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 2/1 1/0 -17/2 1/0 -8/1 -2/1 1/0 -7/1 -2/1 -1/1 -20/3 -1/1 -1/2 -13/2 -1/1 -1/2 -19/3 -1/2 0/1 -25/4 0/1 1/0 -6/1 -1/2 0/1 -17/3 0/1 -11/2 0/1 1/2 -5/1 0/1 1/0 -24/5 -2/1 -1/1 -43/9 -2/1 1/0 -19/4 -1/1 1/0 -33/7 -2/1 1/0 -14/3 -1/1 1/0 -9/2 0/1 1/0 -22/5 0/1 1/0 -35/8 0/1 1/0 -13/3 1/1 1/0 -17/4 1/0 -4/1 -1/1 1/0 -15/4 -1/1 1/0 -11/3 0/1 1/0 -18/5 -2/1 1/0 -7/2 -3/2 -1/1 -17/5 -1/1 -10/3 -1/1 -4/5 -43/13 -4/5 -3/4 -33/10 -3/4 -2/3 -56/17 -3/4 -2/3 -23/7 -2/3 -1/2 -13/4 -1/1 -1/2 -3/1 -1/1 -1/2 -17/6 -1/2 -14/5 -1/2 -1/3 -11/4 -1/2 0/1 -19/7 -1/2 0/1 -8/3 0/1 1/0 -13/5 -1/1 -3/4 -18/7 -2/3 -1/2 -5/2 -1/1 -1/2 -17/7 -1/2 -12/5 -1/2 -2/5 -43/18 -1/2 -2/5 -74/31 -2/5 -3/8 -31/13 -1/2 -1/3 -19/8 -1/2 -1/3 -7/3 -1/3 0/1 -16/7 -1/4 0/1 -9/4 0/1 1/0 -11/5 -1/2 0/1 -13/6 -1/1 -1/2 -15/7 -2/3 -1/2 -17/8 -1/2 -2/1 -1/2 0/1 -17/9 -1/2 -15/8 -1/2 -1/3 -13/7 -1/2 -1/3 -11/6 -1/2 0/1 -9/5 -1/4 0/1 -34/19 0/1 -25/14 0/1 1/4 -16/9 0/1 1/0 -7/4 -1/1 -1/2 -26/15 -2/3 -5/8 -19/11 -2/3 -1/2 -31/18 -2/3 -3/5 -12/7 -4/7 -1/2 -17/10 -1/2 -5/3 -1/2 -2/5 -18/11 -2/5 -3/8 -85/52 -3/8 -67/41 -3/8 -4/11 -49/30 -3/8 -1/3 -31/19 -3/8 -1/3 -13/8 -3/8 -1/3 -34/21 -1/3 -21/13 -1/3 -3/10 -8/5 -1/2 0/1 -27/17 -1/3 0/1 -19/12 -1/2 -1/3 -11/7 -1/2 -2/5 -25/16 -2/5 -3/8 -14/9 -3/8 -1/3 -17/11 -1/3 -3/2 -1/3 0/1 -19/13 -1/4 0/1 -16/11 -1/2 0/1 -13/9 -1/3 -1/4 -23/16 -1/4 0/1 -33/23 -1/4 0/1 -10/7 -1/3 0/1 -17/12 -1/2 -1/4 -7/5 -1/3 0/1 -18/13 -1/2 0/1 -29/21 -1/4 0/1 -11/8 -1/2 0/1 -15/11 -1/2 0/1 -34/25 -1/2 -19/14 -1/2 -1/3 -4/3 -1/2 -1/3 -17/13 -1/3 -13/10 -1/3 -5/16 -48/37 -1/3 -5/16 -35/27 -5/16 -4/13 -57/44 -5/16 -4/13 -136/105 -4/13 -79/61 -4/13 -11/36 -22/17 -4/13 -3/10 -31/24 -1/3 -2/7 -102/79 -1/3 -71/55 -1/3 -5/16 -40/31 -4/13 -3/10 -9/7 -3/10 -2/7 -23/18 -2/7 -1/4 -14/11 -1/3 -1/4 -33/26 -3/10 -2/7 -85/67 -2/7 -52/41 -2/7 -5/18 -19/15 -2/7 -1/4 -43/34 -3/10 -2/7 -24/19 -2/7 -3/11 -29/23 -4/15 -1/4 -34/27 -1/4 -5/4 -1/3 -1/4 -16/13 -2/7 -1/4 -11/9 -4/15 -1/4 -17/14 -1/4 -6/5 -1/4 -2/9 -25/21 -1/4 0/1 -19/16 -1/4 -1/5 -13/11 -1/4 -1/5 -33/28 -1/4 -2/9 -20/17 -3/14 -1/5 -27/23 -6/29 -1/5 -34/29 -1/5 -7/6 -1/5 -1/6 -8/7 -1/8 0/1 -17/15 0/1 -9/8 0/1 1/0 -10/9 -1/3 0/1 -1/1 -1/4 0/1 0/1 0/1 1/1 0/1 1/4 9/8 