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 5/66 -17/2 1/13 -8/1 4/51 -7/1 3/38 -20/3 8/99 -13/2 3/37 -19/3 1/12 -25/4 3/35 -6/1 2/25 -17/3 1/12 -11/2 3/35 -5/1 1/12 -24/5 4/45 -43/9 1/12 -19/4 1/11 -33/7 5/54 -14/3 2/23 -9/2 1/13 -22/5 2/21 -35/8 3/37 -13/3 1/12 -17/4 1/11 -4/1 0/1 -15/4 1/11 -11/3 1/12 -18/5 2/19 -7/2 1/11 -17/5 1/11 -10/3 2/21 -43/13 1/12 -33/10 7/75 -56/17 12/127 -23/7 5/52 -13/4 1/9 -3/1 1/10 -17/6 1/10 -14/5 6/59 -11/4 1/9 -19/7 1/10 -8/3 4/39 -13/5 7/66 -18/7 10/93 -5/2 1/9 -17/7 1/9 -12/5 8/71 -43/18 11/97 -74/31 14/123 -31/13 3/26 -19/8 1/9 -7/3 3/26 -16/7 12/103 -9/4 7/59 -11/5 7/58 -13/6 7/57 -15/7 1/8 -17/8 1/8 -2/1 2/15 -17/9 1/7 -15/8 1/7 -13/7 7/48 -11/6 7/47 -9/5 7/46 -34/19 2/13 -25/14 19/123 -16/9 12/77 -7/4 3/19 -26/15 2/13 -19/11 1/6 -31/18 3/19 -12/7 8/49 -17/10 1/6 -5/3 1/6 -18/11 10/57 -85/52 3/17 -67/41 29/164 -49/30 3/17 -31/19 5/28 -13/8 7/39 -34/21 2/11 -21/13 11/60 -8/5 4/21 -27/17 3/16 -19/12 1/5 -11/7 1/6 -25/16 9/47 -14/9 6/31 -17/11 1/5 -3/2 1/5 -19/13 1/4 -16/11 4/15 -13/9 1/6 -23/16 5/23 -33/23 7/30 -10/7 2/9 -17/12 1/4 -7/5 1/4 -18/13 2/11 -29/21 1/4 -11/8 1/3 -15/11 1/4 -34/25 2/7 -19/14 1/3 -4/3 0/1 -17/13 1/4 -13/10 1/3 -48/37 4/13 -35/27 3/8 -57/44 -1/1 -136/105 0/1 -79/61 1/10 -22/17 2/9 -31/24 3/11 -102/79 2/7 -71/55 7/24 -40/31 4/13 -9/7 1/2 -23/18 3/13 -14/11 2/7 -33/26 5/21 -85/67 1/4 -52/41 8/31 -19/15 1/4 -43/34 1/3 -24/19 4/15 -29/23 5/18 -34/27 2/7 -5/4 1/3 -16/13 4/15 -11/9 3/10 -17/14 1/3 -6/5 2/5 -25/21 3/10 -19/16 1/3 -13/11 3/8 -33/28 13/35 -20/17 8/21 -27/23 11/28 -34/29 2/5 -7/6 3/7 -8/7 4/9 -17/15 1/2 -9/8 5/9 -10/9 2/3 -1/1 1/0 0/1 0/1 1/1 1/30 9/8 5/141 17/15 1/28 8/7 4/111 7/6 3/83 20/17 8/219 13/11 3/82 19/16 1/27 25/21 3/80 6/5 2/55 17/14 1/27 11/9 3/80 5/4 1/27 24/19 4/105 43/34 1/27 19/15 1/26 33/26 5/129 14/11 2/53 9/7 1/28 22/17 2/51 35/27 3/82 13/10 1/27 17/13 1/26 4/3 0/1 15/11 1/26 11/8 1/27 18/13 2/49 7/5 1/26 17/12 1/26 10/7 2/51 43/30 1/27 33/23 7/180 56/39 12/307 23/16 5/127 13/9 1/24 3/2 1/25 17/11 1/25 14/9 6/149 11/7 1/24 19/12 1/25 8/5 4/99 13/8 7/171 18/11 10/243 5/3 1/24 17/10 1/24 12/7 8/191 43/25 11/262 74/43 14/333 31/18 3/71 19/11 1/24 7/4 3/71 16/9 12/283 9/5 7/164 11/6 7/163 13/7 7/162 15/8 1/23 17/9 1/23 2/1 2/45 17/8 1/22 15/7 1/22 13/6 7/153 11/5 7/152 9/4 7/151 34/15 2/43 25/11 19/408 16/7 12/257 7/3 3/64 26/11 2/43 19/8 1/21 31/13 3/64 12/5 8/169 17/7 1/21 5/2 1/21 18/7 10/207 85/33 3/62 67/26 29/599 49/19 3/62 31/12 5/103 13/5 7/144 34/13 2/41 21/8 11/225 8/3 4/81 27/10 3/61 19/7 1/20 11/4 1/21 25/9 9/182 14/5 6/121 17/6 1/20 3/1 1/20 19/6 1/19 16/5 4/75 13/4 1/21 23/7 5/98 33/10 7/135 10/3 2/39 17/5 1/19 7/2 1/19 18/5 2/41 29/8 1/19 11/3 1/18 15/4 1/19 34/9 2/37 19/5 1/18 4/1 0/1 17/4 1/19 13/3 1/18 48/11 4/73 35/8 3/53 57/13 1/14 136/31 0/1 79/18 1/25 22/5 2/39 31/7 3/56 102/23 2/37 71/16 7/129 40/9 4/73 9/2 1/17 23/5 3/58 14/3 2/37 33/7 5/96 85/18 1/19 52/11 8/151 19/4 1/19 43/9 1/18 24/5 4/75 29/6 5/93 34/7 2/37 5/1 1/18 16/3 4/75 11/2 3/55 17/3 1/18 6/1 2/35 25/4 3/55 19/3 1/18 13/2 3/53 33/5 13/230 20/3 8/141 27/4 11/193 34/5 2/35 7/1 3/52 8/1 4/69 17/2 1/17 9/1 5/84 10/1 2/33 1/0 1/15 