -> (9/5,20/11) Hyperbolic Matrix(401,-256,224,-143) (7/11,9/14) -> (25/14,9/5) Hyperbolic Matrix(359,-248,152,-105) (2/3,9/13) -> (7/3,26/11) Hyperbolic Matrix(141,-100,196,-139) (7/10,5/7) -> (5/7,13/18) Parabolic Matrix(1271,-920,344,-249) (13/18,21/29) -> (11/3,37/10) Hyperbolic Matrix(295,-216,56,-41) (8/11,11/15) -> (5/1,16/3) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(141,-196,100,-139) (11/8,7/5) -> (7/5,17/12) Parabolic Matrix(417,-592,112,-159) (17/12,10/7) -> (26/7,15/4) Hyperbolic Matrix(157,-226,66,-95) (10/7,3/2) -> (19/8,12/5) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(639,-1138,178,-317) (16/9,25/14) -> (43/12,18/5) Hyperbolic Matrix(1135,-2066,306,-557) (20/11,31/17) -> (63/17,26/7) Hyperbolic Matrix(1007,-1840,272,-497) (31/17,11/6) -> (37/10,63/17) Hyperbolic Matrix(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(1711,-4050,722,-1709) (26/11,45/19) -> (45/19,64/27) Parabolic Matrix(239,-578,98,-237) (12/5,17/7) -> (17/7,22/9) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(351,-1250,98,-349) (7/2,25/7) -> (25/7,43/12) Parabolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic Matrix(271,-1458,50,-269) (16/3,27/5) -> (27/5,38/7) Parabolic Matrix(15,-98,2,-13) (6/1,7/1) -> (7/1,8/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(89,42,214,101) -> Matrix(5,-2,8,-3) Matrix(239,110,-654,-301) -> Matrix(1,0,0,1) Matrix(31,14,-206,-93) -> Matrix(7,-2,4,-1) Matrix(83,36,-196,-85) -> Matrix(1,0,2,1) Matrix(195,82,302,127) -> Matrix(9,-2,-4,1) Matrix(53,22,330,137) -> Matrix(7,-2,18,-5) Matrix(547,226,-1474,-609) -> Matrix(7,-2,18,-5) Matrix(83,34,-354,-145) -> Matrix(19,-6,16,-5) Matrix(129,50,-338,-131) -> Matrix(5,-2,8,-3) Matrix(127,48,336,127) -> Matrix(3,-2,8,-5) Matrix(409,152,-1784,-663) -> Matrix(11,-2,6,-1) Matrix(687,254,1090,403) -> Matrix(1,0,-2,1) Matrix(605,222,962,353) -> Matrix(1,0,-4,1) Matrix(153,56,-1112,-407) -> Matrix(1,0,-4,1) Matrix(23,8,-72,-25) -> Matrix(5,-2,8,-3) Matrix(209,64,160,49) -> Matrix(5,-4,-6,5) Matrix(47,14,-366,-109) -> Matrix(1,0,0,1) Matrix(385,114,206,61) -> Matrix(1,-2,0,1) Matrix(499,146,270,79) -> Matrix(5,-2,-2,1) Matrix(227,66,-994,-289) -> Matrix(3,-2,2,-1) Matrix(65,18,-242,-67) -> Matrix(5,-2,8,-3) Matrix(241,64,64,17) -> Matrix(1,0,-2,1) Matrix(233,56,104,25) -> Matrix(11,-10,-12,11) Matrix(3737,856,1576,361) -> Matrix(13,-24,-20,37) Matrix(185,42,22,5) -> Matrix(1,-2,0,1) Matrix(19,4,-100,-21) -> Matrix(1,0,2,1) Matrix(173,30,98,17) -> Matrix(1,0,-2,1) Matrix(249,40,56,9) -> Matrix(7,-6,-8,7) Matrix(1001,136,184,25) -> Matrix