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): 864 Minimal number of generators: 145 Number of equivalence classes of cusps: 48 Genus: 49 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 17/15 17/14 17/13 17/12 3/2 17/11 17/10 17/9 2/1 17/8 34/15 12/5 17/7 5/2 85/33 34/13 17/6 3/1 17/5 7/2 11/3 15/4 34/9 4/1 17/4 13/3 136/31 102/23 9/2 14/3 85/18 34/7 5/1 16/3 11/2 17/3 6/1 19/3 13/2 34/5 7/1 8/1 17/2 9/1 10/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 -7/13 -17/2 -1/2 -8/1 -6/13 -7/1 -3/7 -20/3 -2/5 -13/2 -1/2 -19/3 -1/3 -25/4 -1/4 -6/1 -4/11 -17/3 -1/3 -11/2 -7/22 -5/1 -1/3 -24/5 -2/7 -43/9 -1/5 -19/4 -1/4 -33/7 -5/21 -14/3 0/1 -9/2 -5/18 -22/5 -4/15 -35/8 -19/72 -13/3 -7/27 -17/4 -1/4 -4/1 -2/9 -15/4 -1/4 -11/3 -1/3 -18/5 -8/35 -7/2 -3/14 -17/5 -1/5 -10/3 -4/21 -43/13 -13/69 -33/10 -13/70 -56/17 -2/11 -23/7 -3/17 -13/4 -3/16 -3/1 -1/5 -17/6 -1/6 -14/5 -4/25 -11/4 -3/20 -19/7 -1/7 -8/3 -2/15 -13/5 -1/3 -18/7 -4/21 -5/2 -1/6 -17/7 -1/7 -12/5 -2/15 -43/18 -1/6 -74/31 -4/29 -31/13 -3/23 -19/8 -1/8 -7/3 -1/7 -16/7 -2/21 -9/4 -1/4 -11/5 -1/9 -13/6 -1/6 -15/7 -1/7 -17/8 -1/8 -2/1 0/1 -17/9 -1/7 -15/8 -1/8 -13/7 -1/9 -11/6 -1/6 -9/5 -1/11 -34/19 0/1 -25/14 1/2 -16/9 -2/9 -7/4 -1/8 -26/15 -4/29 -19/11 -1/7 -31/18 -3/22 -12/7 -2/15 -17/10 -1/8 -5/3 -1/9 -18/11 -4/39 -85/52 -1/10 -67/41 -11/111 -49/30 -1/10 -31/19 -1/11 -13/8 -1/12 -34/21 0/1 -21/13 -1/3 -8/5 -2/15 -27/17 -1/9 -19/12 -1/8 -11/7 -3/25 -25/16 -7/60 -14/9 -4/35 -17/11 -1/9 -3/2 -1/10 -19/13 -1/9 -16/11 -6/55 -13/9 -3/29 -23/16 -3/28 -33/23 -13/125 -10/7 -4/39 -17/12 -1/10 -7/5 -3/31 -18/13 -8/85 -29/21 -1/11 -11/8 -1/12 -15/11 -1/11 -34/25 0/1 -19/14 -1/10 -4/3 -2/21 -17/13 -1/11 -13/10 -7/78 -48/37 -6/67 -35/27 -19/213 -57/44 -21/236 -136/105 -4/45 -79/61 -23/259 -22/17 -4/45 -31/24 -5/56 -102/79 -4/45 -71/55 -15/169 -40/31 -10/113 -9/7 -5/57 -23/18 -5/58 -14/11 0/1 -33/26 -5/54 -85/67 -1/11 -52/41 -10/111 -19/15 -1/11 -43/34 -1/10 -24/19 -2/23 -29/23 -5/57 -34/27 -2/23 -5/4 -1/12 -16/13 -10/117 -11/9 -7/83 -17/14 -1/12 -6/5 -4/49 -25/21 -1/11 -19/16 -1/12 -13/11 -1/13 -33/28 -13/160 -20/17 -2/25 -27/23 -11/137 -34/29 -2/25 -7/6 -3/38 -8/7 -6/77 -17/15 -1/13 -9/8 -7/92 -10/9 -4/53 -1/1 -1/15 0/1 0/1 1/1 1/15 9/8 7/92 17/15 1/13 8/7 6/77 7/6 3/38 20/17 2/25 13/11 1/13 19/16 1/12 25/21 1/11 6/5 4/49 17/14 1/12 11/9 7/83 5/4 1/12 24/19 2/23 43/34 1/10 19/15 1/11 33/26 5/54 14/11 0/1 9/7 5/57 22/17 4/45 35/27 19/213 13/10 7/78 17/13 1/11 4/3 2/21 15/11 1/11 11/8 1/12 18/13 8/85 7/5 3/31 17/12 1/10 10/7 4/39 43/30 13/126 33/23 13/125 56/39 2/19 23/16 3/28 13/9 3/29 3/2 1/10 17/11 1/9 14/9 4/35 11/7 3/25 19/12 1/8 8/5 2/15 13/8 1/12 18/11 4/39 5/3 1/9 17/10 1/8 12/7 2/15 43/25 1/9 74/43 4/31 31/18 3/22 19/11 1/7 7/4 1/8 16/9 2/9 9/5 1/11 11/6 1/6 13/7 1/9 15/8 1/8 17/9 1/7 2/1 0/1 17/8 1/8 15/7 1/7 13/6 1/6 11/5 1/9 9/4 1/4 34/15 0/1 25/11 1/17 16/7 2/21 7/3 1/7 26/11 4/31 19/8 1/8 31/13 3/23 12/5 2/15 17/7 1/7 5/2 1/6 18/7 4/21 85/33 1/5 67/26 11/54 49/19 1/5 31/12 1/4 13/5 1/3 34/13 0/1 21/8 1/12 8/3 2/15 27/10 1/6 19/7 1/7 11/4 3/20 25/9 7/45 14/5 4/25 17/6 1/6 3/1 1/5 19/6 1/6 16/5 6/35 13/4 3/16 23/7 3/17 33/10 13/70 10/3 4/21 17/5 1/5 7/2 3/14 18/5 8/35 29/8 1/4 11/3 1/3 15/4 1/4 34/9 0/1 19/5 1/5 4/1 2/9 17/4 1/4 13/3 7/27 48/11 6/23 35/8 19/72 57/13 21/79 136/31 4/15 79/18 23/86 22/5 4/15 31/7 5/19 102/23 4/15 71/16 15/56 40/9 10/37 9/2 5/18 