0/1 1/0 17/15 0/1 8/7 0/1 1/8 7/6 1/6 1/5 20/17 1/5 3/14 13/11 1/5 1/4 19/16 1/5 1/4 25/21 0/1 1/4 6/5 2/9 1/4 17/14 1/4 11/9 1/4 4/15 5/4 1/4 1/3 24/19 3/11 2/7 43/34 2/7 3/10 19/15 1/4 2/7 33/26 2/7 3/10 14/11 1/4 1/3 9/7 2/7 3/10 22/17 3/10 4/13 35/27 4/13 5/16 13/10 5/16 1/3 17/13 1/3 4/3 1/3 1/2 15/11 0/1 1/2 11/8 0/1 1/2 18/13 0/1 1/2 7/5 0/1 1/3 17/12 1/4 1/2 10/7 0/1 1/3 43/30 0/1 1/0 33/23 0/1 1/4 56/39 0/1 1/2 23/16 0/1 1/4 13/9 1/4 1/3 3/2 0/1 1/3 17/11 1/3 14/9 1/3 3/8 11/7 2/5 1/2 19/12 1/3 1/2 8/5 0/1 1/2 13/8 1/3 3/8 18/11 3/8 2/5 5/3 2/5 1/2 17/10 1/2 12/7 1/2 4/7 43/25 1/2 2/3 74/43 1/2 4/7 31/18 3/5 2/3 19/11 1/2 2/3 7/4 1/2 1/1 16/9 0/1 1/0 9/5 0/1 1/4 11/6 0/1 1/2 13/7 1/3 1/2 15/8 1/3 1/2 17/9 1/2 2/1 0/1 1/2 17/8 1/2 15/7 1/2 2/3 13/6 1/2 1/1 11/5 0/1 1/2 9/4 0/1 1/0 34/15 0/1 25/11 0/1 1/8 16/7 0/1 1/4 7/3 0/1 1/3 26/11 0/1 1/2 19/8 1/3 1/2 31/13 1/3 1/2 12/5 2/5 1/2 17/7 1/2 5/2 1/2 1/1 18/7 1/2 2/3 85/33 2/3 67/26 2/3 7/10 49/19 2/3 3/4 31/12 2/3 1/1 13/5 3/4 1/1 34/13 1/1 21/8 1/1 3/2 8/3 0/1 1/0 27/10 1/3 1/2 19/7 0/1 1/2 11/4 0/1 1/2 25/9 0/1 1/4 14/5 1/3 1/2 17/6 1/2 3/1 1/2 1/1 19/6 1/2 1/1 16/5 1/2 2/3 13/4 1/2 1/1 23/7 1/2 2/3 33/10 2/3 3/4 10/3 4/5 1/1 17/5 1/1 7/2 1/1 3/2 18/5 2/1 1/0 29/8 3/1 1/0 11/3 0/1 1/0 15/4 1/1 1/0 34/9 1/0 19/5 0/1 1/0 4/1 1/1 1/0 17/4 1/0 13/3 -1/1 1/0 48/11 -1/1 -1/2 35/8 0/1 1/0 57/13 -1/2 0/1 136/31 0/1 79/18 0/1 1/2 22/5 0/1 1/0 31/7 -1/1 1/0 102/23 -1/1 71/16 -1/1 -2/3 40/9 -1/2 0/1 9/2 0/1 1/0 23/5 0/1 1/2 14/3 1/1 1/0 33/7 2/1 1/0 85/18 1/0 52/11 0/1 1/0 19/4 1/1 1/0 43/9 2/1 1/0 24/5 1/1 2/1 29/6 3/1 1/0 34/7 1/0 5/1 0/1 1/0 16/3 -1/2 0/1 11/2 -1/2 0/1 17/3 0/1 6/1 0/1 1/2 25/4 0/1 1/0 19/3 0/1 1/2 13/2 1/2 1/1 33/5 0/1 1/2 20/3 1/2 1/1 27/4 5/6 1/1 34/5 1/1 7/1 1/1 2/1 8/1 2/1 1/0 17/2 1/0 9/1 -2/1 1/0 10/1 -2/1 -1/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(33,340,-10,-103) (-9/1,1/0) -> (-43/13,-33/10) Hyperbolic Matrix(35,306,4,35) (-9/1,-17/2) -> (17/2,9/1) Hyperbolic Matrix(33,272,4,33) (-17/2,-8/1) -> (8/1,17/2) Hyperbolic Matrix(33,238,14,101) (-8/1,-7/1) -> (7/3,26/11) Hyperbolic Matrix(101,680,-86,-579) (-7/1,-20/3) -> (-20/17,-27/23) Hyperbolic Matrix(135,884,-104,-681) (-20/3,-13/2) -> (-13/10,-48/37) Hyperbolic