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 -1/1 -17/2 -1/1 -8/1 -1/1 -1/2 -7/1 -1/1 -20/3 -1/2 0/1 -13/2 -1/1 -1/2 -19/3 -1/3 -25/4 -1/3 0/1 -6/1 -1/1 0/1 -17/3 -1/1 -11/2 -1/1 -1/2 -5/1 -1/3 -24/5 0/1 1/0 -43/9 -1/1 -19/4 -1/1 0/1 -33/7 -1/1 -14/3 -1/2 0/1 -9/2 -1/3 0/1 -22/5 -1/3 -1/4 -35/8 -2/9 -1/5 -13/3 -1/7 -17/4 0/1 -4/1 0/1 1/1 -15/4 0/1 1/1 -11/3 -1/1 -18/5 1/1 1/0 -7/2 0/1 1/0 -17/5 -1/1 1/1 -10/3 0/1 1/0 -43/13 1/1 -33/10 0/1 1/1 -56/17 3/1 1/0 -23/7 -1/1 -13/4 1/1 1/0 -3/1 1/1 -17/6 1/0 -14/5 -6/1 1/0 -11/4 -3/1 1/0 -19/7 -1/1 -8/3 -1/1 1/0 -13/5 -5/1 -18/7 -3/1 -5/2 -5/2 -2/1 -1/1 -17/7 -1/1 -12/5 -1/1 -1/2 -43/18 -1/1 0/1 -74/31 -1/3 0/1 -31/13 1/1 -19/8 -1/1 0/1 -7/3 -1/1 -16/7 -1/1 1/0 -9/4 -1/1 0/1 -11/5 1/1 -13/6 -1/1 1/0 -15/7 1/1 -17/8 1/0 -2/1 -1/1 1/0 -17/9 -1/1 -15/8 -1/1 0/1 -13/7 1/1 -11/6 -1/1 1/0 -9/5 -1/1 -34/19 -1/1 -25/14 -1/1 0/1 -16/9 -1/1 1/0 -7/4 0/1 1/0 -26/15 1/1 1/0 -19/11 1/1 -31/18 3/2 2/1 -12/7 3/1 1/0 -17/10 1/0 -5/3 -3/1 -18/11 -1/1 1/0 -85/52 -1/1 -67/41 -1/1 -49/30 -1/1 0/1 -31/19 1/1 -13/8 1/1 1/0 -34/21 1/0 -21/13 -7/1 -8/5 -3/1 1/0 -27/17 -3/1 -19/12 -3/1 -2/1 -11/7 -1/1 -25/16 -3/1 -2/1 -14/9 -2/1 1/0 -17/11 -3/1 -1/1 -3/2 -2/1 1/0 -19/13 -1/1 -16/11 -1/1 -1/2 -13/9 -3/1 -23/16 -1/1 1/0 -33/23 -3/1 -10/7 -2/1 1/0 -17/12 -2/1 -7/5 -1/1 -18/13 -1/1 1/0 -29/21 -3/1 -11/8 -1/1 1/0 -15/11 -3/1 -34/25 -2/1 -19/14 -2/1 -1/1 -4/3 -2/1 -1/1 -17/13 -1/1 -13/10 -1/1 1/0 -48/37 -2/1 1/0 -35/27 -1/1 -57/44 1/1 1/0 -136/105 1/0 -79/61 -5/1 -22/17 -1/1 1/0 -31/24 -5/2 -2/1 -102/79 -2/1 -71/55 -9/5 -40/31 -2/1 -1/1 -9/7 -1/1 -23/18 -1/1 1/0 -14/11 -4/1 1/0 -33/26 -10/3 -3/1 -85/67 -3/1 -52/41 -3/1 -17/6 -19/15 -3/1 -43/34 -3/1 -2/1 -24/19 -5/2 -2/1 -29/23 -11/5 -34/27 -2/1 -5/4 -2/1 -1/1 -16/13 -3/1 -5/2 -11/9 -7/3 -17/14 -2/1 -6/5 -2/1 -5/3 -25/21 -5/3 -19/16 -2/1 -5/3 -13/11 -1/1 -33/28 -12/7 -5/3 -20/17 -2/1 -3/2 -27/23 -5/3 -34/29 -3/2 -7/6 -2/1 -3/2 -8/7 -3/2 -1/1 -17/15 -1/1 -9/8 -2/1 -1/1 -10/9 -2/1 -3/2 -1/1 -1/1 0/1 -1/1 1/1 -1/1 9/8 -1/1 -2/3 17/15 -1/1 8/7 -1/1 -3/4 7/6 -3/4 -2/3 20/17 -3/4 -2/3 13/11 -1/1 19/16 -5/7 -2/3 25/21 -5/7 6/5 -5/7 -2/3 17/14 -2/3 11/9 -7/11 5/4 -1/1 -2/3 24/19 -2/3 -5/8 43/34 -2/3 -3/5 19/15 -3/5 33/26 -3/5 -10/17 14/11 -4/7 -1/2 9/7 -1/1 22/17 -1/1 -1/2 35/27 -1/1 13/10 -1/1 -1/2 17/13 -1/1 4/3 -1/1 -2/3 15/11 -3/5 11/8 -1/1 -1/2 18/13 -1/1 -1/2 7/5 -1/1 17/12 -2/3 10/7 -2/3 -1/2 43/30 -2/3 -3/5 33/23 -3/5 56/39 -3/5 -1/2 23/16 -1/1 -1/2 13/9 -3/5 3/2 -2/3 -1/2 17/11 -1/1 -3/5 14/9 -2/3 -1/2 11/7 -1/1 19/12 -2/3 -3/5 8/5 -3/5 -1/2 13/8 -1/2 -1/3 18/11 -1/1 -1/2 5/3 -3/5 17/10 -1/2 12/7 -1/2 -3/7 43/25 -3/7 74/43 -3/7 -2/5 31/18 -2/5 -3/8 19/11 -1/3 7/4 -1/2 0/1 16/9 -1/1 -1/2 9/5 -1/1 11/6 -1/1 -1/2 13/7 -1/3 15/8 -1/1 0/1 17/9 -1/1 2/1 -1/1 -1/2 17/8 -1/2 15/7 -1/3 13/6 -1/1 -1/2 11/5 -1/3 9/4 -1/1 0/1 34/15 -1/1 25/11 -1/1 16/7 -1/1 -1/2 7/3 -1/1 26/11 -1/1 -1/2 19/8 -1/1 0/1 31/13 -1/3 12/5 -1/1 1/0 17/7 -1/1 5/2 -1/1 -2/3 18/7 -5/8 -3/5 85/33 -3/5 67/26 -3/5 -22/37 49/19 -3/5 31/12 -4/7 -1/2 13/5 -5/9 34/13 -1/2 21/8 -1/2 -3/7 8/3 -1/1 -1/2 27/10 -2/3 -1/2 19/7 -1/1 11/4 -3/5 -1/2 25/9 -3/5 14/5 -6/11 -1/2 17/6 -1/2 3/1 -1/3 19/6 -1/1 0/1 16/5 -1/1 -1/2 13/4 -1/2 -1/3 23/7 -1/1 33/10 -1/3 0/1 10/3 -1/2 0/1 17/5 -1/1 -1/3 7/2 -1/2 0/1 18/5 -1/2 -1/3 29/8 -1/5 0/1 