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(53,-4,570,-43) Matrix(35,306,4,35) -> Matrix(131,-10,2214,-169) Matrix(33,272,4,33) -> Matrix(103,-8,1764,-137) Matrix(33,238,14,101) -> Matrix(77,-6,1630,-127) Matrix(101,680,-86,-579) -> Matrix(199,-16,510,-41) Matrix(135,884,-104,-681) -> Matrix(49,-4,184,-15) Matrix(69,442,32,205) -> Matrix(121,-10,2650,-219) Matrix(205,1292,162,1021) -> Matrix(47,-4,1234,-105) Matrix(169,1054,38,237) -> Matrix(23,-2,426,-37) Matrix(35,204,6,35) -> Matrix(49,-4,870,-71) Matrix(67,374,12,67) -> Matrix(71,-6,1290,-109) Matrix(69,374,-50,-271) -> Matrix(23,-2,104,-9) Matrix(169,816,-134,-647) -> Matrix(91,-8,330,-29) Matrix(135,646,14,67) -> Matrix(67,-6,1128,-101) Matrix(271,1292,228,1087) -> Matrix(45,-4,1204,-107) Matrix(611,2890,-374,-1769) -> Matrix(157,-14,886,-79) Matrix(441,2074,-340,-1599) -> Matrix(21,-2,74,-7) Matrix(103,476,-66,-305) -> Matrix(43,-4,226,-21) Matrix(239,1054,-100,-441) -> Matrix(67,-6,592,-53) Matrix(613,2686,-186,-815) -> Matrix(27,-2,284,-21) Matrix(203,884,-172,-749) -> Matrix(45,-4,124,-11) Matrix(103,442,24,103) -> Matrix(23,-2,426,-37) Matrix(33,136,8,33) -> Matrix(1,0,8,1) Matrix(35,136,-26,-101) -> Matrix(1,0,-8,1) Matrix(101,374,64,237) -> Matrix(23,-2,564,-49) Matrix(103,374,-84,-305) -> Matrix(21,-2,74,-7) Matrix(67,238,38,135) -> Matrix(25,-2,588,-47) Matrix(69,238,20,69) -> Matrix(23,-2,426,-37) Matrix(101,340,30,101) -> Matrix(43,-4,828,-77) Matrix(441,1462,92,305) -> Matrix(23,-2,426,-37) Matrix(1497,4930,-1156,-3807) -> Matrix(21,-2,158,-15) Matrix(135,442,62,203) -> Matrix(61,-6,1332,-131) Matrix(137,442,-84,-271) -> Matrix(25,-2,138,-11) Matrix(35,102,12,35) -> Matrix(21,-2,410,-39) Matrix(169,476,60,169) -> Matrix(119,-12,2390,-241) Matrix(171,476,-134,-373) -> Matrix(39,-4,166,-17) Matrix(137,374,100,273) -> Matrix(19,-2,504,-53) Matrix(239,646,-138,-373) -> Matrix(59,-6,364,-37) Matrix(103,272,-64,-169) -> Matrix(77,-8,414,-43) Matrix(171,442,-118,-305) -> Matrix(19,-2,48,-5) Matrix(67,170,-54,-137) -> Matrix(19,-2,48,-5) Matrix(69,170,28,69) -> Matrix(55,-6,1146,-125) Matrix(169,408,70,169) -> Matrix(143,-16,3012,-337) Matrix(883,2108,142,339) -> Matrix(35,-4,674,-77) Matrix(2721,6494,-2108,-5031) -> Matrix(193,-22,658,-75) Matrix(885,2108,-542,-1291) -> Matrix(33,-4,190,-23) Matrix(273,646,-172,-407) -> Matrix(53,-6,274,-31) Matrix(103,238,74,171) -> Matrix(17,-2,468,-55) Matrix(239,544,-134,-305) -> Matrix(205,-24,1324,-155) Matrix(169,374,-108,-239) -> Matrix(83,-10,440,-53) Matrix(203,442,62,135) -> Matrix(49,-6,972,-119) Matrix(205,442,32,69) -> Matrix(81,-10,1450,-179) Matrix(239,510,112,239) -> Matrix(209,-26,4590,-571) Matrix(33,68,16,33) -> Matrix(31,-4,690,-89) Matrix(35,68,18,35) -> Matrix(29,-4,660,-91) Matrix(271,510,144,271) -> Matrix(181,-26,4170,-599) Matrix(237,442,200,373) -> Matrix(69,-10,1870,-271) Matrix(239,442,166,307) -> Matrix(41,-6,1032,-151) Matrix(169,306,-132,-239) -> Matrix(13,-2,72,-11) Matrix(645,1156,284,509) -> Matrix(339,-52,7282,-1117) Matrix(647,1156,286,511) -> Matrix(337,-52,7252,-1119) Matrix(135,238,38,67) -> Matrix(13,-2,228,-35) Matrix(137,238,118,205) -> Matrix(37,-6,1030,-167) Matrix(375,646,-256,-441) -> Matrix(13,-2,46,-7) Matrix(613,1054,-474,-815) -> Matrix(37,-6,142,-23) Matrix(239,408,140,239) -> Matrix(97,-16,2322,-383) Matrix(101,170,60,101) -> Matrix(35,-6,846,-145) Matrix(373,612,-270,-443) -> Matrix(23,-4,98,-17) Matrix(3639,5950,770,1259) -> Matrix(193,-34,3650,-643) Matrix(5201,8500,1102,1801) -> Matrix(249,-44,4748,-839) Matrix(713,1156,272,441) -> Matrix(199,-36,4074,-737) Matrix(715,1156,274,443) -> Matrix(197,-36,4044,-739) Matrix(171,272,22,35) -> Matrix(1,0,12,1) Matrix(237,374,64,101) -> Matrix(11,-2,204,-37) Matrix(307,476,198,307) -> Matrix(61,-12,1520,-299) Matrix(67,102,44,67) -> Matrix(9,-2,230,-51) Matrix(885,1292,-698,-1019) -> Matrix(17,-4,64,-15) Matrix(307,442,166,239) -> Matrix(29,-6,672,-139) Matrix(1871,2686,-1444,-2073) -> Matrix(9,-2,14,-3) Matrix(237,340,-214,-307) -> Matrix(17,-4,30,-7) Matrix(239,340,168,239) -> Matrix(17,-4,438,-103) Matrix(169,238,120,169) -> Matrix(7,-2,186,-53) Matrix(171,238,74,103) -> Matrix(5,-2,108,-43) Matrix(273,374,100,137) -> Matrix(7,-2,144,-41) Matrix(849,1156,224,305) -> Matrix(15,-4,274,-73) Matrix(851,1156,226,307) -> Matrix(13,-4,244,-75) Matrix(103,136,78,103) -> Matrix(1,0,22,1) Matrix(339,442,260,339) -> Matrix(7,-2,186,-53) Matrix(14279,18496,3254,4215) -> Matrix(1,0,26,1) Matrix(14281,18496,3256,4217) -> Matrix(1,0,4,1) Matrix(8057,10404,1816,2345) -> Matrix(71,-20,1310,-369) Matrix(8059,10404,1818,2347) -> Matrix(69,-20,1280,-371) Matrix(817,1054,686,885) -> Matrix(7,-2,186,-53) Matrix(883,1122,-750,-953) -> Matrix(11,-2,28,-5) Matrix(6699,8500,2600,3299) -> Matrix(179,-44,3698,-909) Matrix(4691,5950,1822,2311) -> Matrix(133,-34,2750,-703) Matrix(1021,1292,162,205) -> Matrix(15,-4,274,-73) Matrix(1157,1462,808,1021) -> Matrix(7,-2,186,-53) Matrix(917,1156,188,237) -> Matrix(43,-12,792,-221) Matrix(919,1156,190,239) -> Matrix(41,-12,762,-223) Matrix(307,374,252,307) -> Matrix(19,-6,510,-161) Matrix(169,204,140,169) -> Matrix(11,-4,300,-109) Matrix(1769,2108,1028,1225) -> Matrix(17,-4,404,-95) Matrix(1087,1292,228,271) -> Matrix(13,-4,244,-75) Matrix(373,442,200,237) -> Matrix(29,-10,670,-231) Matrix(985,1156,144,169) -> Matrix(71,-28,1240,-489) Matrix(987,1156,146,171) -> Matrix(69,-28,1210,-491) Matrix(237,272,88,101) -> Matrix(1,0,18,1) Matrix(239,272,210,239) -> Matrix(17,-8,474,-223) Matrix(271,306,240,271) -> Matrix(19,-10,534,-281) Matrix(579,646,458,511) -> Matrix(11,-6,288,-157) Matrix(1,0,2,1) -> Matrix(1,0,30,1) Matrix(307,-340,214,-237) -> Matrix(113,-4,2910,-103) Matrix(579,-680,86,-101) -> Matrix(439,-16,7710,-281) Matrix(749,-884,172,-203) -> Matrix(109,-4,2044,-75) Matrix(305,-374,84,-103) -> Matrix(53,-2,1034,-39) Matrix(647,-816,134,-169) -> Matrix(211,-8,3930,-149) Matrix(2279,-2890,884,-1121) -> Matrix(367,-14,7576,-289) Matrix(1633,-2074,374,-475) -> Matrix(51,-2,944,-37) Matrix(373,-476,134,-171) -> Matrix(103,-4,2086,-81) Matrix(815,-1054,474,-613) -> Matrix(157,-6,3742,-143) Matrix(2073,-2686,1444,-1871) -> Matrix(57,-2,1454,-51) Matrix(681,-884,104,-135) -> Matrix(105,-4,1864,-71) Matrix(101,-136,26,-35) -> Matrix(1,0,-8,1) Matrix(271,-374,50,-69) -> Matrix(51,-2,944,-37) Matrix(3433,-4930,782,-1123) -> Matrix(51,-2,1148,-45) Matrix(305,-442,118,-171) -> Matrix(55,-2,1128,-41) Matrix(305,-476,66,-103) -> Matrix(99,-4,1906,-77) Matrix(407,-646,172,-273) -> Matrix(149,-6,3154,-127) Matrix(169,-272,64,-103) -> Matrix(197,-8,4014,-163) Matrix(271,-442,84,-137) -> Matrix(49,-2,858,-35) Matrix(103,-170,20,-33) -> Matrix(49,-2,858,-35) Matrix(3773,-6494,850,-1463) -> Matrix(523,-22,9628,-405) Matrix(1223,-2108,474,-817) -> Matrix(93,-4,1930,-83) Matrix(373,-646,138,-239) -> Matrix(143,-6,2884,-121) Matrix(305,-544,134,-239) -> Matrix(565,-24,12124,-515) Matrix(205,-374,74,-135) -> Matrix(233,-10,4730,-203) Matrix(137,-306,30,-67) -> Matrix(43,-2,882,-41) Matrix(271,-646,86,-205) -> Matrix(43,-2,796,-37) Matrix(441,-1054,100,-239) -> Matrix(127,-6,2392,-113) Matrix(239,-612,66,-169) -> Matrix(83,-4,1598,-77) Matrix(407,-1292,86,-273) -> Matrix(77,-4,1444,-75) Matrix(815,-2686,186,-613) -> Matrix(39,-2,644,-33) Matrix(103,-340,10,-33) -> Matrix(77,-4,1290,-67) Matrix(239,-1122,36,-169) -> Matrix(41,-2,718,-35) 