Matrix(69,442,32,205) (-13/2,-19/3) -> (15/7,13/6) Hyperbolic Matrix(205,1292,162,1021) (-19/3,-25/4) -> (43/34,19/15) Hyperbolic Matrix(169,1054,38,237) (-25/4,-6/1) -> (40/9,9/2) Hyperbolic Matrix(35,204,6,35) (-6/1,-17/3) -> (17/3,6/1) Hyperbolic Matrix(67,374,12,67) (-17/3,-11/2) -> (11/2,17/3) Hyperbolic Matrix(69,374,-50,-271) (-11/2,-5/1) -> (-29/21,-11/8) Hyperbolic Matrix(169,816,-134,-647) (-5/1,-24/5) -> (-24/19,-29/23) Hyperbolic Matrix(135,646,14,67) (-24/5,-43/9) -> (9/1,10/1) Hyperbolic Matrix(271,1292,228,1087) (-43/9,-19/4) -> (19/16,25/21) Hyperbolic Matrix(611,2890,-374,-1769) (-19/4,-33/7) -> (-67/41,-49/30) Hyperbolic Matrix(441,2074,-340,-1599) (-33/7,-14/3) -> (-48/37,-35/27) Hyperbolic Matrix(103,476,-66,-305) (-14/3,-9/2) -> (-25/16,-14/9) Hyperbolic Matrix(239,1054,-100,-441) (-9/2,-22/5) -> (-12/5,-43/18) Hyperbolic Matrix(613,2686,-186,-815) (-22/5,-35/8) -> (-33/10,-56/17) Hyperbolic Matrix(203,884,-172,-749) (-35/8,-13/3) -> (-13/11,-33/28) Hyperbolic Matrix(103,442,24,103) (-13/3,-17/4) -> (17/4,13/3) Hyperbolic Matrix(33,136,8,33) (-17/4,-4/1) -> (4/1,17/4) Hyperbolic Matrix(35,136,-26,-101) (-4/1,-15/4) -> (-19/14,-4/3) Hyperbolic Matrix(101,374,64,237) (-15/4,-11/3) -> (11/7,19/12) Hyperbolic Matrix(103,374,-84,-305) (-11/3,-18/5) -> (-16/13,-11/9) Hyperbolic Matrix(67,238,38,135) (-18/5,-7/2) -> (7/4,16/9) Hyperbolic Matrix(69,238,20,69) (-7/2,-17/5) -> (17/5,7/2) Hyperbolic Matrix(101,340,30,101) (-17/5,-10/3) -> (10/3,17/5) Hyperbolic Matrix(441,1462,92,305) (-10/3,-43/13) -> (43/9,24/5) Hyperbolic Matrix(1497,4930,-1156,-3807) (-56/17,-23/7) -> (-79/61,-22/17) Hyperbolic Matrix(135,442,62,203) (-23/7,-13/4) -> (13/6,11/5) Hyperbolic Matrix(137,442,-84,-271) (-13/4,-3/1) -> (-31/19,-13/8) Hyperbolic Matrix(35,102,12,35) (-3/1,-17/6) -> (17/6,3/1) Hyperbolic Matrix(169,476,60,169) (-17/6,-14/5) -> (14/5,17/6) Hyperbolic Matrix(171,476,-134,-373) (-14/5,-11/4) -> (-23/18,-14/11) Hyperbolic Matrix(137,374,100,273) (-11/4,-19/7) -> (15/11,11/8) Hyperbolic Matrix(239,646,-138,-373) (-19/7,-8/3) -> (-26/15,-19/11) Hyperbolic Matrix(103,272,-64,-169) (-8/3,-13/5) -> (-21/13,-8/5) Hyperbolic Matrix(171,442,-118,-305) (-13/5,-18/7) -> (-16/11,-13/9) Hyperbolic Matrix(67,170,-54,-137) (-18/7,-5/2) -> (-5/4,-16/13) Hyperbolic