11/3 -1/1 15/4 -1/3 0/1 34/9 0/1 19/5 -1/1 4/1 -1/3 0/1 17/4 0/1 13/3 1/5 48/11 0/1 1/4 35/8 1/3 2/5 57/13 3/7 136/31 1/2 79/18 1/2 3/5 22/5 1/2 1/1 31/7 -1/1 102/23 0/1 71/16 0/1 1/6 40/9 0/1 1/3 9/2 0/1 1/1 23/5 -1/1 14/3 0/1 1/0 33/7 -1/1 85/18 -1/1 52/11 -1/1 -1/2 19/4 -1/1 0/1 43/9 -1/1 24/5 -1/2 0/1 29/6 -1/5 0/1 34/7 0/1 5/1 1/1 16/3 -1/1 1/0 11/2 -1/1 1/0 17/3 -1/1 6/1 -1/1 0/1 25/4 0/1 1/1 19/3 1/1 13/2 -1/1 1/0 33/5 -1/1 20/3 0/1 1/0 27/4 0/1 1/0 34/5 1/0 7/1 -1/1 8/1 -1/1 1/0 17/2 -1/1 9/1 -1/1 10/1 0/1 1/0 1/0 -1/1 0/1 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(1,0,2,1) Matrix(35,306,4,35) -> Matrix(1,0,0,1) Matrix(33,272,4,33) -> Matrix(3,2,-2,-1) Matrix(33,238,14,101) -> Matrix(1,0,0,1) Matrix(101,680,-86,-579) -> Matrix(7,2,-4,-1) Matrix(135,884,-104,-681) -> Matrix(3,2,-2,-1) Matrix(69,442,32,205) -> Matrix(1,0,0,1) Matrix(205,1292,162,1021) -> Matrix(9,2,-14,-3) Matrix(169,1054,38,237) -> Matrix(1,0,4,1) Matrix(35,204,6,35) -> Matrix(1,0,0,1) Matrix(67,374,12,67) -> Matrix(3,2,-2,-1) Matrix(69,374,-50,-271) -> Matrix(3,2,-2,-1) Matrix(169,816,-134,-647) -> Matrix(5,-2,-2,1) Matrix(135,646,14,67) -> Matrix(1,0,0,1) Matrix(271,1292,228,1087) -> Matrix(3,-2,-4,3) Matrix(611,2890,-374,-1769) -> Matrix(1,0,0,1) Matrix(441,2074,-340,-1599) -> Matrix(3,2,-2,-1) Matrix(103,476,-66,-305) -> Matrix(3,2,-2,-1) Matrix(239,1054,-100,-441) -> Matrix(1,0,2,1) Matrix(613,2686,-186,-815) -> Matrix(9,2,4,1) Matrix(203,884,-172,-749) -> Matrix(15,2,-8,-1) Matrix(103,442,24,103) -> Matrix(1,0,12,1) Matrix(33,136,8,33) -> Matrix(1,0,-4,1) Matrix(35,136,-26,-101) -> Matrix(1,-2,0,1) Matrix(101,374,64,237) -> Matrix(1,2,-2,-3) Matrix(103,374,-84,-305) -> Matrix(5,-2,-2,1) Matrix(67,238,38,135) -> Matrix(1,0,-2,1) Matrix(69,238,20,69) -> Matrix(1,0,-2,1) Matrix(101,340,30,101) -> Matrix(1,0,-2,1) Matrix(441,1462,92,305) -> Matrix(1,0,-2,1) Matrix(1497,4930,-1156,-3807) -> Matrix(1,-4,0,1) Matrix(135,442,62,203) -> Matrix(1,0,-2,1) Matrix(137,442,-84,-271) -> Matrix(1,0,0,1) Matrix(35,102,12,35) -> Matrix(1,-2,-2,5) Matrix(169,476,60,169) -> Matrix(1,12,-2,-23) Matrix(171,476,-134,-373) -> Matrix(1,2,0,1) Matrix(137,374,100,273) -> Matrix(1,4,-2,-7) Matrix(239,646,-138,-373) -> Matrix(1,2,0,1) Matrix(103,272,-64,-169) -> Matrix(1,-2,0,1) Matrix(171,442,-118,-305) -> Matrix(1,2,0,1) Matrix(67,170,-54,-137) -> Matrix(1,0,0,1) Matrix(69,170,28,69) -> Matrix(3,4,-4,-5) Matrix(169,408,70,169) -> Matrix(3,2,-2,-1) Matrix(883,2108,142,339) -> Matrix(1,0,2,1) Matrix(2721,6494,-2108,-5031) -> Matrix(7,2,-4,-1) Matrix(885,2108,-542,-1291) -> Matrix(1,0,0,1) Matrix(273,646,-172,-407) -> Matrix(1,-2,0,1) Matrix(103,238,74,171) -> Matrix(1,2,-2,-3) Matrix(239,544,-134,-305) -> Matrix(1,0,0,1) Matrix(169,374,-108,-239) -> Matrix(1,-2,0,1) Matrix(203,442,62,135) -> Matrix(1,0,-2,1) Matrix(205,442,32,69) -> Matrix(1,0,0,1) Matrix(239,510,112,239) -> Matrix(1,-2,-2,5) Matrix(33,68,16,33) -> Matrix(1,2,-2,-3) Matrix(35,68,18,35) -> Matrix(1,2,-2,-3) Matrix(271,510,144,271) -> Matrix(1,0,0,1) Matrix(237,442,200,373) -> Matrix(3,-2,-4,3) Matrix(239,442,166,307) -> Matrix(1,2,-2,-3) Matrix(169,306,-132,-239) -> Matrix(1,0,0,1) Matrix(645,1156,284,509) -> Matrix(1,2,-2,-3) Matrix(647,1156,286,511) -> Matrix(1,0,0,1) Matrix(135,238,38,67) -> Matrix(1,0,-2,1) Matrix(137