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 -15/34 -33/85 -13/34 -9/34 -31/136 -23/102 -18/85 -7/34 -3/16 -5/34 -1/10 0/1 1/9 2/17 1/8 1/7 2/13 1/6 3/17 2/11 1/5 3/14 2/9 3/13 4/17 1/4 4/15 3/11 2/7 5/17 1/3 6/17 2/5 7/17 5/12 7/15 8/17 1/2 9/17 10/17 11/17 2/3 12/17 13/17 14/17 15/17 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -8/9 2/3 1/1 -15/17 1/1 -7/8 1/1 2/1 -6/7 1/1 1/0 -17/20 1/1 1/0 -11/13 0/1 1/1 -16/19 1/1 1/0 -21/25 0/1 1/1 -5/6 0/1 1/1 -14/17 1/1 -9/11 1/1 2/1 -4/5 1/1 1/0 -19/24 -2/1 1/0 -34/43 -1/1 0/1 -15/19 -1/1 1/0 -26/33 0/1 1/0 -11/14 1/1 1/0 -7/9 -1/1 0/1 -17/22 -1/1 -1/2 -27/35 -1/1 0/1 -10/13 -1/2 0/1 -13/17 0/1 -3/4 0/1 1/1 -11/15 1/1 1/0 -8/11 0/1 1/0 -13/18 0/1 1/1 -5/7 0/1 1/0 -12/17 0/1 -7/10 0/1 1/2 -30/43 2/3 1/1 -23/33 0/1 1/1 -39/56 1/1 1/0 -16/23 0/1 1/2 -9/13 0/1 1/1 -2/3 0/1 1/1 -11/17 1/1 -9/14 1/1 1/0 -7/11 0/1 1/1 -12/19 1/1 1/0 -5/8 0/1 1/1 -8/13 0/1 1/2 -11/18 0/1 1/1 -3/5 1/2 1/1 -10/17 1/1 -7/12 1/1 3/2 -25/43 1/1 2/1 -43/74 1/1 2/1 -18/31 1/1 2/1 -11/19 1/1 1/0 -4/7 1/1 1/0 -9/16 0/1 1/1 -5/9 1/1 2/1 -6/11 0/1 1/0 -7/13 0/1 1/1 -8/15 1/2 1/1 -9/17 1/1 -1/2 1/1 1/0 -8/17 1/0 -7/15 -1/1 1/0 -6/13 0/1 1/0 -5/11 0/1 1/1 -4/9 0/1 1/1 -15/34 1/1 -11/25 1/1 4/3 -7/16 1/1 2/1 -3/7 2/1 1/0 -11/26 2/1 3/1 -8/19 3/1 1/0 -13/31 3/1 1/0 -5/12 5/1 1/0 -7/17 1/0 -2/5 -1/1 1/0 -7/18 -1/1 0/1 -33/85 0/1 -26/67 0/1 1/0 -19/49 -1/1 1/0 -12/31 -1/1 0/1 -5/13 -1/1 0/1 -13/34 0/1 -8/21 0/1 1/2 -3/8 0/1 1/1 -10/27 1/1 1/0 -7/19 1/1 1/0 -4/11 2/1 1/0 -9/25 3/1 4/1 -5/14 5/1 1/0 -6/17 1/0 -1/3 -1/1 1/0 -6/19 -1/1 1/0 -5/16 -1/1 0/1 -4/13 0/1 1/0 -7/23 -1/1 0/1 -10/33 0/1 1/0 -3/10 0/1 1/0 -5/17 1/0 -2/7 -1/1 1/0 -5/18 -1/1 0/1 -8/29 -1/1 1/0 -3/11 -1/1 0/1 -4/15 -1/1 1/0 -9/34 -1/1 -5/19 -1/1 -1/2 -1/4 -1/1 0/1 -4/17 0/1 -3/13 0/1 1/3 -11/48 1/3 1/2 -8/35 0/1 1/2 -13/57 0/1 1/1 -31/136 0/1 -18/79 0/1 1/4 -5/22 1/3 1/2 -7/31 1/2 1/1 -23/102 1/1 -16/71 0/1 1/1 -9/40 0/1 1/1 -2/9 0/1 1/1 -5/23 0/1 1/1 -3/14 1/1 1/0 -7/33 0/1 1/1 -18/85 1/1 -11/52 1/1 2/1 -4/19 1/1 1/0 -9/43 1/1 2/1 -5/24 2/1 1/0 -6/29 5/1 1/0 -7/34 1/0 -1/5 -1/1 1/0 -3/16 -1/1 0/1 -2/11 -1/2 0/1 -3/17 0/1 -1/6 0/1 1/1 -4/25 0/1 1/1 -3/19 1/1 1/0 -2/13 0/1 1/0 -5/33 0/1 1/1 -3/20 1/1 1/0 -4/27 1/1 1/0 -5/34 1/0 -1/7 0/1 1/0 -1/8 -1/1 0/1 -2/17 0/1 -1/9 0/1 1/1 -1/10 0/1 1/0 0/1 0/1 1/0 1/9 -1/1 0/1 2/17 0/1 1/8 0/1 1/1 1/7 0/1 1/0 3/20 -1/1 1/0 2/13 0/1 1/0 3/19 -1/1 1/0 4/25 -1/1 0/1 1/6 -1/1 0/1 3/17 0/1 2/11 0/1 1/2 1/5 1/1 1/0 5/24 -2/1 1/0 9/43 -2/1 -1/1 4/19 -1/1 1/0 7/33 -1/1 0/1 3/14 -1/1 1/0 2/9 -1/1 0/1 5/22 -1/2 -1/3 8/35 -1/2 0/1 3/13 -1/3 0/1 4/17 0/1 1/4 0/1 1/1 4/15 1/1 1/0 3/11 0/1 1/1 5/18 0/1 1/1 2/7 1/1 1/0 5/17 1/0 3/10 0/1 1/0 13/43 -1/1 0/1 10/33 0/1 1/0 17/56 -1/1 1/0 7/23 0/1 1/1 4/13 0/1 1/0 1/3 1/1 1/0 6/17 1/0 5/14 -5/1 1/0 4/11 -2/1 1/0 7/19 -1/1 1/0 3/8 -1/1 0/1 5/13 0/1 1/1 7/18 0/1 1/1 2/5 1/1 1/0 7/17 1/0 5/12 -5/1 1/0 18/43 -4/1 -3/1 31/74 -4/1 -3/1 13/31 -3/1 1/0 8/19 -3/1 1/0 3/7 -2/1 1/0 7/16 -2/1 -1/1 4/9 -1/1 0/1 5/11 -1/1 0/1 6/13 0/1 1/0 7/15 1/1 1/0 8/17 1/0 1/2 -1/1 1/0 9/17 -1/1 8/15 -1/1 -1/2 7/13 -1/1 0/1 6/11 0/1 1/0 5/9 -2/1 -1/1 19/34 -1/1 14/25 -1/1 -2/3 9/16 -1/1 0/1 4/7 -1/1 1/0 15/26 -1/1 0/1 11/19 -1/1 1/0 18/31 -2/1 -1/1 7/12 -3/2 -1/1 10/17 -1/1 3/5 -1/1 -1/2 11/18 -1/1 0/1 52/85 -1/1 41/67 -1/1 -2/3 30/49 -1/1 -1/2 19/31 -1/1 -1/2 8/13 -1/2 0/1 21/34 0/1 13/21 0/1 1/1 5/8 -1/1 0/1 17/27 0/1 1/0 12/19 -1/1 1/0 7/11 -1/1 0/1 16/25 -1/1 0/1 9/14 -1/1 1/0 11/17 -1/1 2/3 -1/1 0/1 13/19 -1/1 -1/2 11/16 -1/1 0/1 9/13 -1/1 0/1 16/23 -1/2 0/1 23/33 -1/1 0/1 7/10 -1/2 0/1 12/17 0/1 5/7 0/1 1/0 13/18 -1/1 0/1 21/29 -1/1 -1/2 8/11 0/1 1/0 11/15 -1/1 1/0 25/34 -1/1 14/19 -1/1 -1/2 3/4 -1/1 0/1 13/17 0/1 10/13 0/1 1/2 37/48 1/2 1/1 27/35 0/1 1/1 44/57 0/1 1/0 105/136 0/1 61/79 0/1 1/3 17/22 1/2 1/1 24/31 0/1 1/1 79/102 1/1 55/71 1/1 1/0 31/40 0/1 1/1 7/9 0/1 1/1 18/23 2/1 1/0 11/14 -1/1 1/0 26/33 0/1 1/0 67/85 0/1 41/52 0/1 1/1 15/19 1/1 1/0 34/43 0/1 1/1 19/24 2/1 1/0 23/29 5/1 1/0 27/34 1/0 4/5 -1/1 1/0 13/16 -2/1 -1/1 9/11 -2/1 -1/1 14/17 -1/1 5/6 -1/1 0/1 21/25 -1/1 0/1 16/19 -1/1 1/0 11/13 -1/1 0/1 28/33 0/1 1/0 17/20 -1/1 1/0 23/27 0/1 1/0 29/34 1/0 6/7 -1/1 1/0 7/8 -2/1 -1/1 15/17 -1/1 8/9 -1/1 -2/3 9/10 -1/2 0/1 1/1 -1/1 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(237,214,-340,-307) (-1/1,-8/9) -> (-30/43,-23/33) Hyperbolic Matrix(271,240,306,271) (-8/9,-15/17) -> (15/17,8/9) Hyperbolic Matrix(239,210,272,239) (-15/17,-7/8) -> (7/8,15/17) Hyperbolic Matrix(137,118,238,205) (-7/8,-6/7) -> (4/7,15/26) Hyperbolic Matrix(101,86,-680,-579) (-6/7,-17/20) -> (-3/20,-4/27) Hyperbolic Matrix(203,172,-884,-749) (-17/20,-11/13) -> (-3/13,-11/48) Hyperbolic Matrix(237,200,442,373) (-11/13,-16/19) -> (8/15,7/13) Hyperbolic Matrix(271,228,1292,1087) (-16/19,-21/25) -> (9/43,4/19) Hyperbolic