Matrix(69,170,28,69) (-5/2,-17/7) -> (17/7,5/2) Hyperbolic Matrix(169,408,70,169) (-17/7,-12/5) -> (12/5,17/7) Hyperbolic Matrix(883,2108,142,339) (-43/18,-74/31) -> (6/1,25/4) Hyperbolic Matrix(2721,6494,-2108,-5031) (-74/31,-31/13) -> (-71/55,-40/31) Hyperbolic Matrix(885,2108,-542,-1291) (-31/13,-19/8) -> (-49/30,-31/19) Hyperbolic Matrix(273,646,-172,-407) (-19/8,-7/3) -> (-27/17,-19/12) Hyperbolic Matrix(103,238,74,171) (-7/3,-16/7) -> (18/13,7/5) Hyperbolic Matrix(239,544,-134,-305) (-16/7,-9/4) -> (-25/14,-16/9) Hyperbolic Matrix(169,374,-108,-239) (-9/4,-11/5) -> (-11/7,-25/16) Hyperbolic Matrix(203,442,62,135) (-11/5,-13/6) -> (13/4,23/7) Hyperbolic Matrix(205,442,32,69) (-13/6,-15/7) -> (19/3,13/2) Hyperbolic Matrix(239,510,112,239) (-15/7,-17/8) -> (17/8,15/7) Hyperbolic Matrix(33,68,16,33) (-17/8,-2/1) -> (2/1,17/8) Hyperbolic Matrix(35,68,18,35) (-2/1,-17/9) -> (17/9,2/1) Hyperbolic Matrix(271,510,144,271) (-17/9,-15/8) -> (15/8,17/9) Hyperbolic Matrix(237,442,200,373) (-15/8,-13/7) -> (13/11,19/16) Hyperbolic Matrix(239,442,166,307) (-13/7,-11/6) -> (23/16,13/9) Hyperbolic Matrix(169,306,-132,-239) (-11/6,-9/5) -> (-9/7,-23/18) Hyperbolic Matrix(645,1156,284,509) (-9/5,-34/19) -> (34/15,25/11) Hyperbolic Matrix(647,1156,286,511) (-34/19,-25/14) -> (9/4,34/15) Hyperbolic Matrix(135,238,38,67) (-16/9,-7/4) -> (7/2,18/5) Hyperbolic Matrix(137,238,118,205) (-7/4,-26/15) -> (8/7,7/6) Hyperbolic Matrix(375,646,-256,-441) (-19/11,-31/18) -> (-3/2,-19/13) Hyperbolic Matrix(613,1054,-474,-815) (-31/18,-12/7) -> (-22/17,-31/24) Hyperbolic Matrix(239,408,140,239) (-12/7,-17/10) -> (17/10,12/7) Hyperbolic Matrix(101,170,60,101) (-17/10,-5/3) -> (5/3,17/10) Hyperbolic Matrix(373,612,-270,-443) (-5/3,-18/11) -> (-18/13,-29/21) Hyperbolic Matrix(3639,5950,770,1259) (-18/11,-85/52) -> (85/18,52/11) Hyperbolic Matrix(5201,8500,1102,1801) (-85/52,-67/41) -> (33/7,85/18) Hyperbolic Matrix(713,1156,272,441) (-13/8,-34/21) -> (34/13,21/8) Hyperbolic Matrix(715,1156,274,443) (-34/21,-21/13) -> (13/5,34/13) Hyperbolic Matrix(171,272,22,35) (-8/5,-27/17) -> (7/1,8/1) Hyperbolic Matrix(237,374,64,101) (-19/12,-11/7) -> (11/3,15/4) Hyperbolic Matrix(307,476,198,307) (-14/9,-17/11) -> (17/11,14/9) Hyperbolic Matrix(67,102,44,67) (-17/11,-3/2) -> (3/2,17/11) Hyperbolic