,238,118,205) -> Matrix(3,-2,-4,3) Matrix(375,646,-256,-441) -> Matrix(3,-4,-2,3) Matrix(613,1054,-474,-815) -> Matrix(1,-4,0,1) Matrix(239,408,140,239) -> Matrix(1,-6,-2,13) Matrix(101,170,60,101) -> Matrix(1,6,-2,-11) Matrix(373,612,-270,-443) -> Matrix(1,0,0,1) Matrix(3639,5950,770,1259) -> Matrix(1,2,-2,-3) Matrix(5201,8500,1102,1801) -> Matrix(3,2,-2,-1) Matrix(713,1156,272,441) -> Matrix(1,-4,-2,9) Matrix(715,1156,274,443) -> Matrix(1,12,-2,-23) Matrix(171,272,22,35) -> Matrix(1,2,0,1) Matrix(237,374,64,101) -> Matrix(1,2,-2,-3) Matrix(307,476,198,307) -> Matrix(1,4,-2,-7) Matrix(67,102,44,67) -> Matrix(1,4,-2,-7) Matrix(885,1292,-698,-1019) -> Matrix(11,14,-4,-5) Matrix(307,442,166,239) -> Matrix(1,2,-2,-3) Matrix(1871,2686,-1444,-2073) -> Matrix(1,2,0,1) Matrix(237,340,-214,-307) -> Matrix(3,8,-2,-5) Matrix(239,340,168,239) -> Matrix(1,4,-2,-7) Matrix(169,238,120,169) -> Matrix(3,4,-4,-5) Matrix(171,238,74,103) -> Matrix(1,2,-2,-3) Matrix(273,374,100,137) -> Matrix(1,4,-2,-7) Matrix(849,1156,224,305) -> Matrix(1,2,0,1) Matrix(851,1156,226,307) -> Matrix(1,2,-4,-7) Matrix(103,136,78,103) -> Matrix(3,4,-4,-5) Matrix(339,442,260,339) -> Matrix(1,2,-2,-3) Matrix(14279,18496,3254,4215) -> Matrix(1,-4,2,-7) Matrix(14281,18496,3256,4217) -> Matrix(1,8,2,17) Matrix(8057,10404,1816,2345) -> Matrix(1,2,8,17) Matrix(8059,10404,1818,2347) -> Matrix(1,2,-6,-11) Matrix(817,1054,686,885) -> Matrix(7,12,-10,-17) Matrix(883,1122,-750,-953) -> Matrix(3,14,-2,-9) Matrix(6699,8500,2600,3299) -> Matrix(31,96,-52,-161) Matrix(4691,5950,1822,2311) -> Matrix(23,66,-38,-109) Matrix(1021,1292,162,205) -> Matrix(1,2,2,5) Matrix(1157,1462,808,1021) -> Matrix(5,12,-8,-19) Matrix(917,1156,188,237) -> Matrix(1,2,6,13) Matrix(919,1156,190,239) -> Matrix(1,2,-6,-11) Matrix(307,374,252,307) -> Matrix(13,28,-20,-43) Matrix(169,204,140,169) -> Matrix(11,20,-16,-29) Matrix(1769,2108,1028,1225) -> Matrix(9,16,-22,-39) Matrix(1087,1292,228,271) -> Matrix(1,2,-4,-7) Matrix(373,442,200,237) -> Matrix(1,2,-4,-7) Matrix(985,1156,144,169) -> Matrix(5,8,-2,-3) Matrix(987,1156,146,171) -> Matrix(1,2,-2,-3) Matrix(237,272,88,101) -> Matrix(3,4,-4,-5) Matrix(239,272,210,239) -> Matrix(5,6,-6,-7) Matrix(271,306,240,271) -> Matrix(3,4,-4,-5) Matrix(579,646,458,511) -> Matrix(1,4,-2,-7) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(307,-340,214,-237) -> Matrix(11,8,-18,-13) Matrix(579,-680,86,-101) -> Matrix(3,2,4,3) Matrix(749,-884,172,-203) -> Matrix(3,2,16,11) Matrix(305,-374,84,-103) -> Matrix(3,2,-14,-9) Matrix(647,-816,134,-169) -> Matrix(3,2,-14,-9) Matrix(2279,-2890,884,-1121) -> Matrix(59,36,-100,-61) Matrix(1633,-2074,374,-475) -> Matrix(7,4,26,15) Matrix(373,-476,134,-171) -> Matrix(5,2,-8,-3) Matrix(815,-1054,474,-613) -> Matrix(7,4,-16,-9) Matrix(2073,-2686,1444,-1871) -> Matrix(5,2,-8,-3) Matrix(681,-884,104,-135) -> Matrix(3,2,-2,-1) Matrix(101,-136,26,-35) -> Matrix(3,2,-8,-5) Matrix(271,-374,50,-69) -> Matrix(3,2,-2,-1) Matrix(3433,-4930,782,-1123) -> Matrix(7,4,12,7) Matrix(305,-442,118,-171) -> Matrix(5,2,-8,-3) Matrix(305,-476,66,-103) -> Matrix(3,2,-2,-1) Matrix(407,-646,172,-273) -> Matrix(3,2,-8,-5) Matrix(169,-272,64,-103) -> Matrix(3,2,-8,-5) Matrix(271,-442,84,-137) -> Matrix(1,0,0,1) Matrix(103,-170,20,-33) -> Matrix(3,2,-2,-1) Matrix(3773,-6494,850,-1463