(1,0,0,1) Matrix(181,22,74,9) -> Matrix(1,2,-2,-3) Matrix(15,-2,98,-13) -> Matrix(1,0,2,1) Matrix(81,-14,110,-19) -> Matrix(1,0,0,1) Matrix(65,-14,14,-3) -> Matrix(3,-2,-4,3) Matrix(13,-4,36,-11) -> Matrix(1,0,2,1) Matrix(545,-208,752,-287) -> Matrix(5,-2,8,-3) Matrix(289,-112,80,-31) -> Matrix(9,-4,-2,1) Matrix(239,-98,578,-237) -> Matrix(13,-8,18,-11) Matrix(157,-66,226,-95) -> Matrix(3,-2,2,-1) Matrix(77,-34,34,-15) -> Matrix(7,-6,-8,7) Matrix(31,-18,50,-29) -> Matrix(1,-2,0,1) Matrix(391,-248,216,-137) -> Matrix(1,-4,0,1) Matrix(401,-256,224,-143) -> Matrix(1,-2,0,1) Matrix(359,-248,152,-105) -> Matrix(7,-2,-10,3) Matrix(141,-100,196,-139) -> Matrix(1,0,2,1) Matrix(1271,-920,344,-249) -> Matrix(9,-4,-2,1) Matrix(295,-216,56,-41) -> Matrix(1,0,-4,1) Matrix(9,-8,8,-7) -> Matrix(1,-2,0,1) Matrix(127,-162,98,-125) -> Matrix(7,8,-8,-9) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(141,-196,100,-139) -> Matrix(1,0,2,1) Matrix(417,-592,112,-159) -> Matrix(1,-2,0,1) Matrix(157,-226,66,-95) -> Matrix(1,2,-2,-3) Matrix(31,-50,18,-29) -> Matrix(1,-2,0,1) Matrix(639,-1138,178,-317) -> Matrix(1,-10,0,1) Matrix(1135,-2066,306,-557) -> Matrix(5,18,-2,-7) Matrix(1007,-1840,272,-497) -> Matrix(7,18,-2,-5) Matrix(111,-242,50,-109) -> Matrix(19,20,-20,-21) Matrix(1711,-4050,722,-1709) -> Matrix(71,48,-108,-73) Matrix(239,-578,98,-237) -> Matrix(11,8,-18,-13) Matrix(13,-36,4,-11) -> Matrix(1,0,2,1) Matrix(351,-1250,98,-349) -> Matrix(1,-20,0,1) Matrix(79,-338,18,-77) -> Matrix(11,12,-12,-13) Matrix(271,-1458,50,-269) -> Matrix(1,0,4,1) Matrix(15,-98,2,-13) -> 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): 22 Degree of the the map X: 22 Degree of the the map Y: 64 Permutation triple for Y: ((1,6,19,47,56,55,44,62,59,38,34,33,48,20,7,2)(3,11,35,51,26,8,25,46,41,40,58,63,53,36,12,4)(5,15,42,61,64,52,23,22,31,10,9,30,49,27,43,16)(13,29,28,50,21,37,60,32,18,17,45,57,54,24,39,14); (1,4,14,5)(3,10)(6,18)(7,23,24,8)(9,29,40,19)(11,34,42,17)(13,38)(15,41)(20,21)(26,27)(31,59,46,32)(36,44,43,37)(48,63,57,49)(50,64,56,51)(52,53)(54,55); (1,2,8,27,57,45,42,41,59,62,36,52,50,28,9,3)(4,12,37,20,49,30,19,18,46,25,24,55,64,61,34,13)(5,16,44,54,63,58,29,38,31,22,7,21,51,35,17,6)(10,32,60,43,26,56,47,40,15,14,39,23,53,48,33,11)) ----------------------------------------------------------------------- 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): 192 Minimal number of generators: 33 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: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/7 1/3 3/5 5/7 1/1 9/7 7/5 5/3 31/17 2/1 11/5 