23/5 5/17 14/3 0/1 33/7 5/21 85/18 1/4 52/11 10/39 19/4 1/4 43/9 1/5 24/5 2/7 29/6 5/18 34/7 2/7 5/1 1/3 16/3 10/33 11/2 7/22 17/3 1/3 6/1 4/11 25/4 1/4 19/3 1/3 13/2 1/2 33/5 13/35 20/3 2/5 27/4 11/28 34/5 2/5 7/1 3/7 8/1 6/13 17/2 1/2 9/1 7/13 10/1 4/7 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(33,340,-10,-103) (-9/1,1/0) -> (-43/13,-33/10) Hyperbolic Matrix(35,306,4,35) (-9/1,-17/2) -> (17/2,9/1) Hyperbolic Matrix(33,272,4,33) (-17/2,-8/1) -> (8/1,17/2) Hyperbolic Matrix(33,238,14,101) (-8/1,-7/1) -> (7/3,26/11) Hyperbolic Matrix(101,680,-86,-579) (-7/1,-20/3) -> (-20/17,-27/23) Hyperbolic Matrix(135,884,-104,-681) (-20/3,-13/2) -> (-13/10,-48/37) Hyperbolic Matrix(69,442,32,205) (-13/2,-19/3) -> (15/7,13/6) Hyperbolic Matrix(205,1292,162,1021) (-19/3,-25/4) -> (43/34,19/15) Hyperbolic Matrix(169,1054,38,237) (-25/4,-6/1) -> (40/9,9/2) Hyperbolic Matrix(35,204,6,35) (-6/1,-17/3) -> (17/3,6/1) Hyperbolic Matrix(67,374,12,67) (-17/3,-11/2) -> (11/2,17/3) Hyperbolic Matrix(69,374,-50,-271) (-11/2,-5/1) -> (-29/21,-11/8) Hyperbolic Matrix(169,816,-134,-647) (-5/1,-24/5) -> (-24/19,-29/23) Hyperbolic Matrix(135,646,14,67) (-24/5,-43/9) -> (9/1,10/1) Hyperbolic Matrix(271,1292,228,1087) (-43/9,-19/4) -> (19/16,25/21) Hyperbolic Matrix(611,2890,-374,-1769) (-19/4,-33/7) -> (-67/41,-49/30) Hyperbolic Matrix(441,2074,-340,-1599) (-33/7,-14/3) -> (-48/37,-35/27) Hyperbolic Matrix(103,476,-66,-305) (-14/3,-9/2) -> (-25/16,-14/9) Hyperbolic Matrix(239,1054,-100,-441) (-9/2,-22/5) -> (-12/5,-43/18) Hyperbolic Matrix(613,2686,-186,-815) (-22/5,-35/8) -> (-33/10,-56/17) Hyperbolic Matrix(203,884,-172,-749) (-35/8,-13/3) -> (-13/11,-33/28) Hyperbolic Matrix(103,442,24,103) (-13/3,-17/4) -> (17/4,13/3) Hyperbolic Matrix(33,136,8,33) (-17/4,-4/1) -> (4/1,17/4) Hyperbolic Matrix(35,136,-26,-101) (-4/1,-15/4) -> (-19/14,-4/3) Hyperbolic Matrix(101,374,64,237) (-15/4,-11/3) -> (11/7,19/12) Hyperbolic Matrix(103,374,-84,-305) (-11/3,-18/5) -> (-16/13,-11/9) Hyperbolic Matrix(67,238,38,135) (-18/5,-7/2) -> (7/4,16/9) Hyperbolic Matrix(69,238,20,69) (-7/2,-17/5) -> (17/5,7/2) Hyperbolic Matrix(101,340,30,101) (-17/5,-10/3) -> (10/3,17/5) Hyperbolic Matrix(441,1462,92,305) (-10/3,-43/13) -> (43/9,24/5) Hyperbolic Matrix(1497,4930,-1156,-3807) (-56/17,-23/7) -> (-79/61,-22/17) Hyperbolic Matrix(135,442,62,203) (-23/7,-13/4) -> (13/6,11/5) Hyperbolic Matrix(137,442,-84,-271) (-13/4,-3/1) -> (-31/19,-13/8) Hyperbolic Matrix(35,102,12,35) (-3/1,-17/6) -> (17/6,3/1) Hyperbolic Matrix(169,476,60,169) (-17/6,-14/5) -> (14/5,17/6) Hyperbolic Matrix(171,476,-134,-373) (-14/5,-11/4) -> (-23/18,-14/11) Hyperbolic Matrix(137,374,100,273) (-11/4,-19/7) -> (15/11,11/8) Hyperbolic Matrix(239,646,-138,-373) (-19/7,-8/3) -> (-26/15,-19/11) Hyperbolic Matrix(103,272,-64,-169) (-8/3,-13/5) -> (-21/13,-8/5) Hyperbolic Matrix(171,442,-118,-305) (-13/5,-18/7) -> (-16/11,-13/9) Hyperbolic Matrix(67,170,-54,-137) (-18/7,-5/2) -> (-5/4,-16/13) Hyperbolic Matrix(69,170,28,69) (-5/2,-17/7) -> (17/7,5/2) Hyperbolic Matrix(169,408,70,169) (-17/7,-12/5) -> (12/5,17/7) Hyperbolic Matrix(883,2108,142,339) (-43/18,-74/31) -> (6/1,25/4) Hyperbolic Matrix(2721,6494,-2108,-5031) (-74/31,-31/13) -> (-71/55,-40/31) Hyperbolic Matrix(885,2108,-542,-1291) (-31/13,-19/8) -> (-49/30,-31/19) Hyperbolic Matrix(273,646,-172,-407) (-19/8,-7/3) -> (-27/17,-19/12) Hyperbolic