) -> Matrix(5,2,22,9) Matrix(1223,-2108,474,-817) -> Matrix(27,10,-46,-17) Matrix(373,-646,138,-239) -> Matrix(5,2,-8,-3) Matrix(305,-544,134,-239) -> Matrix(1,0,0,1) Matrix(205,-374,74,-135) -> Matrix(5,2,-8,-3) Matrix(137,-306,30,-67) -> Matrix(1,0,2,1) Matrix(271,-646,86,-205) -> Matrix(1,0,0,1) Matrix(441,-1054,100,-239) -> Matrix(1,0,2,1) Matrix(239,-612,66,-169) -> Matrix(3,2,-14,-9) Matrix(407,-1292,86,-273) -> Matrix(1,0,0,1) Matrix(815,-2686,186,-613) -> Matrix(5,2,12,5) Matrix(103,-340,10,-33) -> Matrix(1,0,2,1) Matrix(239,-1122,36,-169) -> 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): 30 Degree of the the map X: 30 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 0/1 1/1 17/15 17/14 17/13 17/12 3/2 17/11 17/10 9/5 17/9 2/1 34/15 17/7 5/2 85/33 17/6 3/1 17/5 7/2 11/3 34/9 19/5 4/1 17/4 13/3 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 -6/1 -1/1 0/1 -5/1 -1/3 -4/1 0/1 1/1 -15/4 0/1 1/1 -11/3 -1/1 -18/5 1/1 1/0 -7/2 0/1 1/0 -3/1 1/1 -17/6 1/0 -14/5 -6/1 1/0 -11/4 -3/1 1/0 -19/7 -1/1 -8/3 -1/1 1/0 -13/5 -5/1 -18/7 -3/1 -5/2 -5/2 -2/1 -1/1 -7/3 -1/1 -16/7 -1/1 1/0 -9/4 -1/1 0/1 -2/1 -1/1 1/0 -13/7 1/1 -11/6 -1/1 1/0 -9/5 -1/1 -34/19 -1/1 -25/14 -1/1 0/1 -16/9 -1/1 1/0 -7/4 0/1 1/0 -26/15 1/1 1/0 -19/11 1/1 -12/7 3/1 1/0 -17/10 1/0 -5/3 -3/1 -8/5 -3/1 1/0 -11/7 -1/1 -3/2 -2/1 1/0 -16/11 -1/1 -1/2 -13/9 -3/1 -10/7 -2/1 1/0 -17/12 -2/1 -7/5 -1/1 -11/8 -1/1 1/0 -15/11 -3/1 -34/25 -2/1 -19/14 -2/1 -1/1 -4/3 -2/1 -1/1 -9/7 -1/1 -23/18 -1/1 1/0 -14/11 -4/1 1/0 -5/4 -2/1 -1/1 -16/13 -3/1 -5/2 -11/9 -7/3 -17/14 -2/1 -6/5 -2/1 -5/3 -7/6 -2/1 -3/2 -8/7 -3/2 -1/1 -1/1 -1/1 0/1 -1/1 1/1 -1/1 9/8 -1/1 -2/3 17/15 -1/1 8/7 -1/1 -3/4 7/6 -3/4 -2/3 13/11 -1/1 6/5 -5/7 -2/3 17/14 -2/3 11/9 -7/11 5/4 -1/1 -2/3 19/15 -3/5 33/26 -3/5 -10/17 14/11 -4/7 -1/2 9/7 -1/1 13/10 -1/1 -1/2 17/13 -1/1 4/3 -1/1 -2/3 15/11 -3/5 11/8 -1/1 -1/2 18/13 -1/1 -1/2 7/5 -1/1 17/12 -2/3 10/7 -2/3 -1/2 23/16 -1/1 -1/2 13/9 -3/5 3/2 -2/3 -1/2 17/11 -1/1 -3/5 14/9 -2/3 -1/2 11/7 -1/1 19/12 -2/3 -3/5 8/5 -3/5 -1/2 13/8 -1/2 -1/3 5/3 -3/5 17/10 -1/2 12/7 -1/2 -3/7 31/18 -2/5 -3/8 19/11 -1/3 7/4 -1/2 0/1 16/9 -1/1 -1/2 9/5 -1/1 11/6 -1/1 -1/2 13/7 -1/3 15/8 -1/1 0/1 17/9 -1/1 2/1 -1/1 -1/2 11/5 -1/3 9/4 -1/1 0/1 34/15 -1/1 25/11 -1/1 16/7 -1/1 -1/2 7/3 -1/1 12/5 -1/1 1/0 17/7 -1/1 5/2 -1/1 -2/3 18/7 -5/8 -3/5 85/33 -3/5 67/26 -3/5 -22/37 49/19 -3/5 31/12 -4/7 -1/2 13/5 -5/9 34/13 -1/2 21/8 -1/2 -3/7 8/3 -1/1 -1/2 27/10 -2/3 -1/2 19/7 -1/1 11/4 -3/5 -1/2 25/9 -3/5 14/5 -6/11 -1/2 17/6 -1/2 3/1 -1/3 10/3 -1/2 0/1 17/5 -1/1 -1/3 7/2 -1/2 0/1 18/5 -1/2 -1/3 29/8 -1/5 0/1 11/3 -1/1 15/4 -1/3 0/1 34/9 0/1 19/5 -1/1 4/1 -1/3 0/1 17/4 0/1 13/3 1/5 9/2 0/1 1/1 5/1 1/1 11/2 -1/1 1/0 17/3 -1/1 6/1 -1/1 0/1 7/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(18,119,13,86) (-6/1,1/0) -> (11/8,18/13) Hyperbolic Matrix(16,85,3,16) (-6/1,-5/1) -> (5/1,11/2) Hyperbolic Matrix(18,85,11,52) (-5/1,-4/1) -> (13/8,5/3) 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(16,51,5,16) (-7/2,-3/1) -> (3/1,10/3) 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(84,221,19,50) (-8/3,-13/5) -> (13/3,9/2) 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(50,119,21,50) (-5/2,-7/3) -> (7/3,12/5) 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(84,187,53,118) (-9/4,-2/1) -> (19/12,8/5) Hyperbolic