Matrix(885,1292,-698,-1019) (-19/13,-16/11) -> (-52/41,-19/15) Hyperbolic Matrix(307,442,166,239) (-13/9,-23/16) -> (11/6,13/7) Hyperbolic Matrix(1871,2686,-1444,-2073) (-23/16,-33/23) -> (-35/27,-57/44) Hyperbolic Matrix(237,340,-214,-307) (-33/23,-10/7) -> (-10/9,-1/1) Hyperbolic Matrix(239,340,168,239) (-10/7,-17/12) -> (17/12,10/7) Hyperbolic Matrix(169,238,120,169) (-17/12,-7/5) -> (7/5,17/12) Hyperbolic Matrix(171,238,74,103) (-7/5,-18/13) -> (16/7,7/3) Hyperbolic Matrix(273,374,100,137) (-11/8,-15/11) -> (19/7,11/4) Hyperbolic Matrix(849,1156,224,305) (-15/11,-34/25) -> (34/9,19/5) Hyperbolic Matrix(851,1156,226,307) (-34/25,-19/14) -> (15/4,34/9) Hyperbolic Matrix(103,136,78,103) (-4/3,-17/13) -> (17/13,4/3) Hyperbolic Matrix(339,442,260,339) (-17/13,-13/10) -> (13/10,17/13) Hyperbolic Matrix(14279,18496,3254,4215) (-57/44,-136/105) -> (136/31,79/18) Hyperbolic Matrix(14281,18496,3256,4217) (-136/105,-79/61) -> (57/13,136/31) Hyperbolic Matrix(8057,10404,1816,2345) (-31/24,-102/79) -> (102/23,71/16) Hyperbolic Matrix(8059,10404,1818,2347) (-102/79,-71/55) -> (31/7,102/23) Hyperbolic Matrix(817,1054,686,885) (-40/31,-9/7) -> (25/21,6/5) Hyperbolic Matrix(883,1122,-750,-953) (-14/11,-33/26) -> (-33/28,-20/17) Hyperbolic Matrix(6699,8500,2600,3299) (-33/26,-85/67) -> (85/33,67/26) Hyperbolic Matrix(4691,5950,1822,2311) (-85/67,-52/41) -> (18/7,85/33) Hyperbolic Matrix(1021,1292,162,205) (-19/15,-43/34) -> (25/4,19/3) Hyperbolic Matrix(1157,1462,808,1021) (-43/34,-24/19) -> (10/7,43/30) Hyperbolic Matrix(917,1156,188,237) (-29/23,-34/27) -> (34/7,5/1) Hyperbolic Matrix(919,1156,190,239) (-34/27,-5/4) -> (29/6,34/7) Hyperbolic Matrix(307,374,252,307) (-11/9,-17/14) -> (17/14,11/9) Hyperbolic Matrix(169,204,140,169) (-17/14,-6/5) -> (6/5,17/14) Hyperbolic Matrix(1769,2108,1028,1225) (-6/5,-25/21) -> (43/25,74/43) Hyperbolic Matrix(1087,1292,228,271) (-25/21,-19/16) -> (19/4,43/9) Hyperbolic Matrix(373,442,200,237) (-19/16,-13/11) -> (13/7,15/8) Hyperbolic Matrix(985,1156,144,169) (-27/23,-34/29) -> (34/5,7/1) Hyperbolic Matrix(987,1156,146,171) (-34/29,-7/6) -> (27/4,34/5) Hyperbolic Matrix(237,272,88,101) (-7/6,-8/7) -> (8/3,27/10) Hyperbolic Matrix(239,272,210,239) (-8/7,-17/15) -> (17/15,8/7) Hyperbolic