7/3 45/19 3/1 25/7 11/3 4/1 13/3 5/1 27/5 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/1 1/7 0/1 1/6 1/2 1/1 1/5 1/2 1/4 1/1 2/1 1/3 0/1 3/8 1/3 2/5 5/13 1/2 2/5 1/2 3/5 7/17 2/3 5/12 2/3 1/1 3/7 3/4 1/2 1/1 2/1 3/5 1/0 5/8 -1/1 0/1 12/19 0/1 1/1 7/11 1/0 2/3 -1/1 0/1 9/13 1/2 7/10 0/1 1/1 5/7 0/1 13/18 0/1 1/3 8/11 0/1 1/1 11/15 1/2 3/4 1/2 1/1 1/1 1/0 5/4 -3/2 -1/1 9/7 -1/1 4/3 -1/1 -1/2 11/8 -1/1 0/1 7/5 0/1 17/12 0/1 1/1 10/7 -1/1 0/1 3/2 0/1 1/1 5/3 1/0 7/4 -2/1 -1/1 9/5 1/0 20/11 -4/1 -3/1 31/17 -3/1 11/6 -3/1 -2/1 2/1 -2/1 -1/1 11/5 -1/1 9/4 -1/1 -8/9 7/3 -3/4 26/11 -9/13 -2/3 45/19 -2/3 19/8 -2/3 -3/5 12/5 -1/1 -2/3 17/7 -2/3 5/2 -3/5 -1/2 3/1 0/1 7/2 3/1 1/0 25/7 1/0 18/5 -7/1 1/0 11/3 1/0 4/1 -2/1 -1/1 13/3 -1/1 9/2 -1/1 -4/5 5/1 -1/2 16/3 -1/3 0/1 27/5 0/1 11/2 -1/1 0/1 6/1 -1/1 -1/2 7/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(42,-5,101,-12) (0/1,1/7) -> (7/17,5/12) Hyperbolic Matrix(56,-9,137,-22) (1/7,1/6) -> (2/5,7/17) Hyperbolic Matrix(81,-14,110,-19) (1/6,1/5) -> (11/15,3/4) Hyperbolic Matrix(65,-14,14,-3) (1/5,1/4) -> (9/2,5/1) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(82,-31,127,-48) (3/8,5/13) -> (7/11,2/3) Hyperbolic Matrix(289,-112,80,-31) (5/13,2/5) -> (18/5,11/3) Hyperbolic Matrix(157,-66,226,-95) (5/12,3/7) -> (9/13,7/10) Hyperbolic Matrix(77,-34,34,-15) (3/7,1/2) -> (9/4,7/3) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(256,-161,353,-222) (5/8,12/19) -> (13/18,8/11) Hyperbolic Matrix(391,-248,216,-137) (12/19,7/11) -> (9/5,20/11) Hyperbolic Matrix(359,-248,152,-105) (2/3,9/13) -> (7/3,26/11) Hyperbolic Matrix(141,-100,196,-139) (7/10,5/7) -> (5/7,13/18) Parabolic Matrix(295,-216,56,-41) (8/11,11/15) -> (5/1,16/3) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(64,-81,49,-62) (5/4,9/7) -> (9/7,4/3) Parabolic Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(141,-196,100,-139) (11/8,7/5) -> (7/5,17/12) Parabolic Matrix(146,-207,79,-112) (17/12,10/7) -> (11/6,2/1) Hyperbolic Matrix(157,-226,66,-95) (10/7,3/2) -> (19/8,12/5) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(64,-113,17,-30) (7/4,9/5) -> (11/3,4/1) Hyperbolic Matrix(528,-961,289,-526) (20/11,31/17) -> (31/17,11/6) Parabolic Matrix(56,-121,25,-54) (2/1,11/5) -> (11/5,9/4) Parabolic Matrix(856,-2025,361,-854) (26/11,45/19) -> (45/19,19/8) Parabolic Matrix(42,-101,5,-12) (12/5,17/7) -> (7/1,1/0) Hyperbolic Matrix(56,-137,9,-22) (17/7,5/2) -> (6/1,7/1) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(176,-625,49,-174) (7/2,25/7) -> (25/7,18/5) Parabolic Matrix(40,-169,9,-38) (4/1,13/3) -> (13/3,9/2) Parabolic