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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 0/1 2/1 17/8 34/15 12/5 17/7 34/13 17/6 3/1 17/5 34/9 4/1 17/4 136/31 102/23 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 0/1 0/1 1/1 1/30 9/8 5/141 17/15 1/28 8/7 4/111 7/6 3/83 20/17 8/219 13/11 3/82 19/16 1/27 25/21 3/80 6/5 2/55 17/14 1/27 11/9 3/80 5/4 1/27 24/19 4/105 43/34 1/27 19/15 1/26 33/26 5/129 14/11 2/53 9/7 1/28 22/17 2/51 35/27 3/82 13/10 1/27 17/13 1/26 4/3 0/1 15/11 1/26 11/8 1/27 18/13 2/49 7/5 1/26 17/12 1/26 10/7 2/51 43/30 1/27 33/23 7/180 56/39 12/307 23/16 5/127 13/9 1/24 3/2 1/25 17/11 1/25 14/9 6/149 11/7 1/24 19/12 1/25 8/5 4/99 13/8 7/171 18/11 10/243 5/3 1/24 17/10 1/24 12/7 8/191 43/25 11/262 74/43 14/333 31/18 3/71 19/11 1/24 7/4 3/71 16/9 12/283 9/5 7/164 11/6 7/163 13/7 7/162 15/8 1/23 17/9 1/23 2/1 2/45 17/8 1/22 15/7 1/22 13/6 7/153 11/5 7/152 9/4 7/151 34/15 2/43 25/11 19/408 16/7 12/257 7/3 3/64 26/11 2/43 19/8 1/21 31/13 3/64 12/5 8/169 17/7 1/21 5/2 1/21 18/7 10/207 85/33 3/62 67/26 29/599 49/19 3/62 31/12 5/103 13/5 7/144 34/13 2/41 21/8 11/225 8/3 4/81 27/10 3/61 19/7 1/20 11/4 1/21 25/9 9/182 14/5 6/121 17/6 1/20 3/1 1/20 19/6 1/19 16/5 4/75 13/4 1/21 23/7 5/98 33/10 7/135 10/3 2/39 17/5 1/19 7/2 1/19 18/5 2/41 29/8 1/19 11/3 1/18 15/4 1/19 34/9 2/37 19/5 1/18 4/1 0/1 17/4 1/19 13/3 1/18 48/11 4/73 35/8 3/53 57/13 1/14 136/31 0/1 79/18 1/25 22/5 2/39 31/7 3/56 102/23 2/37 71/16 7/129 40/9 4/73 9/2 1/17 23/5 3/58 14/3 2/37 33/7 5/96 85/18 1/19 52/11 8/151 19/4 1/19 43/9 1/18 24/5 4/75 29/6 5/93 34/7 2/37 5/1 1/18 16/3 4/75 11/2 3/55 17/3 1/18 6/1 2/35 25/4 3/55 19/3 1/18 13/2 3/53 33/5 13/230 20/3 8/141 27/4 11/193 34/5 2/35 7/1 3/52 8/1 4/69 17/2 1/17 9/1 5/84 10/1 2/33 1/0 1/15 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(307,-340,214,-237) (1/1,9/8) -> (43/30,33/23) Hyperbolic Matrix(271,-306,31,-35) (9/8,17/15) -> (17/2,9/1) Hyperbolic Matrix(239,-272,29,-33) (17/15,8/7) -> (8/1,17/2) Hyperbolic Matrix(205,-238,87,-101) (8/7,7/6) -> (7/3,26/11) 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(373,-442,173,-205) (13/11,19/16) -> (15/7,13/6) Hyperbolic Matrix(1087,-1292,859,-1021) (19/16,25/21) -> (43/34,19/15) Hyperbolic Matrix(885,-1054,199,-237) (25/21,6/5) -> (40/9,9/2) Hyperbolic Matrix(169,-204,29,-35) (6/5,17/14) -> (17/3,6/1) Hyperbolic Matrix(307,-374,55,-67) (17/14,11/9) -> (11/2,17/3) 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(511,-646,53,-67) (24/19,43/34) -> (9/1,10/1) 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(339,-442,79,-103) (13/10,17/13) -> (17/4,13/3) Hyperbolic Matrix(103,-136,25,-33) (17/13,4/3) -> (4/1,17/4) Hyperbolic Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(273,-374,173,-237) (15/11,11/8) -> (11/7,19/12) Hyperbolic Matrix(271,-374,50,-69) (11/8,18/13) -> (16/3,11/2) Hyperbolic Matrix(171,-238,97,-135) (18/13,7/5) -> (7/4,16/9) Hyperbolic Matrix(169,-238,49,-69) (7/5,17/12) -> (17/5,7/2) Hyperbolic