Matrix(271,306,240,271) (-17/15,-9/8) -> (9/8,17/15) Hyperbolic Matrix(579,646,458,511) (-9/8,-10/9) -> (24/19,43/34) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(307,-340,214,-237) (1/1,9/8) -> (43/30,33/23) Hyperbolic Matrix(579,-680,86,-101) (7/6,20/17) -> (20/3,27/4) Hyperbolic Matrix(749,-884,172,-203) (20/17,13/11) -> (13/3,48/11) Hyperbolic Matrix(305,-374,84,-103) (11/9,5/4) -> (29/8,11/3) Hyperbolic Matrix(647,-816,134,-169) (5/4,24/19) -> (24/5,29/6) Hyperbolic Matrix(2279,-2890,884,-1121) (19/15,33/26) -> (67/26,49/19) Hyperbolic Matrix(1633,-2074,374,-475) (33/26,14/11) -> (48/11,35/8) Hyperbolic Matrix(373,-476,134,-171) (14/11,9/7) -> (25/9,14/5) Hyperbolic Matrix(815,-1054,474,-613) (9/7,22/17) -> (12/7,43/25) Hyperbolic Matrix(2073,-2686,1444,-1871) (22/17,35/27) -> (33/23,56/39) Hyperbolic Matrix(681,-884,104,-135) (35/27,13/10) -> (13/2,33/5) Hyperbolic Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(271,-374,50,-69) (11/8,18/13) -> (16/3,11/2) Hyperbolic Matrix(3433,-4930,782,-1123) (56/39,23/16) -> (79/18,22/5) Hyperbolic Matrix(305,-442,118,-171) (13/9,3/2) -> (31/12,13/5) Hyperbolic Matrix(305,-476,66,-103) (14/9,11/7) -> (23/5,14/3) Hyperbolic Matrix(407,-646,172,-273) (19/12,8/5) -> (26/11,19/8) Hyperbolic Matrix(169,-272,64,-103) (8/5,13/8) -> (21/8,8/3) Hyperbolic Matrix(271,-442,84,-137) (13/8,18/11) -> (16/5,13/4) Hyperbolic Matrix(103,-170,20,-33) (18/11,5/3) -> (5/1,16/3) Hyperbolic Matrix(3773,-6494,850,-1463) (74/43,31/18) -> (71/16,40/9) Hyperbolic Matrix(1223,-2108,474,-817) (31/18,19/11) -> (49/19,31/12) Hyperbolic Matrix(373,-646,138,-239) (19/11,7/4) -> (27/10,19/7) Hyperbolic Matrix(305,-544,134,-239) (16/9,9/5) -> (25/11,16/7) Hyperbolic Matrix(205,-374,74,-135) (9/5,11/6) -> (11/4,25/9) Hyperbolic Matrix(137,-306,30,-67) (11/5,9/4) -> (9/2,23/5) Hyperbolic Matrix(271,-646,86,-205) (19/8,31/13) -> (3/1,19/6) Hyperbolic Matrix(441,-1054,100,-239) (31/13,12/5) -> (22/5,31/7) Hyperbolic Matrix(239,-612,66,-169) (5/2,18/7) -> (18/5,29/8) Hyperbolic Matrix(407,-1292,86,-273) (19/6,16/5) -> (52/11,19/4) Hyperbolic Matrix(815,-2686,186,-613) (23/7,33/10) -> (35/8,57/13) Hyperbolic Matrix(103,-340,10,-33) (33/10,10/3) -> (10/1,1/0) Hyperbolic