Matrix(103,238,74,171) (-7/3,-16/7) -> (18/13,7/5) Hyperbolic Matrix(239,544,-134,-305) (-16/7,-9/4) -> (-25/14,-16/9) Hyperbolic Matrix(169,374,-108,-239) (-9/4,-11/5) -> (-11/7,-25/16) Hyperbolic Matrix(203,442,62,135) (-11/5,-13/6) -> (13/4,23/7) Hyperbolic Matrix(205,442,32,69) (-13/6,-15/7) -> (19/3,13/2) Hyperbolic Matrix(239,510,112,239) (-15/7,-17/8) -> (17/8,15/7) Hyperbolic Matrix(33,68,16,33) (-17/8,-2/1) -> (2/1,17/8) Hyperbolic Matrix(35,68,18,35) (-2/1,-17/9) -> (17/9,2/1) Hyperbolic Matrix(271,510,144,271) (-17/9,-15/8) -> (15/8,17/9) Hyperbolic Matrix(237,442,200,373) (-15/8,-13/7) -> (13/11,19/16) Hyperbolic Matrix(239,442,166,307) (-13/7,-11/6) -> (23/16,13/9) Hyperbolic Matrix(169,306,-132,-239) (-11/6,-9/5) -> (-9/7,-23/18) Hyperbolic Matrix(645,1156,284,509) (-9/5,-34/19) -> (34/15,25/11) Hyperbolic Matrix(647,1156,286,511) (-34/19,-25/14) -> (9/4,34/15) Hyperbolic Matrix(135,238,38,67) (-16/9,-7/4) -> (7/2,18/5) Hyperbolic Matrix(137,238,118,205) (-7/4,-26/15) -> (8/7,7/6) Hyperbolic Matrix(375,646,-256,-441) (-19/11,-31/18) -> (-3/2,-19/13) Hyperbolic Matrix(613,1054,-474,-815) (-31/18,-12/7) -> (-22/17,-31/24) Hyperbolic Matrix(239,408,140,239) (-12/7,-17/10) -> (17/10,12/7) Hyperbolic Matrix(101,170,60,101) (-17/10,-5/3) -> (5/3,17/10) Hyperbolic Matrix(373,612,-270,-443) (-5/3,-18/11) -> (-18/13,-29/21) Hyperbolic Matrix(3639,5950,770,1259) (-18/11,-85/52) -> (85/18,52/11) Hyperbolic Matrix(5201,8500,1102,1801) (-85/52,-67/41) -> (33/7,85/18) Hyperbolic Matrix(713,1156,272,441) (-13/8,-34/21) -> (34/13,21/8) Hyperbolic Matrix(715,1156,274,443) (-34/21,-21/13) -> (13/5,34/13) Hyperbolic Matrix(171,272,22,35) (-8/5,-27/17) -> (7/1,8/1) Hyperbolic Matrix(237,374,64,101) (-19/12,-11/7) -> (11/3,15/4) Hyperbolic Matrix(307,476,198,307) (-14/9,-17/11) -> (17/11,14/9) Hyperbolic Matrix(67,102,44,67) (-17/11,-3/2) -> (3/2,17/11) Hyperbolic Matrix(885,1292,-698,-1019) (-19/13,-16/11) -> (-52/41,-19/15) Hyperbolic Matrix(307,442,166,239) (-13/9,-23/16) -> (11/6,13/7) Hyperbolic Matrix(1871,2686,-1444,-2073) (-23/16,-33/23) -> (-35/27,-57/44) Hyperbolic Matrix(237,340,-214,-307) (-33/23,-10/7) -> (-10/9,-1/1) Hyperbolic Matrix(239,340,168,239) (-10/7,-17/12) -> (17/12,10/7) Hyperbolic Matrix(169,238,120,169) (-17/12,-7/5) -> (7/5,17/12) Hyperbolic Matrix(171,238,74,103) (-7/5,-18/13) -> (16/7,7/3) Hyperbolic Matrix(273,374,100,137) (-11/8,-15/11) -> (19/7,11/4) Hyperbolic Matrix(849,1156,224,305) (-15/11,-34/25) -> (34/9,19/5) Hyperbolic Matrix(851,1156,226,307) (-34/25,-19/14) -> (15/4,34/9) Hyperbolic Matrix(103,136,78,103) (-4/3,-17/13) -> (17/13,4/3) Hyperbolic Matrix(339,442,260,339) (-17/13,-13/10) -> (13/10,17/13) Hyperbolic Matrix(14279,18496,3254,4215) (-57/44,-136/105) -> (136/31,79/18) Hyperbolic Matrix(14281,18496,3256,4217) (-136/105,-79/61) -> (57/13,136/31) Hyperbolic Matrix(8057,10404,1816,2345) (-31/24,-102/79) -> (102/23,71/16) Hyperbolic Matrix(8059,10404,1818,2347) (-102/79,-71/55) -> (31/7,102/23) Hyperbolic Matrix(817,1054,686,885) (-40/31,-9/7) -> (25/21,6/5) Hyperbolic Matrix(883,1122,-750,-953) (-14/11,-33/26) -> (-33/28,-20/17) Hyperbolic Matrix(6699,8500,2600,3299) (-33/26,-85/67) -> (85/33,67/26) Hyperbolic Matrix(4691,5950,1822,2311) (-85/67,-52/41) -> (18/7,85/33) Hyperbolic Matrix(1021,1292,162,205) (-19/15,-43/34) -> (25/4,19/3) Hyperbolic Matrix(1157,1462,808,1021) (-43/34,-24/19) -> (10/7,43/30) Hyperbolic