Matrix(817,686,1054,885) (-21/25,-5/6) -> (31/40,7/9) Hyperbolic Matrix(169,140,204,169) (-5/6,-14/17) -> (14/17,5/6) Hyperbolic Matrix(307,252,374,307) (-14/17,-9/11) -> (9/11,14/17) Hyperbolic Matrix(103,84,-374,-305) (-9/11,-4/5) -> (-8/29,-3/11) Hyperbolic Matrix(169,134,-816,-647) (-4/5,-19/24) -> (-5/24,-6/29) Hyperbolic Matrix(579,458,646,511) (-19/24,-34/43) -> (8/9,9/10) Hyperbolic Matrix(205,162,1292,1021) (-34/43,-15/19) -> (3/19,4/25) Hyperbolic Matrix(1121,884,-2890,-2279) (-15/19,-26/33) -> (-26/67,-19/49) Hyperbolic Matrix(475,374,-2074,-1633) (-26/33,-11/14) -> (-11/48,-8/35) Hyperbolic Matrix(171,134,-476,-373) (-11/14,-7/9) -> (-9/25,-5/14) Hyperbolic Matrix(613,474,-1054,-815) (-7/9,-17/22) -> (-7/12,-25/43) Hyperbolic Matrix(1871,1444,-2686,-2073) (-17/22,-27/35) -> (-23/33,-39/56) Hyperbolic Matrix(135,104,-884,-681) (-27/35,-10/13) -> (-2/13,-5/33) Hyperbolic Matrix(339,260,442,339) (-10/13,-13/17) -> (13/17,10/13) Hyperbolic Matrix(103,78,136,103) (-13/17,-3/4) -> (3/4,13/17) Hyperbolic Matrix(35,26,-136,-101) (-3/4,-11/15) -> (-5/19,-1/4) Hyperbolic Matrix(137,100,374,273) (-11/15,-8/11) -> (4/11,7/19) Hyperbolic Matrix(69,50,-374,-271) (-8/11,-13/18) -> (-3/16,-2/11) Hyperbolic Matrix(103,74,238,171) (-13/18,-5/7) -> (3/7,7/16) Hyperbolic Matrix(169,120,238,169) (-5/7,-12/17) -> (12/17,5/7) Hyperbolic Matrix(239,168,340,239) (-12/17,-7/10) -> (7/10,12/17) Hyperbolic Matrix(1157,808,1462,1021) (-7/10,-30/43) -> (34/43,19/24) Hyperbolic Matrix(1123,782,-4930,-3433) (-39/56,-16/23) -> (-18/79,-5/22) Hyperbolic Matrix(239,166,442,307) (-16/23,-9/13) -> (7/13,6/11) Hyperbolic Matrix(171,118,-442,-305) (-9/13,-2/3) -> (-12/31,-5/13) Hyperbolic Matrix(67,44,102,67) (-2/3,-11/17) -> (11/17,2/3) Hyperbolic Matrix(307,198,476,307) (-11/17,-9/14) -> (9/14,11/17) Hyperbolic Matrix(103,66,-476,-305) (-9/14,-7/11) -> (-5/23,-3/14) Hyperbolic Matrix(101,64,374,237) (-7/11,-12/19) -> (4/15,3/11) Hyperbolic Matrix(273,172,-646,-407) (-12/19,-5/8) -> (-11/26,-8/19) Hyperbolic Matrix(103,64,-272,-169) (-5/8,-8/13) -> (-8/21,-3/8) Hyperbolic Matrix(137,84,-442,-271) (-8/13,-11/18) -> (-5/16,-4/13) Hyperbolic Matrix(33,20,-170,-103) (-11/18,-3/5) -> (-1/5,-3/16) Hyperbolic Matrix(101,60,170,101) (-3/5,-10/17) -> (10/17,3/5) Hyperbolic Matrix(239,140,408,239) (-10/17,-7/12) -> (7/12,10/17) Hyperbolic Matrix(1769,1028,2108,1225) (-25/43,-43/74) -> (5/6,21/25) Hyperbolic Matrix(1463,850,-6494,-3773) (-43/74,-18/31) -> (-16/71,-9/40) Hyperbolic Matrix(817,474,-2108,-1223) (-18/31,-11/19) -> (-19/49,-12/31) Hyperbolic Matrix(239,138,-646,-373) (-11/19,-4/7) -> (-10/27,-7/19) Hyperbolic Matrix(67,38,238,135) (-4/7,-9/16) -> (5/18,2/7) Hyperbolic Matrix(239,134,-544,-305) (-9/16,-5/9) -> (-11/25,-7/16) Hyperbolic Matrix(135,74,-374,-205) (-5/9,-6/11) -> (-4/11,-9/25) Hyperbolic Matrix(307,166,442,239) (-6/11,-7/13) -> (9/13,16/23) Hyperbolic Matrix(373,200,442,237) (-7/13,-8/15) -> (16/19,11/13) Hyperbolic Matrix(271,144,510,271) (-8/15,-9/17) -> (9/17,8/15) Hyperbolic Matrix(35,18,68,35) (-9/17,-1/2) -> (1/2,9/17) Hyperbolic Matrix(33,16,68,33) (-1/2,-8/17) -> (8/17,1/2) Hyperbolic Matrix(239,112,510,239) (-8/17,-7/15) -> (7/15,8/17) Hyperbolic Matrix(69,32,442,205) (-7/15,-6/13) -> (2/13,3/19) Hyperbolic Matrix(135,62,442,203) (-6/13,-5/11) -> (7/23,4/13) Hyperbolic Matrix(67,30,-306,-137) (-5/11,-4/9) -> (-2/9,-5/23) Hyperbolic Matrix(647,286,1156,511) (-4/9,-15/34) -> (19/34,14/25) Hyperbolic Matrix(645,284,1156,509) (-15/34,-11/25) -> (5/9,19/34) Hyperbolic Matrix(171,74,238,103) (-7/16,-3/7) -> (5/7,13/18) Hyperbolic Matrix(33,14,238,101) (-3/7,-11/26) -> (1/8,1/7) Hyperbolic Matrix(205,86,-646,-271) (-8/19,-13/31) -> (-1/3,-6/19) Hyperbolic Matrix(239,100,-1054,-441) (-13/31,-5/12) -> (-5/22,-7/31) Hyperbolic Matrix(169,70,408,169) (-5/12,-7/17) -> (7/17,5/12) Hyperbolic Matrix(69,28,170,69) (-7/17,-2/5) -> (2/5,7/17) Hyperbolic Matrix(169,66,-612,-239) (-2/5,-7/18) -> (-5/18,-8/29) Hyperbolic Matrix(4691,1822,5950,2311) (-7/18,-33/85) -> (67/85,41/52) Hyperbolic Matrix(6699,2600,8500,3299) (-33/85,-26/67) -> (26/33,67/85) Hyperbolic Matrix(715,274,1156,443) (-5/13,-13/34) -> (21/34,13/21) Hyperbolic Matrix(713,272,1156,441) (-13/34,-8/21) -> (8/13,21/34) Hyperbolic Matrix(237,88,272,101) (-3/8,-10/27) -> (6/7,7/8) Hyperbolic Matrix(273,100,374,137) (-7/19,-4/11) -> (8/11,11/15) Hyperbolic Matrix(169,60,476,169) (-5/14,-6/17) -> (6/17,5/14) Hyperbolic Matrix(35,12,102,35) (-6/17,-1/3) -> (1/3,6/17) Hyperbolic Matrix(273,86,-1292,-407) (-6/19,-5/16) -> (-11/52,-4/19) Hyperbolic Matrix(203,62,442,135) (-4/13,-7/23) -> (5/11,6/13) Hyperbolic Matrix(613,186,-2686,-815) (-7/23,-10/33) -> (-8/35,-13/57) Hyperbolic Matrix(33,10,-340,-103) (-10/33,-3/10) -> (-1/10,0/1) Hyperbolic Matrix(101,30,340,101) (-3/10,-5/17) -> (5/17,3/10) Hyperbolic