Matrix(136,-729,25,-134) (16/3,27/5) -> (27/5,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(42,-5,101,-12) -> Matrix(1,-2,2,-3) Matrix(56,-9,137,-22) -> Matrix(5,-2,8,-3) Matrix(81,-14,110,-19) -> Matrix(1,0,0,1) Matrix(65,-14,14,-3) -> Matrix(3,-2,-4,3) Matrix(13,-4,36,-11) -> Matrix(1,0,2,1) Matrix(82,-31,127,-48) -> Matrix(5,-2,-2,1) Matrix(289,-112,80,-31) -> Matrix(9,-4,-2,1) Matrix(157,-66,226,-95) -> Matrix(3,-2,2,-1) Matrix(77,-34,34,-15) -> Matrix(7,-6,-8,7) Matrix(31,-18,50,-29) -> Matrix(1,-2,0,1) Matrix(256,-161,353,-222) -> Matrix(1,0,2,1) Matrix(391,-248,216,-137) -> Matrix(1,-4,0,1) Matrix(359,-248,152,-105) -> Matrix(7,-2,-10,3) Matrix(141,-100,196,-139) -> Matrix(1,0,2,1) Matrix(295,-216,56,-41) -> Matrix(1,0,-4,1) Matrix(9,-8,8,-7) -> Matrix(1,-2,0,1) Matrix(64,-81,49,-62) -> Matrix(3,4,-4,-5) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(141,-196,100,-139) -> Matrix(1,0,2,1) Matrix(146,-207,79,-112) -> Matrix(1,-2,0,1) Matrix(157,-226,66,-95) -> Matrix(1,2,-2,-3) Matrix(31,-50,18,-29) -> Matrix(1,-2,0,1) Matrix(64,-113,17,-30) -> Matrix(1,0,0,1) Matrix(528,-961,289,-526) -> Matrix(5,18,-2,-7) Matrix(56,-121,25,-54) -> Matrix(9,10,-10,-11) Matrix(856,-2025,361,-854) -> Matrix(35,24,-54,-37) Matrix(42,-101,5,-12) -> Matrix(3,2,-2,-1) Matrix(56,-137,9,-22) -> Matrix(3,2,-8,-5) Matrix(13,-36,4,-11) -> Matrix(1,0,2,1) Matrix(176,-625,49,-174) -> Matrix(1,-10,0,1) Matrix(40,-169,9,-38) -> Matrix(5,6,-6,-7) Matrix(136,-729,25,-134) -> 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 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): 11 ----------------------------------------------------------------------- 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 -1/1 0/1 2 1 1/1 1/0 2 8 9/7 -1/1 4 1 4/3 (-1/1,-1/2) 0 16 11/8 (-1/1,0/1) 0 16 7/5 0/1 2 4 17/12 (0/1,1/1) 0 16 10/7 (-1/1,0/1) 0 16 3/2 (0/1,1/1) 0 16 5/3 1/0 2 2 7/4 (-2/1,-1/1) 0 16 9/5 1/0 2 8 31/17 -3/1 2 1 11/6 (-3/1,-2/1) 0 16 2/1 (-2/1,-1/1) 0 16 11/5 -1/1 10 1 7/3 -3/4 2 8 45/19 -2/3 6 1 19/8 (-2/3,-3/5) 0 16 12/5 (-1/1,-2/3) 0 16 17/7 -2/3 2 2 5/2 (-3/5,-1/2) 0 16 3/1 0/1 2 4 7/2 (3/1,1/0) 0 16 25/7 1/0 10 1 11/3 1/0 2 8 4/1 (-2/1,-1/1) 0 16 13/3 -1/1 6 1 5/1 -1/2 2 8 27/5 0/1 2 1 11/2 (-1/1,0/1) 0 16 6/1 (-1/1,-1/2) 0 16 7/1 0/1 2 2 1/0 (-1/1,0/1) 0 16 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(8,-9,7,-8) (1/1,9/7) -> (1/1,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(141,-196,100,-139) (11/8,7/5) -> (7/5,17/12) Parabolic Matrix(146,-207,79,-112) (17/12,10/7) -> (11/6,2/1) Hyperbolic