Matrix(917,1156,188,237) (-29/23,-34/27) -> (34/7,5/1) Hyperbolic Matrix(919,1156,190,239) (-34/27,-5/4) -> (29/6,34/7) Hyperbolic Matrix(307,374,252,307) (-11/9,-17/14) -> (17/14,11/9) Hyperbolic Matrix(169,204,140,169) (-17/14,-6/5) -> (6/5,17/14) Hyperbolic Matrix(1769,2108,1028,1225) (-6/5,-25/21) -> (43/25,74/43) Hyperbolic Matrix(1087,1292,228,271) (-25/21,-19/16) -> (19/4,43/9) Hyperbolic Matrix(373,442,200,237) (-19/16,-13/11) -> (13/7,15/8) Hyperbolic Matrix(985,1156,144,169) (-27/23,-34/29) -> (34/5,7/1) Hyperbolic Matrix(987,1156,146,171) (-34/29,-7/6) -> (27/4,34/5) Hyperbolic Matrix(237,272,88,101) (-7/6,-8/7) -> (8/3,27/10) Hyperbolic Matrix(239,272,210,239) (-8/7,-17/15) -> (17/15,8/7) Hyperbolic Matrix(271,306,240,271) (-17/15,-9/8) -> (9/8,17/15) Hyperbolic Matrix(579,646,458,511) (-9/8,-10/9) -> (24/19,43/34) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(307,-340,214,-237) (1/1,9/8) -> (43/30,33/23) Hyperbolic Matrix(579,-680,86,-101) (7/6,20/17) -> (20/3,27/4) Hyperbolic Matrix(749,-884,172,-203) (20/17,13/11) -> (13/3,48/11) Hyperbolic Matrix(305,-374,84,-103) (11/9,5/4) -> (29/8,11/3) Hyperbolic Matrix(647,-816,134,-169) (5/4,24/19) -> (24/5,29/6) Hyperbolic Matrix(2279,-2890,884,-1121) (19/15,33/26) -> (67/26,49/19) Hyperbolic Matrix(1633,-2074,374,-475) (33/26,14/11) -> (48/11,35/8) Hyperbolic Matrix(373,-476,134,-171) (14/11,9/7) -> (25/9,14/5) Hyperbolic Matrix(815,-1054,474,-613) (9/7,22/17) -> (12/7,43/25) Hyperbolic Matrix(2073,-2686,1444,-1871) (22/17,35/27) -> (33/23,56/39) Hyperbolic Matrix(681,-884,104,-135) (35/27,13/10) -> (13/2,33/5) Hyperbolic Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(271,-374,50,-69) (11/8,18/13) -> (16/3,11/2) Hyperbolic Matrix(3433,-4930,782,-1123) (56/39,23/16) -> (79/18,22/5) Hyperbolic Matrix(305,-442,118,-171) (13/9,3/2) -> (31/12,13/5) Hyperbolic Matrix(305,-476,66,-103) (14/9,11/7) -> (23/5,14/3) Hyperbolic Matrix(407,-646,172,-273) (19/12,8/5) -> (26/11,19/8) Hyperbolic Matrix(169,-272,64,-103) (8/5,13/8) -> (21/8,8/3) Hyperbolic Matrix(271,-442,84,-137) (13/8,18/11) -> (16/5,13/4) Hyperbolic Matrix(103,-170,20,-33) (18/11,5/3) -> (5/1,16/3) Hyperbolic Matrix(3773,-6494,850,-1463) (74/43,31/18) -> (71/16,40/9) Hyperbolic Matrix(1223,-2108,474,-817) (31/18,19/11) -> (49/19,31/12) Hyperbolic Matrix(373,-646,138,-239) (19/11,7/4) -> (27/10,19/7) Hyperbolic Matrix(305,-544,134,-239) (16/9,9/5) -> (25/11,16/7) Hyperbolic Matrix(205,-374,74,-135) (9/5,11/6) -> (11/4,25/9) Hyperbolic Matrix(137,-306,30,-67) (11/5,9/4) -> (9/2,23/5) Hyperbolic Matrix(271,-646,86,-205) (19/8,31/13) -> (3/1,19/6) Hyperbolic Matrix(441,-1054,100,-239) (31/13,12/5) -> (22/5,31/7) Hyperbolic Matrix(239,-612,66,-169) (5/2,18/7) -> (18/5,29/8) Hyperbolic Matrix(407,-1292,86,-273) (19/6,16/5) -> (52/11,19/4) Hyperbolic Matrix(815,-2686,186,-613) (23/7,33/10) -> (35/8,57/13) Hyperbolic Matrix(103,-340,10,-33) (33/10,10/3) -> (10/1,1/0) Hyperbolic