Matrix(69,20,238,69) (-5/17,-2/7) -> (2/7,5/17) Hyperbolic Matrix(135,38,238,67) (-2/7,-5/18) -> (9/16,4/7) Hyperbolic Matrix(237,64,374,101) (-3/11,-4/15) -> (12/19,7/11) Hyperbolic Matrix(851,226,1156,307) (-4/15,-9/34) -> (25/34,14/19) Hyperbolic Matrix(849,224,1156,305) (-9/34,-5/19) -> (11/15,25/34) Hyperbolic Matrix(33,8,136,33) (-1/4,-4/17) -> (4/17,1/4) Hyperbolic Matrix(103,24,442,103) (-4/17,-3/13) -> (3/13,4/17) Hyperbolic Matrix(14281,3256,18496,4217) (-13/57,-31/136) -> (105/136,61/79) Hyperbolic Matrix(14279,3254,18496,4215) (-31/136,-18/79) -> (44/57,105/136) Hyperbolic Matrix(8059,1818,10404,2347) (-7/31,-23/102) -> (79/102,55/71) Hyperbolic Matrix(8057,1816,10404,2345) (-23/102,-16/71) -> (24/31,79/102) Hyperbolic Matrix(169,38,1054,237) (-9/40,-2/9) -> (4/25,1/6) Hyperbolic Matrix(169,36,-1122,-239) (-3/14,-7/33) -> (-5/33,-3/20) Hyperbolic Matrix(5201,1102,8500,1801) (-7/33,-18/85) -> (52/85,41/67) Hyperbolic Matrix(3639,770,5950,1259) (-18/85,-11/52) -> (11/18,52/85) Hyperbolic Matrix(1087,228,1292,271) (-4/19,-9/43) -> (21/25,16/19) Hyperbolic Matrix(441,92,1462,305) (-9/43,-5/24) -> (3/10,13/43) Hyperbolic Matrix(919,190,1156,239) (-6/29,-7/34) -> (27/34,4/5) Hyperbolic Matrix(917,188,1156,237) (-7/34,-1/5) -> (23/29,27/34) Hyperbolic Matrix(67,12,374,67) (-2/11,-3/17) -> (3/17,2/11) Hyperbolic Matrix(35,6,204,35) (-3/17,-1/6) -> (1/6,3/17) Hyperbolic Matrix(883,142,2108,339) (-1/6,-4/25) -> (18/43,31/74) Hyperbolic Matrix(1021,162,1292,205) (-4/25,-3/19) -> (15/19,34/43) Hyperbolic Matrix(205,32,442,69) (-3/19,-2/13) -> (6/13,7/15) Hyperbolic Matrix(987,146,1156,171) (-4/27,-5/34) -> (29/34,6/7) Hyperbolic Matrix(985,144,1156,169) (-5/34,-1/7) -> (23/27,29/34) Hyperbolic Matrix(171,22,272,35) (-1/7,-1/8) -> (5/8,17/27) Hyperbolic Matrix(33,4,272,33) (-1/8,-2/17) -> (2/17,1/8) Hyperbolic Matrix(35,4,306,35) (-2/17,-1/9) -> (1/9,2/17) Hyperbolic Matrix(135,14,646,67) (-1/9,-1/10) -> (5/24,9/43) Hyperbolic Matrix(103,-10,340,-33) (0/1,1/9) -> (13/43,10/33) Hyperbolic Matrix(579,-86,680,-101) (1/7,3/20) -> (17/20,23/27) Hyperbolic Matrix(681,-104,884,-135) (3/20,2/13) -> (10/13,37/48) Hyperbolic Matrix(271,-50,374,-69) (2/11,1/5) -> (21/29,8/11) Hyperbolic Matrix(647,-134,816,-169) (1/5,5/24) -> (19/24,23/29) Hyperbolic Matrix(1769,-374,2890,-611) (4/19,7/33) -> (41/67,30/49) Hyperbolic Matrix(1599,-340,2074,-441) (7/33,3/14) -> (37/48,27/35) Hyperbolic Matrix(305,-66,476,-103) (3/14,2/9) -> (16/25,9/14) Hyperbolic Matrix(441,-100,1054,-239) (2/9,5/22) -> (5/12,18/43) Hyperbolic Matrix(815,-186,2686,-613) (5/22,8/35) -> (10/33,17/56) Hyperbolic Matrix(749,-172,884,-203) (8/35,3/13) -> (11/13,28/33) Hyperbolic Matrix(101,-26,136,-35) (1/4,4/15) -> (14/19,3/4) Hyperbolic Matrix(305,-84,374,-103) (3/11,5/18) -> (13/16,9/11) Hyperbolic Matrix(3807,-1156,4930,-1497) (17/56,7/23) -> (61/79,17/22) Hyperbolic Matrix(271,-84,442,-137) (4/13,1/3) -> (19/31,8/13) Hyperbolic Matrix(373,-134,476,-171) (5/14,4/11) -> (18/23,11/14) Hyperbolic Matrix(373,-138,646,-239) (7/19,3/8) -> (15/26,11/19) Hyperbolic Matrix(169,-64,272,-103) (3/8,5/13) -> (13/21,5/8) Hyperbolic Matrix(305,-118,442,-171) (5/13,7/18) -> (11/16,9/13) Hyperbolic Matrix(137,-54,170,-67) (7/18,2/5) -> (4/5,13/16) Hyperbolic Matrix(5031,-2108,6494,-2721) (31/74,13/31) -> (55/71,31/40) Hyperbolic Matrix(1291,-542,2108,-885) (13/31,8/19) -> (30/49,19/31) Hyperbolic Matrix(407,-172,646,-273) (8/19,3/7) -> (17/27,12/19) Hyperbolic Matrix(305,-134,544,-239) (7/16,4/9) -> (14/25,9/16) Hyperbolic Matrix(239,-108,374,-169) (4/9,5/11) -> (7/11,16/25) Hyperbolic Matrix(239,-132,306,-169) (6/11,5/9) -> (7/9,18/23) Hyperbolic Matrix(441,-256,646,-375) (11/19,18/31) -> (2/3,13/19) Hyperbolic Matrix(815,-474,1054,-613) (18/31,7/12) -> (17/22,24/31) Hyperbolic Matrix(443,-270,612,-373) (3/5,11/18) -> (13/18,21/29) Hyperbolic Matrix(1019,-698,1292,-885) (13/19,11/16) -> (41/52,15/19) Hyperbolic Matrix(2073,-1444,2686,-1871) (16/23,23/33) -> (27/35,44/57) Hyperbolic Matrix(307,-214,340,-237) (23/33,7/10) -> (9/10,1/1) Hyperbolic Matrix(953,-750,1122,-883) (11/14,26/33) -> (28/33,17/20) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(237,214,-340,-307) -> Matrix(1,0,0,1) Matrix(271,240,306,271) -> Matrix(5,-4,-6,5) Matrix(239,210,272,239) -> Matrix(3,-4,-2,3) Matrix(137,118,238,205) -> Matrix(1,-2,0,1) Matrix(101,86,-680,-579) -> Matrix(1,0,0,1) Matrix(203,172,-884,-749) -> Matrix(1,0,2,1) Matrix(237,200,442,373) -> Matrix(1,0,-2,1) Matrix(271,228,1292,1087) -> Matrix(1,-2,0,1) Matrix(817,686,1054,885) -> Matrix(1,0,0,1) Matrix(169,140,204,169) -> Matrix(1,0,-2,1) Matrix(307,252,374,307) -> Matrix(3,-4,-2,3) Matrix(103,84,-374,-305) -> Matrix(1,-2,0,1) Matrix(169,134,-816,-647) -> Matrix(1,4,0,1) Matrix(579,458,646,511) -> Matrix(1,2,-2,-