Matrix(239,-1122,36,-169) (14/3,33/7) -> (33/5,20/3) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(33,340,-10,-103) -> Matrix(3,-2,-4,3) Matrix(35,306,4,35) -> Matrix(1,-4,0,1) Matrix(33,272,4,33) -> Matrix(1,4,0,1) Matrix(33,238,14,101) -> Matrix(1,2,2,5) Matrix(101,680,-86,-579) -> Matrix(5,4,-24,-19) Matrix(135,884,-104,-681) -> Matrix(3,4,-10,-13) Matrix(69,442,32,205) -> Matrix(3,2,4,3) Matrix(205,1292,162,1021) -> Matrix(3,2,10,7) Matrix(169,1054,38,237) -> Matrix(1,0,0,1) Matrix(35,204,6,35) -> Matrix(1,0,4,1) Matrix(67,374,12,67) -> Matrix(1,0,-4,1) Matrix(69,374,-50,-271) -> Matrix(1,0,-4,1) Matrix(169,816,-134,-647) -> Matrix(1,4,-4,-15) Matrix(135,646,14,67) -> Matrix(1,0,0,1) Matrix(271,1292,228,1087) -> Matrix(1,2,4,9) Matrix(611,2890,-374,-1769) -> Matrix(3,2,-8,-5) Matrix(441,2074,-340,-1599) -> Matrix(5,6,-16,-19) Matrix(103,476,-66,-305) -> Matrix(3,2,-8,-5) Matrix(239,1054,-100,-441) -> Matrix(1,-2,-2,5) Matrix(613,2686,-186,-815) -> Matrix(3,-2,-4,3) Matrix(203,884,-172,-749) -> Matrix(1,-2,-4,9) Matrix(103,442,24,103) -> Matrix(1,-2,0,1) Matrix(33,136,8,33) -> Matrix(1,2,0,1) Matrix(35,136,-26,-101) -> Matrix(1,0,-2,1) Matrix(101,374,64,237) -> Matrix(1,2,2,5) Matrix(103,374,-84,-305) -> Matrix(1,4,-4,-15) Matrix(67,238,38,135) -> Matrix(1,2,0,1) Matrix(69,238,20,69) -> Matrix(5,6,4,5) Matrix(101,340,30,101) -> Matrix(9,8,10,9) Matrix(441,1462,92,305) -> Matrix(3,2,4,3) Matrix(1497,4930,-1156,-3807) -> Matrix(25,18,-82,-59) Matrix(135,442,62,203) -> Matrix(3,2,4,3) Matrix(137,442,-84,-271) -> Matrix(5,4,-14,-11) Matrix(35,102,12,35) -> Matrix(3,2,4,3) Matrix(169,476,60,169) -> Matrix(5,2,12,5) Matrix(171,476,-134,-373) -> Matrix(5,2,-18,-7) Matrix(137,374,100,273) -> Matrix(1,0,4,1) Matrix(239,646,-138,-373) -> Matrix(5,2,-8,-3) Matrix(103,272,-64,-169) -> Matrix(1,0,-2,1) Matrix(171,442,-118,-305) -> Matrix(3,2,-8,-5) Matrix(67,170,-54,-137) -> Matrix(1,0,-2,1) Matrix(69,170,28,69) -> Matrix(3,2,4,3) Matrix(169,408,70,169) -> Matrix(9,4,20,9) Matrix(883,2108,142,339) -> Matrix(5,2,2,1) Matrix(2721,6494,-2108,-5031) -> Matrix(17,6,-54,-19) Matrix(885,2108,-542,-1291) -> Matrix(7,2,-18,-5) Matrix(273,646,-172,-407) -> Matrix(1,0,0,1) Matrix(103,238,74,171) -> Matrix(1,0,6,1) Matrix(239,544,-134,-305) -> Matrix(1,0,4,1) Matrix(169,374,