Matrix(239,-1122,36,-169) (14/3,33/7) -> (33/5,20/3) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(33,340,-10,-103) -> Matrix(13,8,-70,-43) Matrix(35,306,4,35) -> Matrix(27,14,52,27) Matrix(33,272,4,33) -> Matrix(25,12,52,25) Matrix(33,238,14,101) -> Matrix(5,2,42,17) Matrix(101,680,-86,-579) -> Matrix(41,16,-510,-199) Matrix(135,884,-104,-681) -> Matrix(23,8,-256,-89) Matrix(69,442,32,205) -> Matrix(5,2,32,13) Matrix(205,1292,162,1021) -> Matrix(1,0,14,1) Matrix(169,1054,38,237) -> Matrix(3,2,10,7) Matrix(35,204,6,35) -> Matrix(23,8,66,23) Matrix(67,374,12,67) -> Matrix(43,14,132,43) Matrix(69,374,-50,-271) -> Matrix(19,6,-206,-65) Matrix(169,816,-134,-647) -> Matrix(29,8,-330,-91) Matrix(135,646,14,67) -> Matrix(23,6,42,11) Matrix(271,1292,228,1087) -> Matrix(1,0,16,1) Matrix(611,2890,-374,-1769) -> Matrix(23,6,-234,-61) Matrix(441,2074,-340,-1599) -> Matrix(29,6,-324,-67) Matrix(103,476,-66,-305) -> Matrix(13,4,-114,-35) Matrix(239,1054,-100,-441) -> Matrix(7,2,-60,-17) Matrix(613,2686,-186,-815) -> Matrix(113,30,-614,-163) Matrix(203,884,-172,-749) -> Matrix(31,8,-376,-97) Matrix(103,442,24,103) -> Matrix(55,14,216,55) Matrix(33,136,8,33) -> Matrix(17,4,72,17) Matrix(35,136,-26,-101) -> Matrix(1,0,-6,1) Matrix(101,374,64,237) -> Matrix(9,2,76,17) Matrix(103,374,-84,-305) -> Matrix(25,6,-296,-71) Matrix(67,238,38,135) -> Matrix(9,2,58,13) Matrix(69,238,20,69) -> Matrix(29,6,140,29) Matrix(101,340,30,101) -> Matrix(41,8,210,41) Matrix(441,1462,92,305) -> Matrix(53,10,196,37) Matrix(1497,4930,-1156,-3807) -> Matrix(185,34,-2084,-383) Matrix(135,442,62,203) -> Matrix(11,2,82,15) Matrix(137,442,-84,-271) -> Matrix(11,2,-116,-21) Matrix(35,102,12,35) -> Matrix(11,2,60,11) Matrix(169,476,60,169) -> Matrix(49,8,300,49) Matrix(171,476,-134,-373) -> Matrix(25,4,-294,-47) Matrix(137,374,100,273) -> Matrix(13,2,136,21) Matrix(239,646,-138,-373) -> Matrix(13,2,-98,-15) Matrix(103,272,-64,-169) -> Matrix(1,0,0,1) Matrix(171,442,-118,-305) -> Matrix(9,2,-86,-19) Matrix(67,170,-54,-137) -> Matrix(13,2,-150,-23) Matrix(69,170,28,69) -> Matrix(13,2,84,13) Matrix(169,408,70,169) -> Matrix(29,4,210,29) Matrix(883,2108,142,339) -> Matrix(1,0,10,1) Matrix(2721,6494,-2108,-5031) -> Matrix(133,18,-1500,-203) Matrix(885,2108,-542,-1291) -> Matrix(31,4,-318,-41) Matrix(273,646,-172,-407) -> Matrix(15,2,-128,-17) Matrix(103,238,74,171) -> Matrix(17,2,178,21) Matrix(239,544,-134,-305) -> Matrix(1,0,6,1) Matrix(169,374,-108,-239) -> Matrix(15,2,-128,-17) Matrix(203,442,62,135) -> Matrix(15,2,82,11) Matrix(205,442,32,69) -> Matrix(13,2,32,5) Matrix(239,510,112,239) -> Matrix(15,2,112,15) Matrix(33,68,16,33) -> Matrix(1,0,16,1) Matrix(35,68,18,35) -> Matrix(1,0,14,1) Matrix(271,510,144,271) -> Matrix(15,2,112,15) Matrix(237,442,200,373) -> Matrix(17,2,212,25) Matrix(239,442,166,307) -> Matrix(15,2,142,19) Matrix(169,306,-132,-239) -> Matrix(17,2,-196,-23) Matrix(645,1156,284,509) -> Matrix(1,0,28,1) Matrix(647,1156,286,511) -> Matrix(1,0,2,1) Matrix(135,238,38,67) -> Matrix(13,2,58,9) Matrix(137,238,118,205) -> Matrix(13,2,162,25) Matrix(375,646,-256,-441) -> Matrix(15,2,-128,-17) Matrix(613,1054,-474,-815) -> Matrix(13,2,-150,-23) Matrix(239,408,140,239) -> Matrix(31,4,240,31) Matrix(101,170,60,101) -> Matrix(17,2,144,17) Matrix(373,612,-270,-443) -> Matrix(37,4,-398,-43) Matrix(3639,5950,770,1259) -> Matrix(139,14,546,55) Matrix(5201,8500,1102,1801) -> Matrix(161,16,654,65) Matrix(713,1156,272,441) -> Matrix(1,0,24,1) Matrix(715,1156,274,443) -> Matrix(1,0,6,1) Matrix(171,272,22,35) -> Matrix(33,4,74,9) Matrix(237,374,64,101) -> Matrix(17,2,76,9) Matrix(307,476,198,307) -> Matrix(71,8,630,71) Matrix(67,102,44,67) -> Matrix(19,2,180,19) Matrix(885,1292,-698,-1019) -> Matrix(35,4,-394,-45) Matrix(307,442,166,239) -> Matrix(19,2,142,15) Matrix(1871,2686