3) Matrix(205,162,1292,1021) -> Matrix(1,0,0,1) Matrix(1121,884,-2890,-2279) -> Matrix(1,0,0,1) Matrix(475,374,-2074,-1633) -> Matrix(1,0,2,1) Matrix(171,134,-476,-373) -> Matrix(1,4,0,1) Matrix(613,474,-1054,-815) -> Matrix(1,2,0,1) Matrix(1871,1444,-2686,-2073) -> Matrix(1,0,2,1) Matrix(135,104,-884,-681) -> Matrix(1,0,2,1) Matrix(339,260,442,339) -> Matrix(1,0,4,1) Matrix(103,78,136,103) -> Matrix(1,0,-2,1) Matrix(35,26,-136,-101) -> Matrix(1,0,-2,1) Matrix(137,100,374,273) -> Matrix(1,-2,0,1) Matrix(69,50,-374,-271) -> Matrix(1,0,-2,1) Matrix(103,74,238,171) -> Matrix(1,-2,0,1) Matrix(169,120,238,169) -> Matrix(1,0,0,1) Matrix(239,168,340,239) -> Matrix(1,0,-4,1) Matrix(1157,808,1462,1021) -> Matrix(3,-2,2,-1) Matrix(1123,782,-4930,-3433) -> Matrix(1,0,2,1) Matrix(239,166,442,307) -> Matrix(1,0,-2,1) Matrix(171,118,-442,-305) -> Matrix(1,0,-2,1) Matrix(67,44,102,67) -> Matrix(1,0,-2,1) Matrix(307,198,476,307) -> Matrix(1,-2,0,1) Matrix(103,66,-476,-305) -> Matrix(1,0,0,1) Matrix(101,64,374,237) -> Matrix(1,0,0,1) Matrix(273,172,-646,-407) -> Matrix(1,2,0,1) Matrix(103,64,-272,-169) -> Matrix(1,0,0,1) Matrix(137,84,-442,-271) -> Matrix(1,0,-2,1) Matrix(33,20,-170,-103) -> Matrix(1,0,-2,1) Matrix(101,60,170,101) -> Matrix(3,-2,-4,3) Matrix(239,140,408,239) -> Matrix(5,-6,-4,5) Matrix(1769,1028,2108,1225) -> Matrix(1,-2,0,1) Matrix(1463,850,-6494,-3773) -> Matrix(1,-2,2,-3) Matrix(817,474,-2108,-1223) -> Matrix(1,-2,0,1) Matrix(239,138,-646,-373) -> Matrix(1,0,0,1) Matrix(67,38,238,135) -> Matrix(1,0,0,1) Matrix(239,134,-544,-305) -> Matrix(3,-2,2,-1) Matrix(135,74,-374,-205) -> Matrix(1,2,0,1) Matrix(307,166,442,239) -> Matrix(1,0,-2,1) Matrix(373,200,442,237) -> Matrix(1,0,-2,1) Matrix(271,144,510,271) -> Matrix(3,-2,-4,3) Matrix(35,18,68,35) -> Matrix(1,-2,0,1) Matrix(33,16,68,33) -> Matrix(1,-2,0,1) Matrix(239,112,510,239) -> Matrix(1,2,0,1) Matrix(69,32,442,205) -> Matrix(1,0,0,1) Matrix(135,62,442,203) -> Matrix(1,0,0,1) Matrix(67,30,-306,-137) -> Matrix(1,0,0,1) Matrix(647,286,1156,511) -> Matrix(3,-2,-4,3) Matrix(645,284,1156,509) -> Matrix(5,-6,-4,5) Matrix(171,74,238,103) -> Matrix(1,-2,0,1) Matrix(33,14,238,101) -> Matrix(1,-2,0,1) Matrix(205,86,-646,-271) -> Matrix(1,-4,0,1) Matrix(239,100,-1054,-441) -> Matrix(1,-4,2,-7) Matrix(169,70,408,169) -> Matrix(1,-10,0,1) Matrix(69,28,170,69) -> Matrix(1,2,0,1) Matrix(169,66,-612,-239) -> Matrix(1,0,0,1) Matrix(4691,1822,5950,2311) -> Matrix(1,0,2,1) Matrix(6699,2600,8500,3299) -> Matrix(1,0,0,1) Matrix(715,274,1156,443) -> Matrix(1,0,2,1) Matrix(713,272,1156,441) -> Matrix(1,0,-4,1) Matrix(237,88,272,101) -> Matrix(1,-2,0,1) Matrix(273,100,374,137) -> Matrix(1,-2,0,1) Matrix(169,60,476,169) -> Matrix(1,-10,0,1) Matrix(35,12,102,35) -> Matrix(1,2,0,1) Matrix(273,86,-1292,-407) -> Matrix(1,2,0,1) Matrix(203,62,442,135) -> Matrix(1,0,0,1) Matrix(613,186,-2686,-815) -> Matrix(1,0,2,1) Matrix(33,10,-340,-103) -> Matrix(1,0,0,1) Matrix(101,30,340,101) -> Matrix(1,0,0,1) Matrix(69,20,238,69) -> Matrix(1,2,0,1) Matrix(135,38,238,67) -> Matrix(1,0,0,1) Matrix(237,64,374,101) -> Matrix(1,0,0,1) Matrix(851,226,1156,307) -> Matrix(1,2,-2,-3) Matrix(849,224,1156,305) -> Matrix(3,2,-2,-1) Matrix(33,8,136,33) -> Matrix(1,0,2,1) Matrix(103,24,442,103) -> Matrix(1,0,-6,1) Matrix(14281,3256,18496,4217) -> Matrix(1,0,2,1) Matrix(14279,3254,18496,4215) -> Matrix(1,0,-4,1) Matrix(8059,1818,10404,2347) -> Matrix(3,-2,2,-1) Matrix(8057,1816,10404,2345) -> Matrix(1,0,0,1) Matrix(169,38,1054,237) -> Matrix(1,0,-2,1) Matrix(169,36,-1122,-239) -> Matrix(1,0,0,1) Matrix(5201,1102,8500,1801) -> Matrix(3,-2,-4,3) Matrix(3639,770,5950,1259) -> Matrix(1,-2,0,1) Matrix(1087,228,1292,271) -> Matrix(1,-2,0,1) Matrix(441,92,1462,305) -> Matrix(1,-2,0,1) Matrix(919,190,1156,239) -> Matrix(1,-6,0,1) Matrix(917,188,1156,237) -> Matrix(1,6,0,1) Matrix(67,12,374,67) -> Matrix(1,0,4,1) Matrix(35,6,204,35) -> Matrix(1,0,-2,1) Matrix(883,142,2108,339) -> Matrix(1,-4,0,1) Matrix(1021,162,1292,205) -> Matrix(1,0,0,1) Matrix(205,32,442,69) -> Matrix(1,0,0,1) Matrix(987,146,1156,171) -> Matrix(1,-2,0,1) Matrix(985,144,1156,169) -> Matrix(1,0,0,1) Matrix(171,22,272,35) -> Matrix(1,0,0,1) Matrix(33,4,272,33) -> Matrix(1,0,2,1) Matrix(35,4,306,35) -> Matrix(1,0,-2,1) Matrix(135,14,646,67) -> Matrix(1,-2,0,1) Matrix(103,-10,340,-33) -> Matrix(1,0,0,1) Matrix(579,-86,680,-101) -> Matrix(1,0,0,1) Matrix(681,-104,884,-135) -> Matrix(1,0,2,1) Matrix(271,-50,374,-69) -> Matrix(1,0,-2,1) Matrix(647,-134,816,-169) -> Matrix(1,4,0,1) Matrix(1769,-374,2890,-611) -> Matrix(1,2,-2,-3) Matrix(1599,-340,2074,-441) -> Matrix(1,0,2,1) Matrix(305,-66,476,-103) -> Matrix(1,0,0,1) Matrix(441,-100,1054,-239) -> Matrix(7,4,-2,-1) Matrix(815,-186,2686,-613) -> Matrix(1,0,2,1) Matrix(749,-172,884,-203) -> Matrix(1,0,2,1) Matrix(101,-26,136