Matrix(239,-340,71,-101) (17/12,10/7) -> (10/3,17/5) Hyperbolic Matrix(1021,-1462,213,-305) (10/7,43/30) -> (43/9,24/5) Hyperbolic Matrix(3433,-4930,782,-1123) (56/39,23/16) -> (79/18,22/5) Hyperbolic Matrix(307,-442,141,-203) (23/16,13/9) -> (13/6,11/5) Hyperbolic Matrix(305,-442,118,-171) (13/9,3/2) -> (31/12,13/5) Hyperbolic Matrix(67,-102,23,-35) (3/2,17/11) -> (17/6,3/1) Hyperbolic Matrix(307,-476,109,-169) (17/11,14/9) -> (14/5,17/6) 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(101,-170,41,-69) (5/3,17/10) -> (17/7,5/2) Hyperbolic Matrix(239,-408,99,-169) (17/10,12/7) -> (12/5,17/7) Hyperbolic Matrix(1225,-2108,197,-339) (43/25,74/43) -> (6/1,25/4) 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(239,-442,73,-135) (11/6,13/7) -> (13/4,23/7) Hyperbolic Matrix(237,-442,37,-69) (13/7,15/8) -> (19/3,13/2) Hyperbolic Matrix(271,-510,127,-239) (15/8,17/9) -> (17/8,15/7) Hyperbolic Matrix(35,-68,17,-33) (17/9,2/1) -> (2/1,17/8) Parabolic Matrix(137,-306,30,-67) (11/5,9/4) -> (9/2,23/5) Hyperbolic Matrix(511,-1156,225,-509) (9/4,34/15) -> (34/15,25/11) Parabolic Matrix(103,-238,29,-67) (16/7,7/3) -> (7/2,18/5) 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(2311,-5950,489,-1259) (18/7,85/33) -> (85/18,52/11) Hyperbolic Matrix(3299,-8500,699,-1801) (85/33,67/26) -> (33/7,85/18) Hyperbolic Matrix(443,-1156,169,-441) (13/5,34/13) -> (34/13,21/8) Parabolic Matrix(101,-272,13,-35) (8/3,27/10) -> (7/1,8/1) Hyperbolic Matrix(137,-374,37,-101) (19/7,11/4) -> (11/3,15/4) 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(307,-1156,81,-305) (15/4,34/9) -> (34/9,19/5) Parabolic Matrix(4217,-18496,961,-4215) (57/13,136/31) -> (136/31,79/18) Parabolic Matrix(2347,-10404,529,-2345) (31/7,102/23) -> (102/23,71/16) Parabolic Matrix(239,-1122,36,-169) (14/3,33/7) -> (33/5,20/3) Hyperbolic Matrix(271,-1292,43,-205) (19/4,43/9) -> (25/4,19/3) Hyperbolic Matrix(239,-1156,49,-237) (29/6,34/7) -> (34/7,5/1) Parabolic Matrix(171,-1156,25,-169) (27/4,34/5) -> (34/5,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,15,1) Matrix(307,-340,214,-237) -> Matrix(113,-4,2910,-103) Matrix(271,-306,31,-35) -> Matrix(281,-10,4749,-169) Matrix(239,-272,29,-33) -> Matrix(223,-8,3819,-137) Matrix(205,-238,87,-101) -> Matrix(167,-6,3535,-127) Matrix(579,-680,86,-101) -> Matrix(439,-16,7710,-281) Matrix(749,-884,172,-203) -> Matrix(109,-4,2044,-75) Matrix(373,-442,173,-205) -> Matrix(271,-10,5935,-219) Matrix(1087,-1292,859,-1021) -> Matrix(107,-4,2809,-105) Matrix(885,-1054,199,-237) -> Matrix(53,-2,981,-37) Matrix(169,-204,29,-35) -> Matrix(109,-4,1935,-71) Matrix(307,-374,55,-67) -> Matrix(161,-6,2925,-109) Matrix(305,-374,84,-103) -> Matrix(53,-2,1034,-39) Matrix(647,-816,134,-169) -> Matrix(211,-8,3930,-149) Matrix(511,-646,53,-67) -> Matrix(157,-6,2643,-101) Matrix(2279,-2890,884,-1121) -> Matrix(367,-14,7576,-289) Matrix(1633,-2074,374,-475) -> Matrix(51,-2,944,-37) Matrix(373,-476,134,-171) -> Matrix(103,-4,2086,-81) Matrix(815,-1054,474,-613) -> Matrix(157,-6,3742,-143) Matrix(2073,-2686,1444,-1871) -> Matrix(57,-2,1454,-51) Matrix(681,-884,104,-135) -> Matrix(105,-4,1864,-71) Matrix(339,-442,79,-103) -> Matrix(53,-2,981,-37) Matrix(103,-136,25,-33) -> Matrix(1,0,-7,1) Matrix(101,-136,26,-35) -> Matrix(1,0,-8,1) Matrix(273,-374,173,-237) -> Matrix(53,-2,1299,-49) Matrix(271,-374,50,-69) -> Matrix(51,-2,944,-37) Matrix(171,-238,97,-135) -> Matrix(55,-2,1293,-47) Matrix(169,-238,49,-69) -> Matrix(53,-2,981,-37) Matrix(239,-340,71,-101) -> Matrix(103,-4,1983,-77) Matrix(1021,-1462,213,-305) -> Matrix(53,-2,981,-37) Matrix(3433,-4930,782,-1123) -> Matrix(51,-2,1148,-45) Matrix(307,-442,141,-203) -> Matrix(151,-6,3297,-131) Matrix(305,-442,118,-171) -> Matrix(55,-2,1128,-41) Matrix(67,-102,23,-35) -> Matrix(51,-2,995,-39) Matrix(307,-476,109,-169) -> Matrix(299,-12,6005,-241) Matrix(305,-476,66,-103) -> Matrix(99,-4,1906,-77) Matrix(407,-646,172,-273) -> Matrix(149,-6,3154,-127) Matrix(169,-272,64,-103) -> Matrix(197,-8,4014,-163) Matrix(271,-442,84,-137) -> Matrix(49,-2,858,-35) Matrix(103,-170,20,-33) -> Matrix(49,-2,858,-35) Matrix(101,-170,41,-69) -> Matrix(145,-6,3021,-125) Matrix(239,-408,99,-169) -> Matrix(383,-16,8067,-337) Matrix(1225,-2108,197,-339) -> Matrix(95,-4,1829,-77) Matrix(3773,-6494,850,-1463) -> Matrix(523,-22,9628,-405) Matrix(1223,-2108,474,-817) -> Matrix(93,-4,1930,-83) Matrix(373,-646,138,-239) -> Matrix(143,-6,2884,-121) Matrix(305,-544,134,-239) -> Matrix(565,-24,12124,-515) Matrix(205,-374,74,-135) -> Matrix(233,-10,4730,-203) Matrix(239,-442,73,-135) -> Matrix(139,-6,2757,-119) Matrix(237,-442,37,-69) -> Matrix(231,-10,4135,-179) Matrix(271,-510,127,-239) -> Matrix(599,-26,13155,-571) Matrix(35,-68,17,-33) -> Matrix(91,-4,2025,-89) Matrix(137,-306,30,-67) -> Matrix(43,-2,882,-41) Matrix(511,-1156,225,-509) -> Matrix(1119,-52,24037,-1117) Matrix(103,-238,29,-67) -> Matrix(43,-2,753,-35) Matrix(271,-646,86,-205) -> Matrix(43,-2,796,-37) Matrix(441,-1054,100,-239) -> Matrix(127,-6,2392,-113) Matrix(239,-612,66,-169) -> Matrix(83,-4,1598,-77) Matrix(2311,-5950,489,-1259) -> Matrix(703,-34,13295,-643) Matrix(3299,-8500,699,-1801) -> Matrix(909,-44,17333,-839) Matrix(443,-1156,169,-441) -> Matrix(739,-36,15129,-737) Matrix(101,-272,13,-35) -> Matrix(1,0,-3,1) Matrix(137,-374,37,-101) -> Matrix(41,-2,759,-37) Matrix(407,-1292,86,-273) -> Matrix(77,-4,1444,-75) Matrix(815,-2686,186,-613) -> Matrix(39,-2,644,-33) Matrix(103,-340,10,-33) -> Matrix(77,-4,1290,-67) Matrix(307,-1156,81,-305) -> Matrix(75,-4,1369,-73) Matrix(4217,-18496,961,-4215) -> Matrix(1,0,11,1) Matrix(2347,-10404,529,-2345) -> Matrix(371,-20,6845,-369) Matrix(239,-1122,36,-169) -> Matrix(41,-2,718,-35) Matrix(271,-1292,43,-205) -> Matrix(75,-4,1369,-73) Matrix(239,-1156,49,-237) -> Matrix(223,-12,4107,-221) Matrix(171,-1156,25,-169) -> Matrix(491,-28,8575,-489) 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): 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 2/45 1 17 17/8 1/22 15 1 15/7 1/22 1 17 13/6 7/153 1 17 11/5 7/152 1 17 9/4 7/151 1 17 34/15 2/43 13 1 16/7 12/257 1 17 7/3 3/64 1 17 19/8 1/21 1 17 31/13 3/64 1 17 12/5 8/169 1 17 17/7 1/21 11 1 5/2 1/21 1 17 18/7 10/207 1 17 13/5 7/144 1 17 34/13 2/41 9 1 8/3 4/81 1 17 27/10 3/61 1 17 19/7 1/20 1 17 11/4 1/21 1 17 14/5 6/121 1 17 17/6 1/20 7 1 3/1 1/20 1 17 19/6 1/19 1 17 16/5 4/75 1 17 13/4 1/21 1 17 23/7 5/98 1 17 33/10 7/135 1 17 10/3 2/39 1 17 17/5 1/19 3 1 7/2 1/19 1 17 18/5 2/41 1 17 11/3 1/18 1 17 15/4 1/19 1 17 34/9 2/37 1 1 4/1 0/1 1 17 17/4 1/19 1 1 13/3 1/18 1 17 35/8 3/53 1 17 57/13 1/14 1 17 136/31 0/1 11 1 22/5 2/39 1 17 31/7 3/56 1 17 102/23 2/37 5 1 40/9 4/73 1 17 9/2 1/17 1 17 23/5 3/58 1 17 14/3 2/37 1 17 33/7 5/96 1 17 85/18 1/19 13 1 52/11 8/151 1 17 19/4 1/19 1 17 43/9 1/18 1 17 24/5 4/75 1 17 34/7 2/37 3 1 