Matrix(118,221,63,118) (-2/1,-13/7) -> (13/7,15/8) 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(188,323,149,256) (-19/11,-12/7) -> (5/4,19/15) 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(52,85,11,18) (-5/3,-8/5) -> (9/2,5/1) Hyperbolic Matrix(118,187,53,84) (-8/5,-11/7) -> (11/5,9/4) Hyperbolic Matrix(120,187,77,120) (-11/7,-3/2) -> (14/9,11/7) Hyperbolic Matrix(220,323,173,254) (-3/2,-16/11) -> (33/26,14/11) Hyperbolic Matrix(154,221,131,188) (-13/9,-10/7) -> (7/6,13/11) 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(86,119,13,18) (-7/5,-11/8) -> (6/1,7/1) 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(118,153,91,118) (-4/3,-9/7) -> (9/7,13/10) Hyperbolic Matrix(256,323,149,188) (-14/11,-5/4) -> (12/7,31/18) 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(188,221,131,154) (-6/5,-7/6) -> (10/7,23/16) Hyperbolic Matrix(237,272,88,101) (-7/6,-8/7) -> (8/3,27/10) Hyperbolic Matrix(16,17,15,16) (-8/7,-1/1) -> (1/1,9/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(256,-289,225,-254) (9/8,17/15) -> (17/15,8/7) Parabolic Matrix(186,-221,101,-120) (13/11,6/5) -> (11/6,13/7) Hyperbolic Matrix(305,-374,84,-103) (11/9,5/4) -> (29/8,11/3) Hyperbolic Matrix(2279,-2890,884,-1121) (19/15,33/26) -> (67/26,49/19) Hyperbolic Matrix(373,-476,134,-171) (14/11,9/7) -> (25/9,14/5) Hyperbolic Matrix(222,-289,169,-220) (13/10,17/13) -> (17/13,4/3) Parabolic Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(305,-442,118,-171) (13/9,3/2) -> (31/12,13/5) Hyperbolic Matrix(188,-289,121,-186) (3/2,17/11) -> (17/11,14/9) Parabolic Matrix(169,-272,64,-103) (8/5,13/8) -> (21/8,8/3) 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(154,-289,81,-152) (15/8,17/9) -> (17/9,2/1) Parabolic Matrix(86,-187,23,-50) (2/1,11/5) -> (11/3,15/4) Hyperbolic 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(239,-612,66,-169) (5/2,18/7) -> (18/5,29/8) Hyperbolic Matrix(2806,-7225,1089,-2804) (18/7,85/33) -> (85/33,67/26) Parabolic Matrix(254,-663,59,-154) (13/5,34/13) -> (17/4,13/3) Hyperbolic Matrix(188,-493,45,-118) (34/13,21/8) -> (4/1,17/4) Hyperbolic Matrix(86,-289,25,-84) (10/3,17/5) -> (17/5,7/2) Parabolic Matrix(52,-289,9,-50) (11/2,17/3) -> (17/3,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(18,119,13,86) -> Matrix(0,-1,1,2) Matrix(16,85,3,16) -> Matrix(2,1,-1,0) Matrix(18,85,11,52) -> Matrix(0,-1,1,2) Matrix(35,136,-26,-101) -> Matrix(1,-2,0,1) Matrix(101,374,64,237) -> Matrix(1,2,-2,-3) Matrix(103,374,-84,-305) -> Matrix(5,-2,-2,1) Matrix(67,238,38,135) -> Matrix(1,0,-2,1) Matrix(16,51,5,16) -> Matrix(0,-1,1,2) Matrix(35,102,12,35) -> Matrix(1,-2,-2,5) Matrix(169,476,60,169) -> Matrix(1,12,-2,-23) Matrix(171,476,-134,-373) -> Matrix(1,2,0,1) Matrix(137,374,100,273) -> Matrix(1,4,-2,-7) Matrix(239,646,-138,-373) -> Matrix(1,2,0,1) Matrix(84,221,19,50) -> Matrix(0,-1,1,0) Matrix(171,442,-118,-305) -> Matrix(1,2,0,1) Matrix(67,170,-54,-137) -> Matrix(1,0,0,1) Matrix(50,119,21,50) -> Matrix(0,-1,1,2) Matrix(103,238,74,171) -> Matrix(1,2,-2,-3) Matrix(239,544,-134,-305) -> Matrix(1,0,0,1) Matrix(84,187,53,118) -> Matrix(2,-1,-3,2) Matrix(118,221,63,118) -> Matrix(0,-1,1,2) Matrix(239,442,166,307) -> Matrix(1,2,-2,-3) Matrix(169,306,-132,-239) -> Matrix(1,0,0,1) Matrix(645,1156,284,509) -> Matrix(1,2,-2,-3) Matrix(647,1156,286,511) -> Matrix(1,0,0,1) Matrix(135,238,38,67) -> Matrix(1,0,-2,1) Matrix(137,238,118,205) -> Matrix(3,-2,-4,3) Matrix(188,323,149,256) -> Matrix(2,-5,-3,8) Matrix(239,408,140,239) -> Matrix(1,-6,-2,13) Matrix(101,170,60,101) -> Matrix(1,6,-2,-11) Matrix(52,85,11,18) -> Matrix(0,-1,1,2) Matrix(118,187,53,84) -> Matrix(0,-1,1,4) Matrix(120,187,77,120) -> Matrix(2,3,-3,-4) Matrix(220,323,173,254) -> Matrix(4,7,-7,-12) Matrix(154,221,131,188) -> Matrix(2,7,-3,-10) Matrix(239,340,168,239) -> Matrix(1,4,-2,-7) Matrix(169,238,120,169) -> Matrix(3,4,-4,-5) Matrix(86,119,13,18) -> Matrix(0,-1,1,2) Matrix(273,374,100,137) -> Matrix(1,4,-2,-7) Matrix(849,1156,224,305) -> Matrix(1,2,0,1) Matrix(851,1156,226,307) -> Matrix(1,2,-4,-7) Matrix(118,153,91,118) -> Matrix(2,3,-3,-4) Matrix(256,323,149,188) -> Matrix(2,5,-5,-12) Matrix(307,374,252,307) -> Matrix(13,28,-20,-43) Matrix(169,204,140,169) -> Matrix(11,20,-16,-29) Matrix(188,221,131,154) -> Matrix(4,7,-7,-12) Matrix(237,272,88,101) -> Matrix(3,4,-4,-5) Matrix(16,17,15,16) -> Matrix(4,5,-5,-6) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(256,-289,225,-254) -> Matrix(0,-1,1,2) Matrix(186,-221,101,-120) -> Matrix(4,3,-11,-8) Matrix(305,-374,84,-103) -> Matrix(3,2,-14,-9) Matrix(2279,-2890,884,-1121) -> Matrix(59,36,-100,-61) Matrix(373,-476,134,-171) -> Matrix(5,2,-8,-3) Matrix(222,-289,169,-220) -> Matrix(0,-1,1,2) Matrix(101,-136,26,-35) -> Matrix(3,2,-8,-5) Matrix(305,-442,118,-171) -> Matrix(5,2,-8,-3) Matrix(188,-289,121,-186) -> Matrix(8,5,-13,-8) Matrix(169,-272,64,-103) -> Matrix(3,2,-8,-5) Matrix(1223,-2108,474,-817) -> Matrix(27,10,-46,-17) Matrix(373,-646,138,-239) -> Matrix(5,2,-8,-3) Matrix(305,-544,134,-239) -> Matrix(1,0,0,1) Matrix(205,-374,74,-135) -> Matrix(5,2,-8,-3) Matrix(154,-289,81,-152) -> Matrix(0,-1,1,2) Matrix(86,-187,23,-50) -> Matrix(2,1,-5,-2) Matrix(52,-119,7,-16) -> Matrix(2,1,-1,0) Matrix(120,-289,49,-118) -> Matrix(2,3,-3,-4) Matrix(239,-612,66,-169) -> Matrix(3,2,-14,-9) Matrix(2806,-7225,1089,-2804) -> Matrix(134,81,-225,-136) Matrix(254,-663,59,-154) -> Matrix(2,1,19,10) Matrix(188,-493,45,-118) -> Matrix(2,1,-13,-6) Matrix(86,-289,25,-84) -> Matrix(2,1,-5,-2) Matrix(52,-289,9,-50) -> Matrix(0,-1,1,2) 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): 30 ----------------------------------------------------------------------- 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 -1/1 2 1 1/1 -1/1 1 17 17/15 -1/1 1 1 8/7 (-1/1,-3/4) 0 17 7/6 (-3/4,-2/3) 0 17 6/5 (-5/7,-2/3) 0 17 17/14 -2/3 6 1 11/9 -7/11 1 17 5/4 (-1/1,-2/3) 0 17 14/11 (-4/7,-1/2) 0 17 9/7 -1/1 1 17 17/13 -1/1 1 1 4/3 (-1/1,-2/3) 0 17 15/11 -3/5 1 17 11/8 (-1/1,-1/2) 0 17 7/5 -1/1 1 17 17/12 -2/3 1 1 10/7 (-2/3,-1/2) 0 17 23/16 (-1/1,-1/2) 0 17 13/9 -3/5 1 17 3/2 (-2/3,-1/2) 0 17 17/11 (-1/1,-3/5).(-2/3,-1/2) 0 1 11/7 -1/1 1 17 8/5 (-3/5,-1/2) 0 17 5/3 -3/5 1 17 17/10 -1/2 6 1 12/7 (-1/2,-3/7) 0 17 31/18 (-2/5,-3/8) 0 17 19/11 -1/3 1 17 7/4 (-1/2,0/1) 0 17 16/9 (-1/1,-1/2) 0 17 9/5 -1/1 1 17 11/6 (-1/1,-1/2) 0 17 13/7 -1/3 1 17 17/9 -1/1 1 1 2/1 (-1/1,-1/2) 0 17 11/5 -1/3 1 17 9/4 (-1/1,0/1) 0 17 34/15 -1/1 1 1 25/11 -1/1 1 17 16/7 (-1/1,-1/2) 0 17 7/3 -1/1 1 17 17/7 -1/1 3 1 5/2 (-1/1,-2/3) 0 17 18/7 (-5/8,-3/5) 0 17 85/33 -3/5 9 1 49/19 -3/5 1 17 31/12 (-4/7,-1/2) 0 17 13/5 -5/9 1 17 8/3 (-1/1,-1/2) 0 17 27/10 (-2/3,-1/2) 0 17 19/7 -1/1 1 17 11/4 (-3/5,-1/2) 0 17 25/9 -3/5 1 17 14/5 (-6/11,-1/2) 0 17 17/6 -1/2 7 1 3/1 -1/3 1 17 17/5 (-1/1,-1/3).