,-1444,-2073) -> Matrix(287,30,-3224,-337) Matrix(237,340,-214,-307) -> Matrix(77,8,-1030,-107) Matrix(239,340,168,239) -> Matrix(79,8,780,79) Matrix(169,238,120,169) -> Matrix(61,6,620,61) Matrix(171,238,74,103) -> Matrix(21,2,178,17) Matrix(273,374,100,137) -> Matrix(21,2,136,13) Matrix(849,1156,224,305) -> Matrix(1,0,16,1) Matrix(851,1156,226,307) -> Matrix(1,0,14,1) Matrix(103,136,78,103) -> Matrix(43,4,462,43) Matrix(339,442,260,339) -> Matrix(155,14,1716,155) Matrix(14279,18496,3254,4215) -> Matrix(1979,176,7410,659) Matrix(14281,18496,3256,4217) -> Matrix(1981,176,7440,661) Matrix(8057,10404,1816,2345) -> Matrix(899,80,3360,299) Matrix(8059,10404,1818,2347) -> Matrix(901,80,3390,301) Matrix(817,1054,686,885) -> Matrix(23,2,310,27) Matrix(883,1122,-750,-953) -> Matrix(19,2,-238,-25) Matrix(6699,8500,2600,3299) -> Matrix(175,16,864,79) Matrix(4691,5950,1822,2311) -> Matrix(155,14,786,71) Matrix(1021,1292,162,205) -> Matrix(1,0,14,1) Matrix(1157,1462,808,1021) -> Matrix(113,10,1096,97) Matrix(917,1156,188,237) -> Matrix(137,12,468,41) Matrix(919,1156,190,239) -> Matrix(139,12,498,43) Matrix(307,374,252,307) -> Matrix(167,14,1992,167) Matrix(169,204,140,169) -> Matrix(97,8,1176,97) Matrix(1769,2108,1028,1225) -> Matrix(1,0,20,1) Matrix(1087,1292,228,271) -> Matrix(1,0,16,1) Matrix(373,442,200,237) -> Matrix(25,2,212,17) Matrix(985,1156,144,169) -> Matrix(349,28,860,69) Matrix(987,1156,146,171) -> Matrix(351,28,890,71) Matrix(237,272,88,101) -> Matrix(51,4,344,27) Matrix(239,272,210,239) -> Matrix(155,12,2002,155) Matrix(271,306,240,271) -> Matrix(183,14,2392,183) Matrix(579,646,458,511) -> Matrix(79,6,882,67) Matrix(1,0,2,1) -> Matrix(1,0,30,1) Matrix(307,-340,214,-237) -> Matrix(107,-8,1030,-77) Matrix(579,-680,86,-101) -> Matrix(199,-16,510,-41) Matrix(749,-884,172,-203) -> Matrix(97,-8,376,-31) Matrix(305,-374,84,-103) -> Matrix(71,-6,296,-25) Matrix(647,-816,134,-169) -> Matrix(91,-8,330,-29) Matrix(2279,-2890,884,-1121) -> Matrix(67,-6,324,-29) Matrix(1633,-2074,374,-475) -> Matrix(61,-6,234,-23) Matrix(373,-476,134,-171) -> Matrix(47,-4,294,-25) Matrix(815,-1054,474,-613) -> Matrix(23,-2,150,-13) Matrix(2073,-2686,1444,-1871) -> Matrix(337,-30,3224,-287) Matrix(681,-884,104,-135) -> Matrix(89,-8,256,-23) Matrix(101,-136,26,-35) -> Matrix(1,0,-6,1) Matrix(271,-374,50,-69) -> Matrix(65,-6,206,-19) Matrix(3433,-4930,782,-1123) -> Matrix(325,-34,1214,-127) Matrix(305,-442,118,-171) -> Matrix(19,-2,86,-9) Matrix(305,-476,66,-103) -> Matrix(35,-4,114,-13) Matrix(407,-646,172,-273) -> Matrix(17,-2,128,-15) Matrix(169,-272,64,-103) -> Matrix(1,0,0,1) Matrix(271,-442,84,-137) -> Matrix(21,-2,116,-11) Matrix(103,-170,20,-33) -> Matrix(17,-2,60,-7) Matrix(3773,-6494,850,-1463) -> Matrix(137,-18,510,-67) Matrix(1223,-2108,474,-817) -> Matrix(29,-4,138,-19) Matrix(373,-646,138,-239) -> Matrix(15,-2,98,-13) Matrix(305,-544,134,-239) -> Matrix(1,0,6,1) Matrix(205,-374,74,-135) -> Matrix(15,-2,98,-13) Matrix(137,-306,30,-67) -> Matrix(13,-2,46,-7) Matrix(271,-646,86,-205) -> Matrix(15,-2,98,-13) Matrix(441,-1054,100,-239) -> Matrix(17,-2,60,-7) Matrix(239,-612,66,-169) -> Matrix(23,-4,98,-17) Matrix(407,-1292,86,-273) -> Matrix(25,-4,94,-15) Matrix(815,-2686,186,-613) -> Matrix(163,-30,614,-113) Matrix(103,-340,10,-33) -> Matrix(43,-8,70,-13) Matrix(239,-1122,36,-169) -> Matrix(11,-2,28,-5) 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): 72 Degree of the the map X: 72 Degree of the the map Y: 144 ----------------------------------------------------------------------- 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): 144 Minimal number of generators: 25 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: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 34/15 17/7 17/6 3/1 17/5 34/9 4/1 17/4 9/2 5/1 17/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/15 6/5 4/49 5/4 1/12 4/3 2/21 15/11 1/11 11/8 1/12 18/13 8/85 7/5 3/31 3/2 1/10 11/7 3/25 8/5 2/15 13/8 1/12 5/3 1/9 7/4 1/8 16/9 2/9 9/5 1/11 11/6 1/6 2/1 0/1 9/4 1/4 34/15 0/1 25/11 1/17 16/7 2/21 7/3 1/7 12/5 2/15 17/7 1/7 5/2 1/6 8/3 2/15 19/7 1/7 11/4 3/20 14/5 4/25 17/6 1/6 3/1 1/5 10/3 4/21 17/5 1/5 7/2 3/14 11/3 1/3 15/4 1/4 34/9 0/1 19/5 1/5 4/1 2/9 17/4 1/4 13/3 7/27 9/2 5/18 5/1 1/3 11/2 7/22 17/3 1/3 6/1 4/11 7/1 3/7 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(101,-119,73,-86) (1/1,6/5) -> (11/8,18/13) Hyperbolic Matrix(69,-85,13,-16) (6/5,5/4) -> (5/1,11/2) Hyperbolic Matrix(67,-85,41,-52) (5/4,4/3) -> (13/8,5/3) Hyperbolic Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(137,-187,74,-101) (15/11,11/8) -> (11/6,2/1) Hyperbolic Matrix(171,-238,97,-135) (18/13,7/5) -> (7/4,16/9) Hyperbolic Matrix(35,-51,11,-16) (7/5,3/2) -> (3/1,10/3) Hyperbolic Matrix(120,-187,43,-67) (3/2,11/7) -> (11/4,14/5) Hyperbolic Matrix(118,-187,65,-103) (11/7,8/5) -> (9/5,11/6) Hyperbolic Matrix(137,-221,31,-50) (8/5,13/8) -> (13/3,9/2) Hyperbolic Matrix(69,-119,29,-50) (5/3,7/4) -> (7/3,12/5) Hyperbolic Matrix(305,-544,134,-239) (16/9,9/5) -> (25/11,16/7) Hyperbolic Matrix(84,-187,31,-69) (2/1,9/4) -> (8/3,19/7) Hyperbolic Matrix(511,-1156,225,-509) (9/4,34/15) -> (34/15,25/11) Parabolic Matrix(52,-119,7,-16) (16/7,7/3) -> (7/1,1/0) Hyperbolic Matrix(120,-289,49,-118) (12/5,17/7) -> (17/7,5/2) Parabolic Matrix(33,-85,7,-18) (5/2,8/3) -> (9/2,5/1) Hyperbolic Matrix(137,-374,37,-101) (19/7,11/4) -> (11/3,15/4) Hyperbolic Matrix(103,-289,36,-101) (14/5,17/6) -> (17/6,3/1) Parabolic Matrix(86,-289,25,-84) (10/3,17/5) -> (17/5,7/2) Parabolic Matrix(33,-119,5,-18) (7/2,11/3) -> (6/1,7/1) Hyperbolic Matrix(307,-1156,81,-305) (15/4,34/9) -> (34/9,19/5) Parabolic Matrix(69,-289,16,-67) (4/1,17/4) -> (17/4,13/3) Parabolic Matrix(52,-289,9,-50) (11/2,17/3) -> (17/3,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,15,1) Matrix(101,-119,73,-86) -> Matrix(37,-3,395,-32) Matrix(69,-85,13,-16) -> Matrix(35,-3,117,-10) Matrix(67,-85,41,-52) -> Matrix(11,-1,111,-10) Matrix(101,-136,26,-35) -> Matrix(1,0,-6,1) Matrix(137,-187,74,-101) -> Matrix(11,-1,78,-7) Matrix(171,-238,97,-135) -> Matrix(21,-2,137,-13) Matrix(35,-51,11,-16) -> Matrix(9,-1,55,-6) Matrix(120,-187,43,-67) -> Matrix(26,-3,165,-19) Matrix(118,-187,65,-103) -> Matrix(8,-1,73,-9) Matrix(137,-221,31,-50) -> Matrix(5,-1,21,-4) Matrix(69,-119,29,-50) -> Matrix(7,-1,57,-8) Matrix(305,-544,134,-239) -> Matrix(1,0,6,1) Matrix(84,-187,31,-69) -> Matrix(6,-1,43,-7) Matrix(511,-1156,225,-509) -> Matrix(1,0,13,1) Matrix(52,-119,7,-16) -> Matrix(10,-1,21,-2) Matrix(120,-289,49,-118) -> Matrix(22,-3,147,-20) Matrix(33,-85,7,-18) -> Matrix(5,-1,21,-4) Matrix(137,-374,37,-101) -> Matrix(13,-2,59,-9) Matrix(103,-289,36,-101) -> Matrix(31,-5,180,-29) Matrix(86,-289,25,-84) -> Matrix(36,-7,175,-34) Matrix(33,-119,5,-18) -> Matrix(13,-3,35,-8) Matrix(307,-1156,81,-305) -> Matrix(1,0,1,1) Matrix(69,-289,16,-67) -> Matrix(37,-9,144,-35) Matrix(52,-289,9,-50) -> Matrix(34,-11,99,-32) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 