5/1 1/18 1 17 16/3 4/75 1 17 11/2 3/55 1 17 17/3 1/18 5 1 6/1 2/35 1 17 25/4 3/55 1 17 19/3 1/18 1 17 13/2 3/53 1 17 33/5 13/230 1 17 20/3 8/141 1 17 34/5 2/35 7 1 7/1 3/52 1 17 8/1 4/69 1 17 17/2 1/17 9 1 9/1 5/84 1 17 10/1 2/33 1 17 1/0 1/15 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(33,-68,16,-33) (2/1,17/8) -> (2/1,17/8) Reflection Matrix(239,-510,112,-239) (17/8,15/7) -> (17/8,15/7) Reflection Matrix(205,-442,32,-69) (15/7,13/6) -> (19/3,13/2) Glide Reflection Matrix(203,-442,62,-135) (13/6,11/5) -> (13/4,23/7) Glide Reflection Matrix(137,-306,30,-67) (11/5,9/4) -> (9/2,23/5) 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(103,-238,29,-67) (16/7,7/3) -> (7/2,18/5) Hyperbolic Matrix(273,-646,101,-239) (7/3,19/8) -> (27/10,19/7) Glide Reflection 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(169,-408,70,-169) (12/5,17/7) -> (12/5,17/7) Reflection Matrix(69,-170,28,-69) (17/7,5/2) -> (17/7,5/2) Reflection Matrix(67,-170,13,-33) (5/2,18/7) -> (5/1,16/3) Glide Reflection Matrix(171,-442,53,-137) (18/7,13/5) -> (16/5,13/4) Glide Reflection Matrix(339,-884,130,-339) (13/5,34/13) -> (13/5,34/13) Reflection Matrix(103,-272,39,-103) (34/13,8/3) -> (34/13,8/3) Reflection Matrix(101,-272,13,-35) (8/3,27/10) -> (7/1,8/1) Hyperbolic Matrix(137,-374,37,-101) (19/7,11/4) -> (11/3,15/4) Hyperbolic Matrix(171,-476,37,-103) (11/4,14/5) -> (23/5,14/3) Glide Reflection Matrix(169,-476,60,-169) (14/5,17/6) -> (14/5,17/6) Reflection Matrix(35,-102,12,-35) (17/6,3/1) -> (17/6,3/1) Reflection 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(101,-340,30,-101) (10/3,17/5) -> (10/3,17/5) Reflection Matrix(69,-238,20,-69) (17/5,7/2) -> (17/5,7/2) Reflection Matrix(103,-374,19,-69) (18/5,11/3) -> (16/3,11/2) Glide Reflection 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(103,-442,24,-103) (17/4,13/3) -> (17/4,13/3) Reflection Matrix(203,-884,31,-135) (13/3,35/8) -> (13/2,33/5) Glide Reflection Matrix(3535,-15504,806,-3535) (57/13,136/31) -> (57/13,136/31) Reflection Matrix(681,-2992,155,-681) (136/31,22/5) -> (136/31,22/5) Reflection Matrix(1427,-6324,322,-1427) (31/7,102/23) -> (31/7,102/23) Reflection Matrix(919,-4080,207,-919) (102/23,40/9) -> (102/23,40/9) Reflection Matrix(237,-1054,38,-169) (40/9,9/2) -> (6/1,25/4) Glide Reflection Matrix(239,-1122,36,-169) (14/3,33/7) -> (33/5,20/3) Hyperbolic Matrix(1189,-5610,252,-1189) (33/7,85/18) -> (33/7,85/18) Reflection Matrix(1871,-8840,396,-1871) (85/18,52/11) -> (85/18,52/11) Reflection Matrix(271,-1292,43,-205) (19/4,43/9) -> (25/4,19/3) Hyperbolic Matrix(135,-646,14,-67) (43/9,24/5) -> (9/1,10/1) Glide Reflection Matrix(169,-816,35,-169) (24/5,34/7) -> (24/5,34/7) Reflection Matrix(69,-340,14,-69) (34/7,5/1) -> (34/7,5/1) Reflection Matrix(67,-374,12,-67) (11/2,17/3) -> (11/2,17/3) Reflection Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) Reflection Matrix(101,-680,15,-101) (20/3,34/5) -> (20/3,34/5) Reflection Matrix(69,-476,10,-69) (34/5,7/1) -> (34/5,7/1) Reflection Matrix(33,-272,4,-33) (8/1,17/2) -> (8/1,17/2) Reflection Matrix(35,-306,4,-35) (17/2,9/1) -> (17/2,9/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,30,-1) (0/1,1/0) -> (0/1,1/15) Matrix(1,0,1,-1) -> Matrix(1,0,45,-1) (0/1,2/1) -> (0/1,2/45) Matrix(33,-68,16,-33) -> Matrix(89,-4,1980,-89) (2/1,17/8) -> (2/45,1/22) Matrix(239,-510,112,-239) -> Matrix(571,-26,12540,-571) (17/8,15/7) -> (1/22,13/285) Matrix(205,-442,32,-69) -> Matrix(219,-10,3920,-179) Matrix(203,-442,62,-135) -> Matrix(131,-6,2598,-119) Matrix(137,-306,30,-67) -> Matrix(43,-2,882,-41) 