,-35) -> Matrix(1,0,-2,1) Matrix(305,-84,374,-103) -> Matrix(1,-2,0,1) Matrix(3807,-1156,4930,-1497) -> Matrix(1,0,2,1) Matrix(271,-84,442,-137) -> Matrix(1,0,-2,1) Matrix(373,-134,476,-171) -> Matrix(1,4,0,1) Matrix(373,-138,646,-239) -> Matrix(1,0,0,1) Matrix(169,-64,272,-103) -> Matrix(1,0,0,1) Matrix(305,-118,442,-171) -> Matrix(1,0,-2,1) Matrix(137,-54,170,-67) -> Matrix(1,-2,0,1) Matrix(5031,-2108,6494,-2721) -> Matrix(1,4,0,1) Matrix(1291,-542,2108,-885) -> Matrix(1,4,-2,-7) Matrix(407,-172,646,-273) -> Matrix(1,2,0,1) Matrix(305,-134,544,-239) -> Matrix(1,2,-2,-3) Matrix(239,-108,374,-169) -> Matrix(1,0,0,1) Matrix(239,-132,306,-169) -> Matrix(1,2,0,1) Matrix(441,-256,646,-375) -> Matrix(1,2,-2,-3) Matrix(815,-474,1054,-613) -> Matrix(1,2,0,1) Matrix(443,-270,612,-373) -> Matrix(1,0,0,1) Matrix(1019,-698,1292,-885) -> Matrix(1,0,2,1) Matrix(2073,-1444,2686,-1871) -> Matrix(1,0,2,1) Matrix(307,-214,340,-237) -> Matrix(1,0,0,1) Matrix(953,-750,1122,-883) -> Matrix(1,0,0,1) 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): 22 Degree of the the map X: 22 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): 432 Minimal number of generators: 73 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: 32 Genus: 21 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -15/34 -9/34 0/1 2/17 1/8 1/7 2/13 1/6 3/17 2/11 1/5 2/9 4/17 1/4 4/15 2/7 5/17 1/3 6/17 2/5 7/17 8/17 1/2 10/17 52/85 11/17 2/3 12/17 13/17 14/17 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -5/6 0/1 1/1 -4/5 1/1 1/0 -3/4 0/1 1/1 -11/15 1/1 1/0 -8/11 0/1 1/0 -13/18 0/1 1/1 -5/7 0/1 1/0 -2/3 0/1 1/1 -11/17 1/1 -9/14 1/1 1/0 -7/11 0/1 1/1 -12/19 1/1 1/0 -5/8 0/1 1/1 -8/13 0/1 1/2 -11/18 0/1 1/1 -3/5 1/2 1/1 -4/7 1/1 1/0 -9/16 0/1 1/1 -5/9 1/1 2/1 -1/2 1/1 1/0 -6/13 0/1 1/0 -5/11 0/1 1/1 -4/9 0/1 1/1 -15/34 1/1 -11/25 1/1 4/3 -7/16 1/1 2/1 -3/7 2/1 1/0 -11/26 2/1 3/1 -8/19 3/1 1/0 -5/12 5/1 1/0 -7/17 1/0 -2/5 -1/1 1/0 -3/8 0/1 1/1 -4/11 2/1 1/0 -1/3 -1/1 1/0 -5/16 -1/1 0/1 -4/13 0/1 1/0 -3/10 0/1 1/0 -5/17 1/0 -2/7 -1/1 1/0 -3/11 -1/1 0/1 -4/15 -1/1 1/0 -9/34 -1/1 -5/19 -1/1 -1/2 -1/4 -1/1 0/1 -2/9 0/1 1/1 -5/23 0/1 1/1 -3/14 1/1 1/0 -1/5 -1/1 1/0 -3/16 -1/1 0/1 -2/11 -1/2 0/1 -3/17 0/1 -1/6 0/1 1/1 -1/7 0/1 1/0 -1/8 -1/1 0/1 0/1 0/1 1/0 1/9 -1/1 0/1 2/17 0/1 1/8 0/1 1/1 1/7 0/1 1/0 2/13 0/1 1/0 1/6 -1/1 0/1 3/17 0/1 2/11 0/1 1/2 1/5 1/1 1/0 4/19 -1/1 1/0 7/33 -1/1 0/1 3/14 -1/1 1/0 2/9 -1/1 0/1 3/13 -1/3 0/1 4/17 0/1 1/4 0/1 1/1 4/15 1/1 1/0 3/11 0/1 1/1 5/18 0/1 1/1 2/7 1/1 1/0 5/17 1/0 3/10 0/1 1/0 7/23 0/1 1/1 4/13 0/1 1/0 1/3 1/1 1/0 6/17 1/0 5/14 -5/1 1/0 4/11 -2/1 1/0 7/19 -1/1 1/0 3/8 -1/1 0/1 5/13 0/1 1/1 2/5 1/1 1/0 7/17 1/0 5/12 -5/1 1/0 13/31 -3/1 1/0 8/19 -3/1 1/0 3/7 -2/1 1/0 7/16 -2/1 -1/1 4/9 -1/1 0/1 5/11 -1/1 0/1 6/13 0/1 1/0 7/15 1/1 1/0 8/17 1/0 1/2 -1/1 1/0 6/11 0/1 1/0 5/9 -2/1 -1/1 19/34 -1/1 14/25 -1/1 -2/3 9/16 -1/1 0/1 4/7 -1/1 1/0 7/12 -3/2 -1/1 10/17 -1/1 3/5 -1/1 -1/2 11/18 -1/1 0/1 52/85 -1/1 41/67 -1/1 -2/3 30/49 -1/1 -1/2 19/31 -1/1 -1/2 8/13 -1/2 0/1 21/34 0/1 13/21 0/1 1/1 5/8 -1/1 0/1 17/27 0/1 1/0 12/19 -1/1 1/0 7/11 -1/1 0/1 16/25 -1/1 0/1 9/14 -1/1 1/0 11/17 -1/1 2/3 -1/1 0/1 7/10 -1/2 0/1 12/17 0/1 5/7 0/1 1/0 13/18 -1/1 0/1 21/29 -1/1 -1/2 8/11 0/1 1/0 11/15 -1/1 1/0 25/34 -1/1 14/19 -1/1 -1/2 3/4 -1/1 0/1 13/17 0/1 10/13 0/1 1/2 7/9 0/1 1/1 4/5 -1/1 1/0 9/11 -2/1 -1/1 14/17 -1/1 5/6 -1/1 0/1 6/7 -1/1 1/0 1/1 -1/1 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(33,28,119,101) (-1/1,-5/6) -> (3/11,5/18) Hyperbolic Matrix(69,56,85,69) (-5/6,-4/5) -> (4/5,9/11) Hyperbolic Matrix(33,26,85,67) (-4/5,-3/4) -> (5/13,2/5) Hyperbolic Matrix(35,26,-136,-101) (-3/4,-11/15) -> (-5/19,-1/4) Hyperbolic Matrix(137,100,374,273) (-11/15,-8/11) -> (4/11,7/19) Hyperbolic Matrix(69,50,-374,-271) (-8/11,-13/18) -> (-3/16,-2/11) Hyperbolic Matrix(103,74,238,171) (-13/18,-5/7) -> (3/7,7/16) Hyperbolic Matrix(35,24,51,35) (-5/7,-2/3) -> (2/3,7/10) Hyperbolic Matrix(67,44,102,67) (-2/3,-11/17) -> (11/17,2/3) Hyperbolic Matrix(307,198,476,307) (-11/17,-9/14) -> (9/14,11/17) Hyperbolic Matrix(103,66,-476,-305) (-9/14,-7/11) -> (-5/23,-3/14) Hyperbolic Matrix(101,64,374,237) (-7/11,-12/19) -> (4/15,3/11) Hyperbolic Matrix(273,172,-646,-407) (-12/19,-5/8) -> (-11/26,-8/19) Hyperbolic Matrix(171,106,221,137) (-5/8,-8/13) -> (10/13,7/9) Hyperbolic Matrix(137,84,-442,-271) (-8/13,-11/18) -> (-5/16,-4/13) Hyperbolic Matrix(33,20,-170,-103) (-11/18,-3/5) -> (-1/5,-3/16) Hyperbolic Matrix(69,40,119,69) (-3/5,-4/7) -> (4/7,7/12) Hyperbolic Matrix(67,38,238,135) (-4/7,-9/16) -> (5/18,2/7) Hyperbolic Matrix(239,134,-544,-305) (-9/16,-5/9) -> (-11/25,-7/16) Hyperbolic Matrix(69,38,187,103) (-5/9,-1/2) -> (7/19,3/8) Hyperbolic Matrix(103,48,221,103) (-1/2,-6/13) -> (6/13,7/15) Hyperbolic Matrix(135,62,442,203) (-6/13,-5/11) -> (7/23,4/13) Hyperbolic