(-1/2,0/1) 0 1 7/2 (-1/2,0/1) 0 17 18/5 (-1/2,-1/3) 0 17 29/8 (-1/5,0/1) 0 17 11/3 -1/1 1 17 15/4 (-1/3,0/1) 0 17 34/9 0/1 2 1 19/5 -1/1 1 17 4/1 (-1/3,0/1) 0 17 17/4 0/1 8 1 13/3 1/5 1 17 9/2 (0/1,1/1) 0 17 5/1 1/1 1 17 17/3 -1/1 1 1 6/1 (-1/1,0/1) 0 17 7/1 -1/1 1 17 1/0 (-1/1,0/1) 0 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(16,-17,15,-16) (1/1,17/15) -> (1/1,17/15) Reflection Matrix(239,-272,210,-239) (17/15,8/7) -> (17/15,8/7) Reflection Matrix(237,-272,88,-101) (8/7,7/6) -> (8/3,27/10) Glide Reflection Matrix(188,-221,131,-154) (7/6,6/5) -> (10/7,23/16) Glide Reflection Matrix(169,-204,140,-169) (6/5,17/14) -> (6/5,17/14) Reflection Matrix(307,-374,252,-307) (17/14,11/9) -> (17/14,11/9) Reflection Matrix(305,-374,84,-103) (11/9,5/4) -> (29/8,11/3) Hyperbolic Matrix(256,-323,149,-188) (5/4,14/11) -> (12/7,31/18) Glide Reflection Matrix(373,-476,134,-171) (14/11,9/7) -> (25/9,14/5) Hyperbolic Matrix(118,-153,91,-118) (9/7,17/13) -> (9/7,17/13) Reflection Matrix(103,-136,78,-103) (17/13,4/3) -> (17/13,4/3) Reflection Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(273,-374,100,-137) (15/11,11/8) -> (19/7,11/4) Glide Reflection Matrix(86,-119,13,-18) (11/8,7/5) -> (6/1,7/1) Glide Reflection Matrix(169,-238,120,-169) (7/5,17/12) -> (7/5,17/12) Reflection Matrix(239,-340,168,-239) (17/12,10/7) -> (17/12,10/7) Reflection Matrix(307,-442,166,-239) (23/16,13/9) -> (11/6,13/7) Glide Reflection Matrix(305,-442,118,-171) (13/9,3/2) -> (31/12,13/5) Hyperbolic Matrix(67,-102,44,-67) (3/2,17/11) -> (3/2,17/11) Reflection Matrix(120,-187,77,-120) (17/11,11/7) -> (17/11,11/7) Reflection Matrix(118,-187,53,-84) (11/7,8/5) -> (11/5,9/4) Glide Reflection Matrix(52,-85,11,-18) (8/5,5/3) -> (9/2,5/1) Glide Reflection Matrix(101,-170,60,-101) (5/3,17/10) -> (5/3,17/10) Reflection Matrix(239,-408,140,-239) (17/10,12/7) -> (17/10,12/7) Reflection 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(135,-238,38,-67) (7/4,16/9) -> (7/2,18/5) Glide Reflection 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(118,-221,63,-118) (13/7,17/9) -> (13/7,17/9) Reflection Matrix(35,-68,18,-35) (17/9,2/1) -> (17/9,2/1) Reflection Matrix(86,-187,23,-50) (2/1,11/5) -> (11/3,15/4) Hyperbolic Matrix(271,-612,120,-271) (9/4,34/15) -> (9/4,34/15) Reflection Matrix(749,-1700,330,-749) (34/15,25/11) -> (34/15,25/11) 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(239,-612,66,-169) (5/2,18/7) -> (18/5,29/8) Hyperbolic Matrix(1189,-3060,462,-1189) (18/7,85/33) -> (18/7,85/33) Reflection Matrix(1616,-4165,627,-1616) (85/33,49/19) -> (85/33,49/19) Reflection Matrix(84,-221,19,-50) (13/5,8/3) -> (13/3,9/2) 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(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(271,-1020,72,-271) (15/4,34/9) -> (15/4,34/9) Reflection Matrix(341,-1292,90,-341) (34/9,19/5) -> (34/9,19/5) 