Matrix(157,-226,66,-95) (10/7,3/2) -> (19/8,12/5) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(64,-113,17,-30) (7/4,9/5) -> (11/3,4/1) Hyperbolic Matrix(154,-279,85,-154) (9/5,31/17) -> (9/5,31/17) Reflection Matrix(373,-682,204,-373) (31/17,11/6) -> (31/17,11/6) Reflection Matrix(21,-44,10,-21) (2/1,11/5) -> (2/1,11/5) Reflection Matrix(34,-77,15,-34) (11/5,7/3) -> (11/5,7/3) Reflection Matrix(134,-315,57,-134) (7/3,45/19) -> (7/3,45/19) Reflection Matrix(721,-1710,304,-721) (45/19,19/8) -> (45/19,19/8) Reflection Matrix(42,-101,5,-12) (12/5,17/7) -> (7/1,1/0) Hyperbolic Matrix(56,-137,9,-22) (17/7,5/2) -> (6/1,7/1) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(99,-350,28,-99) (7/2,25/7) -> (7/2,25/7) Reflection Matrix(76,-275,21,-76) (25/7,11/3) -> (25/7,11/3) Reflection Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(14,-65,3,-14) (13/3,5/1) -> (13/3,5/1) Reflection Matrix(26,-135,5,-26) (5/1,27/5) -> (5/1,27/5) Reflection Matrix(109,-594,20,-109) (27/5,11/2) -> (27/5,11/2) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(0,1,1,0) -> Matrix(1,0,0,-1) (-1/1,1/1) -> (0/1,1/0) Matrix(8,-9,7,-8) -> Matrix(1,2,0,-1) (1/1,9/7) -> (-1/1,1/0) Matrix(55,-72,42,-55) -> Matrix(3,2,-4,-3) (9/7,4/3) -> (-1/1,-1/2) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(141,-196,100,-139) -> Matrix(1,0,2,1) 0/1 Matrix(146,-207,79,-112) -> Matrix(1,-2,0,1) 1/0 Matrix(157,-226,66,-95) -> Matrix(1,2,-2,-3) -1/1 Matrix(31,-50,18,-29) -> Matrix(1,-2,0,1) 1/0 Matrix(64,-113,17,-30) -> Matrix(1,0,0,1) Matrix(154,-279,85,-154) -> Matrix(1,6,0,-1) (9/5,31/17) -> (-3/1,1/0) Matrix(373,-682,204,-373) -> Matrix(5,12,-2,-5) (31/17,11/6) -> (-3/1,-2/1) Matrix(21,-44,10,-21) -> Matrix(3,4,-2,-3) (2/1,11/5) -> (-2/1,-1/1) Matrix(34,-77,15,-34) -> Matrix(7,6,-8,-7) (11/5,7/3) -> (-1/1,-3/4) Matrix(134,-315,57,-134) -> Matrix(17,12,-24,-17) (7/3,45/19) -> (-3/4,-2/3) Matrix(721,-1710,304,-721) -> Matrix(19,12,-30,-19) (45/19,19/8) -> (-2/3,-3/5) Matrix(42,-101,5,-12) -> Matrix(3,2,-2,-1) -1/1 Matrix(56,-137,9,-22) -> Matrix(3,2,-8,-5) -1/2 Matrix(13,-36,4,-11) -> Matrix(1,0,2,1) 0/1 Matrix(99,-350,28,-99) -> Matrix(-1,6,0,1) (7/2,25/7) -> (3/1,1/0) Matrix(76,-275,21,-76) -> Matrix(1,4,0,-1) (25/7,11/3) -> (-2/1,1/0) Matrix(25,-104,6,-25) -> Matrix(3,4,-2,-3) (4/1,13/3) -> (-2/1,-1/1) Matrix(14,-65,3,-14) -> Matrix(3,2,-4,-3) (13/3,5/1) -> (-1/1,-1/2) Matrix(26,-135,5,-26) -> Matrix(-1,0,4,1) (5/1,27/5) -> (-1/2,0/1) Matrix(109,-594,20,-109) -> Matrix(-1,0,2,1) (27/5,11/2) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.