Matrix(67,30,-306,-137) (-5/11,-4/9) -> (-2/9,-5/23) Hyperbolic Matrix(647,286,1156,511) (-4/9,-15/34) -> (19/34,14/25) Hyperbolic Matrix(645,284,1156,509) (-15/34,-11/25) -> (5/9,19/34) Hyperbolic Matrix(171,74,238,103) (-7/16,-3/7) -> (5/7,13/18) Hyperbolic Matrix(33,14,238,101) (-3/7,-11/26) -> (1/8,1/7) Hyperbolic Matrix(67,28,323,135) (-8/19,-5/12) -> (1/5,4/19) Hyperbolic Matrix(169,70,408,169) (-5/12,-7/17) -> (7/17,5/12) Hyperbolic Matrix(69,28,170,69) (-7/17,-2/5) -> (2/5,7/17) Hyperbolic Matrix(67,26,85,33) (-2/5,-3/8) -> (7/9,4/5) Hyperbolic Matrix(103,38,187,69) (-3/8,-4/11) -> (6/11,5/9) Hyperbolic Matrix(67,24,187,67) (-4/11,-1/3) -> (5/14,4/11) Hyperbolic Matrix(69,22,323,103) (-1/3,-5/16) -> (7/33,3/14) Hyperbolic Matrix(33,10,221,67) (-4/13,-3/10) -> (1/7,2/13) Hyperbolic Matrix(101,30,340,101) (-3/10,-5/17) -> (5/17,3/10) Hyperbolic Matrix(69,20,238,69) (-5/17,-2/7) -> (2/7,5/17) Hyperbolic Matrix(101,28,119,33) (-2/7,-3/11) -> (5/6,6/7) Hyperbolic Matrix(237,64,374,101) (-3/11,-4/15) -> (12/19,7/11) Hyperbolic Matrix(851,226,1156,307) (-4/15,-9/34) -> (25/34,14/19) Hyperbolic Matrix(849,224,1156,305) (-9/34,-5/19) -> (11/15,25/34) Hyperbolic Matrix(35,8,153,35) (-1/4,-2/9) -> (2/9,3/13) Hyperbolic Matrix(135,28,323,67) (-3/14,-1/5) -> (5/12,13/31) Hyperbolic Matrix(67,12,374,67) (-2/11,-3/17) -> (3/17,2/11) Hyperbolic Matrix(35,6,204,35) (-3/17,-1/6) -> (1/6,3/17) Hyperbolic Matrix(67,10,221,33) (-1/6,-1/7) -> (3/10,7/23) Hyperbolic Matrix(171,22,272,35) (-1/7,-1/8) -> (5/8,17/27) Hyperbolic Matrix(1,0,17,1) (-1/8,0/1) -> (0/1,1/9) Parabolic Matrix(35,-4,289,-33) (1/9,2/17) -> (2/17,1/8) Parabolic Matrix(101,-16,221,-35) (2/13,1/6) -> (5/11,6/13) Hyperbolic Matrix(271,-50,374,-69) (2/11,1/5) -> (21/29,8/11) Hyperbolic Matrix(1769,-374,2890,-611) (4/19,7/33) -> (41/67,30/49) Hyperbolic Matrix(305,-66,476,-103) (3/14,2/9) -> (16/25,9/14) Hyperbolic Matrix(69,-16,289,-67) (3/13,4/17) -> (4/17,1/4) Parabolic Matrix(101,-26,136,-35) (1/4,4/15) -> (14/19,3/4) Hyperbolic Matrix(271,-84,442,-137) (4/13,1/3) -> (19/31,8/13) Hyperbolic Matrix(103,-36,289,-101) (1/3,6/17) -> (6/17,5/14) Parabolic Matrix(169,-64,272,-103) (3/8,5/13) -> (13/21,5/8) Hyperbolic Matrix(1291,-542,2108,-885) (13/31,8/19) -> (30/49,19/31) Hyperbolic Matrix(407,-172,646,-273) (8/19,3/7) -> (17/27,12/19) Hyperbolic Matrix(305,-134,544,-239) (7/16,4/9) -> (14/25,9/16) Hyperbolic Matrix(239,-108,374,-169) (4/9,5/11) -> (7/11,16/25) Hyperbolic Matrix(137,-64,289,-135) (7/15,8/17) -> (8/17,1/2) Parabolic Matrix(137,-74,187,-101) (1/2,6/11) -> (8/11,11/15) Hyperbolic Matrix(103,-58,119,-67) (9/16,4/7) -> (6/7,1/1) Hyperbolic Matrix(171,-100,289,-169) (7/12,10/17) -> (10/17,3/5) Parabolic Matrix(443,-270,612,-373) (3/5,11/18) -> (13/18,21/29) Hyperbolic Matrix(4421,-2704,7225,-4419) (11/18,52/85) -> (52/85,41/67) Parabolic Matrix(509,-314,663,-409) (8/13,21/34) -> (13/17,10/13) Hyperbolic Matrix(375,-232,493,-305) (21/34,13/21) -> (3/4,13/17) Hyperbolic Matrix(205,-144,289,-203) (7/10,12/17) -> (12/17,5/7) Parabolic Matrix(239,-196,289,-237) (9/11,14/17) -> (14/17,5/6) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(33,28,119,101) -> Matrix(1,0,0,1) Matrix(69,56,85,69) -> Matrix(1,-2,0,1) Matrix(33,26,85,67) -> Matrix(1,0,0,1) Matrix(35,26,-136,-101) -> Matrix(1,0,-2,1) Matrix(137,100,374,273) -> Matrix(1,-2,0,1) Matrix(69,50,-374,-271) -> Matrix(1,0,-2,1) Matrix(103,74,238,171) -> Matrix(1,-2,0,1) Matrix(35,24,51,35) -> Matrix(1,0,-2,1) Matrix(67,44,102,67) -> Matrix(1,0,-2,1) Matrix(307,198,476,307) -> Matrix(1,-2,0,1) Matrix(103,66,-476,-305) -> Matrix(1,0,0,1) Matrix(101,64,374,237) -> Matrix(1,0,0,1) Matrix(273,172,-646,-407) -> Matrix(1,2,0,1) Matrix(171,106,221,137) -> Matrix(1,0,0,1) Matrix(137,84,-442,-271) -> Matrix(1,0,-2,1) Matrix(33,20,-170,-103) -> Matrix(1,0,-2,1) Matrix(69,40,119,69) -> Matrix(1,-2,0,1) Matrix(67,38,238,135) -> Matrix(1,0,0,1) Matrix(239,134,-544,-305) -> Matrix(3,-2,2,-1) Matrix(69,38,187,103) -> Matrix(1,-2,0,1) Matrix(103,48,221,103) -> Matrix(1,0,0,1) Matrix(135,62,442,203) -> Matrix(1,0,0,1) Matrix(67,30,-306,-137) -> Matrix(1,0,0,1) Matrix(647,286,1156,511) -> Matrix(3,-2,-4,3) Matrix(645,284,1156,509) -> Matrix(5,-6,-4,5) Matrix(171,74,238,103) -> Matrix(1,-2,0,1) Matrix(33,14,238,101) -> Matrix(1,-2,0,1) Matrix(67,28,323,135) -> Matrix(1,-4,0,1) Matrix(169,70,408,169) -> Matrix(1,-10,0,1) Matrix(69,28,170,69) -> Matrix(1,2,0,1) Matrix(67,26,85,33) -> Matrix(1,0,0,1) Matrix(103,38,187,69) -> Matrix(1,-2,0,1) Matrix(67,24,187,67) -> Matrix(1,-4,0,1) Matrix(69,22,323,103) -> Matrix(1,0,0,1) Matrix(33,10,221,67) -> Matrix(1,0,0,1) Matrix(101,30,340,101) -> Matrix(1,0,0,1) Matrix(69,20,238,69) -> Matrix(1,2,0,1) Matrix(101,28,119,33) -> Matrix(1,0,0,1) Matrix(237,64,374,101) -> Matrix(1,0,0,1) Matrix(851,226,1156,307) -> Matrix(1,2,-2,-3) Matrix(849,224,1156,305) -> Matrix(3,2,-2,-1) Matrix(35,8,153,35) -> Matrix(1,0,-2,1) Matrix(135,28,323,67) -> Matrix(1,-4,0,1) Matrix(67,12,374,67) -> Matrix(1,0,4,1) Matrix(35,6,204,35) -> Matrix(1,0,-2,1) Matrix(67,10,221,33) -> Matrix(1,0,0,1) Matrix(171,22,272,35) -> Matrix(1,0,0,1) Matrix(1,0,17,1) -> Matrix(1,0,0,1) Matrix(35,-4,289,-33) -> Matrix(1,0,2,1) Matrix(101,-16,221,-35) -> Matrix(1,0,0,1) Matrix(271,-50,374,-69) -> Matrix(1,0,-2,1) Matrix(1769,-374,2890,-611) -> Matrix(1,2,-2,-3) Matrix(305,-66,476,-103) -> Matrix(1,0,0,1) Matrix(69,-16,289,-67) -> Matrix(1,0,4,1) Matrix(101,-26,136,-35) -> Matrix(1,0,-2,1) Matrix(271,-84,442,-137) -> Matrix(1,0,-2,1) Matrix(103,-36,289,-101) -> Matrix(1,-6,0,1) Matrix(169,-64,272,-103) -> Matrix(1,0,0,1) Matrix(1291,-542,2108,-885) -> Matrix(1,4,-2,-7) Matrix(407,-172,646,-273) -> Matrix(1,2,0,1) Matrix(305,-134,544,-239) -> Matrix(1,2,-2,-3) Matrix(239,-108,374,-169) -> Matrix(1,0,0,1) Matrix(137,-64,289,-135) -> Matrix(1,-2,0,1) Matrix(137,-74,187,-101) -> Matrix(1,0,0,1) Matrix(103,-58,119,-67) -> Matrix(1,0,0,1) Matrix(171,-100,289,-169) -> Matrix(3,4,-4,-5) Matrix(443,-270,612,-373) -> Matrix(1,0,0,1) Matrix(4421,-2704,7225,-4419) -> Matrix(1,2,-2,-3) Matrix(509,-314,663,-409) -> Matrix(1,0,4,1) Matrix(375,-232,493,-305) -> Matrix(1,0,-2,1) Matrix(205,-144,289,-203) -> Matrix(1,0,2,1) Matrix(239,-196,289,-237) -> Matrix(1,2,-2,-3) 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 0/1 (0/1,1/0) 0 17 2/17 0/1 2 1 1/8 (0/1,1/1) 0 17 1/7 (0/1,1/0) 0 17 1/6 (-1/1,0/1) 0 17 3/17 0/1 3 1 2/11 (0/1,1/2) 0 17 1/5 (1/1,1/0) 0 17 3/14 (-1/1,1/0) 0 17 2/9 (-1/1,0/1) 0 17 4/17 0/1 4 1 1/4 (0/1,1/1) 0 17 4/15 (1/1,1/0) 0 17 3/11 (0/1,1/1) 0 17 2/7 (1/1,1/0) 0 17 5/17 1/0 1 1 3/10 (0/1,1/0) 0 17 7/23 (0/1,1/1) 0 17 4/13 (0/1,1/0) 0 17 1/3 (1/1,1/0) 0 17 6/17 1/0 6 1 4/11 (-2/1,1/0) 0 17 3/8 (-1/1,0/1) 0 17 2/5 (1/1,1/0) 0 17 7/17 1/0 6 1 5/12 (-5/1,1/0) 0 17 13/31 (-3/1,1/0) 0 17 8/19 (-3/1,1/0) 0 17 3/7 (-2/1,1/0) 0 17 7/16 (-2/1,-1/1) 0 17 4/9 (-1/1,0/1) 0 17 5/11 (-1/1,0/1) 0 17 6/13 (0/1,1/0) 0 17 8/17 1/0 2 1 1/2 (-1/1,1/0) 0 17 6/11 (0/1,1/0) 0 17 5/9 (-2/1,-1/1) 0 17 19/34 -1/1 4 1 14/25 (-1/1,-2/3) 0 17 9/16 (-1/1,0/1) 0 17 4/7 (-1/1,1/0) 0 17 10/17 -1/1 4 1 3/5 (-1/1,-1/2) 0 17 11/18 (-1/1,0/1) 0 17 52/85 -1/1 2 1 30/49 (-1/1,-1/2) 0 17 19/31 (-1/1,-1/2) 0 17 8/13 (-1/2,0/1) 0 17 5/8 (-1/1,0/1) 0 17 17/27 (0/1,1/0) 0 17 12/19 (-1/1,1/0) 0 17 7/11 (-1/1,0/1) 0 17 16/25 (-1/1,0/1) 0 17 9/14 (-1/1,1/0) 0 17 11/17 -1/1 1 1 2/3 (-1/1,0/1) 0 17 12/17 0/1 2 1 5/7 (0/1,1/0) 0 17 13/18 (-1/1,0/1) 0 17 21/29 (-1/1,-1/2) 0 17 8/11 (0/1,1/0) 0 17 11/15 (-1/1,1/0) 0 17 25/34 -1/1 2 1 14/19 (-1/1,-1/2) 0 17 3/4 (-1/1,0/1) 0 17 13/17 0/1 3 1 10/13 (0/1,1/2) 0 17 7/9 (0/1,1/1) 0 17 4/5 (-1/1,1/0) 0 17 14/17 -1/1 2 1 5/6 (-1/1,0/1) 0 17 6/7 (-1/1,1/0) 0 17 1/1 (-1/1,0/1) 0 17 1/0 0/1 1 1 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(1,0,17,-1) (0/1,2/17) -> (0/1,2/17) Reflection Matrix(33,-4,272,-33) (2/17,1/8) -> (2/17,1/8) Reflection Matrix(171,-22,272,-35) (1/8,1/7) -> (5/8,17/27) Glide Reflection Matrix(67,-10,221,-33) (1/7,1/6) -> (3/10,7/23) Glide Reflection Matrix(35,-6,204,-35) (1/6,3/17) -> (1/6,3/17) Reflection Matrix(67,-12,374,-67) (3/17,2/11) -> (3/17,2/11) Reflection Matrix(271,-50,374,-69) (2/11,1/5) -> (21/29,8/11) Hyperbolic Matrix(135,-28,323,-67) (1/5,3/14) -> (5/12,13/31) Glide Reflection Matrix(305,-66,476,-103) (3/14,2/9) -> (16/25,9/14) Hyperbolic Matrix(35,-8,153,-35) (2/9,4/17) -> (2/9,4/17) Reflection Matrix(33,-8,136,-33) (4/17,1/4) -> (4/17,1/4) Reflection Matrix(101,-26,136,-35) (1/4,4/15) -> (14/19,3/4) Hyperbolic Matrix(237,-64,374,-101) (4/15,3/11) -> (12/19,7/11) Glide Reflection Matrix(101,-28,119,-33) (3/11,2/7) -> (5/6,6/7) Glide Reflection Matrix(69,-20,238,-69) (2/7,5/17) -> (2/7,5/17) Reflection Matrix(101,-30,340,-101) (5/17,3/10) -> (5/17,3/10) Reflection Matrix(203,-62,442,-135) (7/23,4/13) -> (5/11,6/13) Glide Reflection Matrix(271,-84,442,-137) (4/13,1/3) -> (19/31,8/13) Hyperbolic Matrix(35,-12,102,-35) (1/3,6/17) -> (1/3,6/17) Reflection Matrix(67,-24,187,-67) (6/17,4/11) -> (6/17,4/11) Reflection Matrix(103,-38,187,-69) (4/11,3/8) -> (6/11,5/9) Glide Reflection Matrix(67,-26,85,-33) (3/8,2/5) -> (7/9,4/5) Glide Reflection Matrix(69,-28,170,-69) (2/5,7/17) -> (2/5,7/17) Reflection Matrix(169,-70,408,-169) (7/17,5/12) -> (7/17,5/12) Reflection Matrix(1291,-542,2108,-885) (13/31,8/19) -> (30/49,19/31) Hyperbolic Matrix(407,-172,646,-273) (8/19,3/7) -> (17/27,12/19) Hyperbolic Matrix(171,-74,238,-103) (3/7,7/16) -> (5/7,13/18) Glide Reflection Matrix(305,-134,544,-239) (7/16,4/9) -> (14/25,9/16) Hyperbolic Matrix(239,-108,374,-169) (4/9,5/11) -> (7/11,16/25) Hyperbolic