1/21 Matrix(271,-612,120,-271) -> Matrix(603,-28,12986,-603) (9/4,34/15) -> (7/151,2/43) Matrix(239,-544,105,-239) -> Matrix(515,-24,11051,-515) (34/15,16/7) -> (2/43,12/257) Matrix(103,-238,29,-67) -> Matrix(43,-2,753,-35) Matrix(273,-646,101,-239) -> Matrix(127,-6,2561,-121) Matrix(271,-646,86,-205) -> Matrix(43,-2,796,-37) Matrix(441,-1054,100,-239) -> Matrix(127,-6,2392,-113) Matrix(169,-408,70,-169) -> Matrix(337,-16,7098,-337) (12/5,17/7) -> (8/169,1/21) Matrix(69,-170,28,-69) -> Matrix(125,-6,2604,-125) (17/7,5/2) -> (1/21,3/62) Matrix(67,-170,13,-33) -> Matrix(41,-2,717,-35) Matrix(171,-442,53,-137) -> Matrix(41,-2,717,-35) Matrix(339,-884,130,-339) -> Matrix(575,-28,11808,-575) (13/5,34/13) -> (7/144,2/41) Matrix(103,-272,39,-103) -> Matrix(163,-8,3321,-163) (34/13,8/3) -> (2/41,4/81) Matrix(101,-272,13,-35) -> Matrix(1,0,-3,1) 0/1 Matrix(137,-374,37,-101) -> Matrix(41,-2,759,-37) Matrix(171,-476,37,-103) -> Matrix(81,-4,1559,-77) Matrix(169,-476,60,-169) -> Matrix(241,-12,4840,-241) (14/5,17/6) -> (6/121,1/20) Matrix(35,-102,12,-35) -> Matrix(39,-2,760,-39) (17/6,3/1) -> (1/20,1/19) Matrix(407,-1292,86,-273) -> Matrix(77,-4,1444,-75) 1/19 Matrix(815,-2686,186,-613) -> Matrix(39,-2,644,-33) Matrix(103,-340,10,-33) -> Matrix(77,-4,1290,-67) Matrix(101,-340,30,-101) -> Matrix(77,-4,1482,-77) (10/3,17/5) -> (2/39,1/19) Matrix(69,-238,20,-69) -> Matrix(37,-2,684,-37) (17/5,7/2) -> (1/19,1/18) Matrix(103,-374,19,-69) -> Matrix(39,-2,721,-37) Matrix(271,-1020,72,-271) -> Matrix(75,-4,1406,-75) (15/4,34/9) -> (1/19,2/37) Matrix(35,-136,9,-35) -> Matrix(1,0,37,-1) (34/9,4/1) -> (0/1,2/37) Matrix(33,-136,8,-33) -> Matrix(1,0,38,-1) (4/1,17/4) -> (0/1,1/19) Matrix(103,-442,24,-103) -> Matrix(37,-2,684,-37) (17/4,13/3) -> (1/19,1/18) Matrix(203,-884,31,-135) -> Matrix(75,-4,1331,-71) Matrix(3535,-15504,806,-3535) -> Matrix(1,0,28,-1) (57/13,136/31) -> (0/1,1/14) Matrix(681,-2992,155,-681) -> Matrix(1,0,39,-1) (136/31,22/5) -> (0/1,2/39) Matrix(1427,-6324,322,-1427) -> Matrix(223,-12,4144,-223) (31/7,102/23) -> (3/56,2/37) Matrix(919,-4080,207,-919) -> Matrix(147,-8,2701,-147) (102/23,40/9) -> (2/37,4/73) Matrix(237,-1054,38,-169) -> Matrix(37,-2,684,-37) *** -> (1/19,1/18) Matrix(239,-1122,36,-169) -> Matrix(41,-2,718,-35) Matrix(1189,-5610,252,-1189) -> Matrix(191,-10,3648,-191) (33/7,85/18) -> (5/96,1/19) Matrix(1871,-8840,396,-1871) -> Matrix(303,-16,5738,-303) (85/18,52/11) -> (1/19,8/151) Matrix(271,-1292,43,-205) -> Matrix(75,-4,1369,-73) 2/37 Matrix(135,-646,14,-67) -> Matrix(113,-6,1902,-101) Matrix(169,-816,35,-169) -> Matrix(149,-8,2775,-149) (24/5,34/7) -> (4/75,2/37) Matrix(69,-340,14,-69) -> Matrix(73,-4,1332,-73) (34/7,5/1) -> (2/37,1/18) Matrix(67,-374,12,-67) -> Matrix(109,-6,1980,-109) (11/2,17/3) -> (3/55,1/18) Matrix(35,-204,6,-35) -> Matrix(71,-4,1260,-71) (17/3,6/1) -> (1/18,2/35) Matrix(101,-680,15,-101) -> Matrix(281,-16,4935,-281) (20/3,34/5) -> (8/141,2/35) Matrix(69,-476,10,-69) -> Matrix(209,-12,3640,-209) (34/5,7/1) -> (2/35,3/52) Matrix(33,-272,4,-33) -> Matrix(137,-8,2346,-137) (8/1,17/2) -> (4/69,1/17) Matrix(35,-306,4,-35) -> Matrix(169,-10,2856,-169) (17/2,9/1) -> (1/17,5/84) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.