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(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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(5,4,-6,-5) (0/1,1/1) -> (-1/1,-2/3) Matrix(16,-17,15,-16) -> Matrix(6,5,-7,-6) (1/1,17/15) -> (-1/1,-5/7) Matrix(239,-272,210,-239) -> Matrix(7,6,-8,-7) (17/15,8/7) -> (-1/1,-3/4) Matrix(237,-272,88,-101) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(188,-221,131,-154) -> Matrix(10,7,-17,-12) Matrix(169,-204,140,-169) -> Matrix(29,20,-42,-29) (6/5,17/14) -> (-5/7,-2/3) Matrix(307,-374,252,-307) -> Matrix(43,28,-66,-43) (17/14,11/9) -> (-2/3,-7/11) Matrix(305,-374,84,-103) -> Matrix(3,2,-14,-9) Matrix(256,-323,149,-188) -> Matrix(8,5,-19,-12) Matrix(373,-476,134,-171) -> Matrix(5,2,-8,-3) -1/2 Matrix(118,-153,91,-118) -> Matrix(4,3,-5,-4) (9/7,17/13) -> (-1/1,-3/5) Matrix(103,-136,78,-103) -> Matrix(5,4,-6,-5) (17/13,4/3) -> (-1/1,-2/3) Matrix(101,-136,26,-35) -> Matrix(3,2,-8,-5) -1/2 Matrix(273,-374,100,-137) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(86,-119,13,-18) -> Matrix(2,1,-3,-2) *** -> (-1/1,-1/3) Matrix(169,-238,120,-169) -> Matrix(5,4,-6,-5) (7/5,17/12) -> (-1/1,-2/3) Matrix(239,-340,168,-239) -> Matrix(7,4,-12,-7) (17/12,10/7) -> (-2/3,-1/2) Matrix(307,-442,166,-239) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(305,-442,118,-171) -> Matrix(5,2,-8,-3) -1/2 Matrix(67,-102,44,-67) -> Matrix(7,4,-12,-7) (3/2,17/11) -> (-2/3,-1/2) Matrix(120,-187,77,-120) -> Matrix(4,3,-5,-4) (17/11,11/7) -> (-1/1,-3/5) Matrix(118,-187,53,-84) -> Matrix(2,1,-7,-4) Matrix(52,-85,11,-18) -> Matrix(2,1,-3,-2) *** -> (-1/1,-1/3) Matrix(101,-170,60,-101) -> Matrix(11,6,-20,-11) (5/3,17/10) -> (-3/5,-1/2) Matrix(239,-408,140,-239) -> Matrix(13,6,-28,-13) (17/10,12/7) -> (-1/2,-3/7) Matrix(1223,-2108,474,-817) -> Matrix(27,10,-46,-17) Matrix(373,-646,138,-239) -> Matrix(5,2,-8,-3) -1/2 Matrix(135,-238,38,-67) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(305,-544,134,-239) -> Matrix(1,0,0,1) Matrix(205,-374,74,-135) -> Matrix(5,2,-8,-3) -1/2 Matrix(118,-221,63,-118) -> Matrix(2,1,-3,-2) (13/7,17/9) -> (-1/1,-1/3) Matrix(35,-68,18,-35) -> Matrix(3,2,-4,-3) (17/9,2/1) -> (-1/1,-1/2) Matrix(86,-187,23,-50) -> Matrix(2,1,-5,-2) (-1/1,-1/3).(-1/2,0/1) Matrix(271,-612,120,-271) -> Matrix(-1,0,2,1) (9/4,34/15) -> (-1/1,0/1) Matrix(749,-1700,330,-749) -> Matrix(3,2,-4,-3) (34/15,25/11) -> (-1/1,-1/2) Matrix(52,-119,7,-16) -> Matrix(2,1,-1,0) -1/1 Matrix(50,-119,21,-50) -> Matrix(2,1,-3,-2) (7/3,17/7) -> (-1/1,-1/3) Matrix(69,-170,28,-69) -> Matrix(5,4,-6,-5) (17/7,5/2) -> (-1/1,-2/3) Matrix(239,-612,66,-169) -> Matrix(3,2,-14,-9) Matrix(1189,-3060,462,-1189) -> Matrix(49,30,-80,-49) (18/7,85/33) -> (-5/8,-3/5) Matrix(1616,-4165,627,-1616) -> Matrix(86,51,-145,-86) (85/33,49/19) -> (-3/5,-17/29) Matrix(84,-221,19,-50) -> Matrix(2,1,1,0) Matrix(169,-476,60,-169) -> Matrix(23,12,-44,-23) (14/5,17/6) -> (-6/11,-1/2) Matrix(35,-102,12,-35) -> Matrix(5,2,-12,-5) (17/6,3/1) -> (-1/2,-1/3) Matrix(16,-51,5,-16) -> Matrix(2,1,-3,-2) (3/1,17/5) -> (-1/1,-1/3) Matrix(69,-238,20,-69) -> Matrix(-1,0,4,1) (17/5,7/2) -> (-1/2,0/1) Matrix(271,-1020,72,-271) -> Matrix(-1,0,6,1) (15/4,34/9) -> (-1/3,0/1) Matrix(341,-1292,90,-341) -> Matrix(-1,0,2,1) (34/9,19/5) -> (-1/1,0/1) Matrix(33,-136,8,-33) -> Matrix(-1,0,6,1) (4/1,17/4) -> (-1/3,0/1) Matrix(103,-442,24,-103) -> Matrix(1,0,10,-1) (17/4,13/3) -> (0/1,1/5) Matrix(16,-85,3,-16) -> Matrix(0,1,1,0) (5/1,17/3) -> (-1/1,1/1) Matrix(35,-204,6,-35) -> Matrix(-1,0,2,1) (17/3,6/1) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.