Matrix(103,-48,221,-103) (6/13,8/17) -> (6/13,8/17) Reflection Matrix(33,-16,68,-33) (8/17,1/2) -> (8/17,1/2) Reflection Matrix(137,-74,187,-101) (1/2,6/11) -> (8/11,11/15) Hyperbolic Matrix(341,-190,612,-341) (5/9,19/34) -> (5/9,19/34) Reflection Matrix(951,-532,1700,-951) (19/34,14/25) -> (19/34,14/25) Reflection Matrix(103,-58,119,-67) (9/16,4/7) -> (6/7,1/1) Hyperbolic Matrix(69,-40,119,-69) (4/7,10/17) -> (4/7,10/17) Reflection Matrix(101,-60,170,-101) (10/17,3/5) -> (10/17,3/5) Reflection Matrix(443,-270,612,-373) (3/5,11/18) -> (13/18,21/29) Hyperbolic Matrix(1871,-1144,3060,-1871) (11/18,52/85) -> (11/18,52/85) Reflection Matrix(2549,-1560,4165,-2549) (52/85,30/49) -> (52/85,30/49) Reflection Matrix(171,-106,221,-137) (8/13,5/8) -> (10/13,7/9) Glide Reflection Matrix(307,-198,476,-307) (9/14,11/17) -> (9/14,11/17) Reflection Matrix(67,-44,102,-67) (11/17,2/3) -> (11/17,2/3) Reflection Matrix(35,-24,51,-35) (2/3,12/17) -> (2/3,12/17) Reflection Matrix(169,-120,238,-169) (12/17,5/7) -> (12/17,5/7) Reflection Matrix(749,-550,1020,-749) (11/15,25/34) -> (11/15,25/34) Reflection Matrix(951,-700,1292,-951) (25/34,14/19) -> (25/34,14/19) Reflection Matrix(103,-78,136,-103) (3/4,13/17) -> (3/4,13/17) Reflection Matrix(339,-260,442,-339) (13/17,10/13) -> (13/17,10/13) Reflection Matrix(69,-56,85,-69) (4/5,14/17) -> (4/5,14/17) Reflection Matrix(169,-140,204,-169) (14/17,5/6) -> (14/17,5/6) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(1,0,17,-1) -> Matrix(1,0,0,-1) (0/1,2/17) -> (0/1,1/0) Matrix(33,-4,272,-33) -> Matrix(1,0,2,-1) (2/17,1/8) -> (0/1,1/1) Matrix(171,-22,272,-35) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(67,-10,221,-33) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(35,-6,204,-35) -> Matrix(-1,0,2,1) (1/6,3/17) -> (-1/1,0/1) Matrix(67,-12,374,-67) -> Matrix(1,0,4,-1) (3/17,2/11) -> (0/1,1/2) Matrix(271,-50,374,-69) -> Matrix(1,0,-2,1) 0/1 Matrix(135,-28,323,-67) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(305,-66,476,-103) -> Matrix(1,0,0,1) Matrix(35,-8,153,-35) -> Matrix(-1,0,2,1) (2/9,4/17) -> (-1/1,0/1) Matrix(33,-8,136,-33) -> Matrix(1,0,2,-1) (4/17,1/4) -> (0/1,1/1) Matrix(101,-26,136,-35) -> Matrix(1,0,-2,1) 0/1 Matrix(237,-64,374,-101) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(101,-28,119,-33) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(69,-20,238,-69) -> Matrix(-1,2,0,1) (2/7,5/17) -> (1/1,1/0) Matrix(101,-30,340,-101) -> Matrix(1,0,0,-1) (5/17,3/10) -> (0/1,1/0) Matrix(203,-62,442,-135) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(271,-84,442,-137) -> Matrix(1,0,-2,1) 0/1 Matrix(35,-12,102,-35) -> Matrix(-1,2,0,1) (1/3,6/17) -> (1/1,1/0) Matrix(67,-24,187,-67) -> Matrix(1,4,0,-1) (6/17,4/11) -> (-2/1,1/0) Matrix(103,-38,187,-69) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(67,-26,85,-33) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(69,-28,170,-69) -> Matrix(-1,2,0,1) (2/5,7/17) -> (1/1,1/0) Matrix(169,-70,408,-169) -> Matrix(1,10,0,-1) (7/17,5/12) -> (-5/1,1/0) Matrix(1291,-542,2108,-885) -> Matrix(1,4,-2,-7) Matrix(407,-172,646,-273) -> Matrix(1,2,0,1) 1/0 Matrix(171,-74,238,-103) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(305,-134,544,-239) -> Matrix(1,2,-2,-3) -1/1 Matrix(239,-108,374,-169) -> Matrix(1,0,0,1) Matrix(103,-48,221,-103) -> Matrix(1,0,0,-1) (6/13,8/17) -> (0/1,1/0) Matrix(33,-16,68,-33) -> Matrix(1,2,0,-1) (8/17,1/2) -> (-1/1,1/0) Matrix(137,-74,187,-101) -> Matrix(1,0,0,1) Matrix(341,-190,612,-341) -> Matrix(3,4,-2,-3) (5/9,19/34) -> (-2/1,-1/1) Matrix(951,-532,1700,-951) -> Matrix(5,4,-6,-5) (19/34,14/25) -> (-1/1,-2/3) Matrix(103,-58,119,-67) -> Matrix(1,0,0,1) Matrix(69,-40,119,-69) -> Matrix(1,2,0,-1) (4/7,10/17) -> (-1/1,1/0) Matrix(101,-60,170,-101) -> Matrix(3,2,-4,-3) (10/17,3/5) -> (-1/1,-1/2) Matrix(443,-270,612,-373) -> Matrix(1,0,0,1) Matrix(1871,-1144,3060,-1871) -> Matrix(-1,0,2,1) (11/18,52/85) -> (-1/1,0/1) Matrix(2549,-1560,4165,-2549) -> Matrix(3,2,-4,-3) (52/85,30/49) -> (-1/1,-1/2) Matrix(171,-106,221,-137) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(307,-198,476,-307) -> Matrix(1,2,0,-1) (9/14,11/17) -> (-1/1,1/0) Matrix(67,-44,102,-67) -> Matrix(-1,0,2,1) (11/17,2/3) -> (-1/1,0/1) Matrix(35,-24,51,-35) -> Matrix(-1,0,2,1) (2/3,12/17) -> (-1/1,0/1) Matrix(169,-120,238,-169) -> Matrix(1,0,0,-1) (12/17,5/7) -> (0/1,1/0) Matrix(749,-550,1020,-749) -> Matrix(1,2,0,-1) (11/15,25/34) -> (-1/1,1/0) Matrix(951,-700,1292,-951) -> Matrix(3,2,-4,-3) (25/34,14/19) -> (-1/1,-1/2) Matrix(103,-78,136,-103) -> Matrix(-1,0,2,1) (3/4,13/17) -> (-1/1,0/1) Matrix(339,-260,442,-339) -> Matrix(1,0,4,-1) (13/17,10/13) -> (0/1,1/2) Matrix(69,-56,85,-69) -> Matrix(1,2,0,-1) (4/5,14/17) -> (-1/1,1/0) Matrix(169,-140,204,-169) -> Matrix(-1,0,2,1) (14/17,5/6) -> (-1/1,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.