1 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: 1 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 72 ----------------------------------------------------------------------- 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 15 1 2/1 0/1 1 17 9/4 1/4 1 17 34/15 0/1 13 1 16/7 2/21 1 17 7/3 1/7 1 17 17/7 1/7 3 1 5/2 1/6 1 17 8/3 2/15 1 17 19/7 1/7 1 17 11/4 3/20 1 17 17/6 1/6 5 1 3/1 1/5 1 17 17/5 1/5 7 1 7/2 3/14 1 17 11/3 1/3 1 17 15/4 1/4 1 17 34/9 0/1 1 1 4/1 2/9 1 17 17/4 1/4 9 1 9/2 5/18 1 17 5/1 1/3 1 17 17/3 1/3 11 1 6/1 4/11 1 17 7/1 3/7 1 17 1/0 1/0 1 17 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(84,-187,31,-69) (2/1,9/4) -> (8/3,19/7) Hyperbolic Matrix(271,-612,120,-271) (9/4,34/15) -> (9/4,34/15) Reflection Matrix(239,-544,105,-239) (34/15,16/7) -> (34/15,16/7) Reflection Matrix(52,-119,7,-16) (16/7,7/3) -> (7/1,1/0) Hyperbolic Matrix(50,-119,21,-50) (7/3,17/7) -> (7/3,17/7) Reflection Matrix(69,-170,28,-69) (17/7,5/2) -> (17/7,5/2) Reflection Matrix(33,-85,7,-18) (5/2,8/3) -> (9/2,5/1) Hyperbolic Matrix(137,-374,37,-101) (19/7,11/4) -> (11/3,15/4) Hyperbolic Matrix(67,-187,24,-67) (11/4,17/6) -> (11/4,17/6) Reflection Matrix(35,-102,12,-35) (17/6,3/1) -> (17/6,3/1) Reflection Matrix(16,-51,5,-16) (3/1,17/5) -> (3/1,17/5) Reflection Matrix(69,-238,20,-69) (17/5,7/2) -> (17/5,7/2) Reflection Matrix(33,-119,5,-18) (7/2,11/3) -> (6/1,7/1) Hyperbolic Matrix(271,-1020,72,-271) (15/4,34/9) -> (15/4,34/9) Reflection Matrix(35,-136,9,-35) (34/9,4/1) -> (34/9,4/1) Reflection Matrix(33,-136,8,-33) (4/1,17/4) -> (4/1,17/4) Reflection Matrix(35,-153,8,-35) (17/4,9/2) -> (17/4,9/2) Reflection Matrix(16,-85,3,-16) (5/1,17/3) -> (5/1,17/3) Reflection Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) 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(1,0,1,-1) -> Matrix(1,0,15,-1) (0/1,2/1) -> (0/1,2/15) Matrix(84,-187,31,-69) -> Matrix(6,-1,43,-7) (1/7,1/5).(0/1,2/13).(1/8,1/6) Matrix(271,-612,120,-271) -> Matrix(1,0,8,-1) (9/4,34/15) -> (0/1,1/4) Matrix(239,-544,105,-239) -> Matrix(1,0,21,-1) (34/15,16/7) -> (0/1,2/21) Matrix(52,-119,7,-16) -> Matrix(10,-1,21,-2) Matrix(50,-119,21,-50) -> Matrix(8,-1,63,-8) (7/3,17/7) -> (1/9,1/7) Matrix(69,-170,28,-69) -> Matrix(13,-2,84,-13) (17/7,5/2) -> (1/7,1/6) Matrix(33,-85,7,-18) -> Matrix(5,-1,21,-4) (1/5,1/3).(0/1,2/9).(1/6,1/4) Matrix(137,-374,37,-101) -> Matrix(13,-2,59,-9) Matrix(67,-187,24,-67) -> Matrix(19,-3,120,-19) (11/4,17/6) -> (3/20,1/6) Matrix(35,-102,12,-35) -> Matrix(11,-2,60,-11) (17/6,3/1) -> (1/6,1/5) Matrix(16,-51,5,-16) -> Matrix(6,-1,35,-6) (3/1,17/5) -> (1/7,1/5) Matrix(69,-238,20,-69) -> Matrix(29,-6,140,-29) (17/5,7/2) -> (1/5,3/14) Matrix(33,-119,5,-18) -> Matrix(13,-3,35,-8) Matrix(271,-1020,72,-271) -> Matrix(1,0,8,-1) (15/4,34/9) -> (0/1,1/4) Matrix(35,-136,9,-35) -> Matrix(1,0,9,-1) (34/9,4/1) -> (0/1,2/9) Matrix(33,-136,8,-33) -> Matrix(17,-4,72,-17) (4/1,17/4) -> (2/9,1/4) Matrix(35,-153,8,-35) -> Matrix(19,-5,72,-19) (17/4,9/2) -> (1/4,5/18) Matrix(16,-85,3,-16) -> Matrix(10,-3,33,-10) (5/1,17/3) -> (3/11,1/3) Matrix(35,-204,6,-35) -> Matrix(23,-8,66,-23) (17/3,6/1) -> (1/3,4/11) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.