-108,-239) -> Matrix(3,2,-8,-5) Matrix(203,442,62,135) -> Matrix(3,2,4,3) Matrix(205,442,32,69) -> Matrix(3,2,4,3) Matrix(239,510,112,239) -> Matrix(7,4,12,7) Matrix(33,68,16,33) -> Matrix(1,0,4,1) Matrix(35,68,18,35) -> Matrix(1,0,4,1) Matrix(271,510,144,271) -> Matrix(5,2,12,5) Matrix(237,442,200,373) -> Matrix(5,2,22,9) Matrix(239,442,166,307) -> Matrix(1,0,6,1) Matrix(169,306,-132,-239) -> Matrix(5,2,-18,-7) Matrix(645,1156,284,509) -> Matrix(1,0,12,1) Matrix(647,1156,286,511) -> Matrix(1,0,-4,1) Matrix(135,238,38,67) -> Matrix(1,2,0,1) Matrix(137,238,118,205) -> Matrix(3,2,16,11) Matrix(375,646,-256,-441) -> Matrix(3,2,-14,-9) Matrix(613,1054,-474,-815) -> Matrix(13,8,-44,-27) Matrix(239,408,140,239) -> Matrix(15,8,28,15) Matrix(101,170,60,101) -> Matrix(9,4,20,9) Matrix(373,612,-270,-443) -> Matrix(5,2,-18,-7) Matrix(3639,5950,770,1259) -> Matrix(5,2,-8,-3) Matrix(5201,8500,1102,1801) -> Matrix(27,10,8,3) Matrix(713,1156,272,441) -> Matrix(17,6,14,5) Matrix(715,1156,274,443) -> Matrix(19,6,22,7) Matrix(171,272,22,35) -> Matrix(5,2,2,1) Matrix(237,374,64,101) -> Matrix(5,2,2,1) Matrix(307,476,198,307) -> Matrix(17,6,48,17) Matrix(67,102,44,67) -> Matrix(1,0,6,1) Matrix(885,1292,-698,-1019) -> Matrix(9,2,-32,-7) Matrix(307,442,166,239) -> Matrix(1,0,6,1) Matrix(1871,2686,-1444,-2073) -> Matrix(11,4,-36,-13) Matrix(237,340,-214,-307) -> Matrix(1,0,0,1) Matrix(239,340,168,239) -> Matrix(1,0,6,1) Matrix(169,238,120,169) -> Matrix(1,0,6,1) Matrix(171,238,74,103) -> Matrix(1,0,6,1) Matrix(273,374,100,137) -> Matrix(1,0,4,1) Matrix(849,1156,224,305) -> Matrix(1,0,2,1) Matrix(851,1156,226,307) -> Matrix(5,2,2,1) Matrix(103,136,78,103) -> Matrix(5,2,12,5) Matrix(339,442,260,339) -> Matrix(31,10,96,31) Matrix(14279,18496,3254,4215) -> Matrix(13,4,42,13) Matrix(14281,18496,3256,4217) -> Matrix(13,4,-62,-19) Matrix(8057,10404,1816,2345) -> Matrix(1,0,2,1) Matrix(8059,10404,1818,2347) -> Matrix(19,6,-16,-5) Matrix(817,1054,686,885) -> Matrix(7,2,38,11) Matrix(883,1122,-750,-953) -> Matrix(13,4,-62,-19) Matrix(6699,8500,2600,3299) -> Matrix(69,20,100,29) Matrix(4691,5950,1822,2311) -> Matrix(43,12,68,19) Matrix(1021,1292,162,205) -> Matrix(7,2,10,3) Matrix(1157,1462,808,1021) -> Matrix(7,2,10,3) Matrix(917,1156,188,237) -> Matrix(15,4,-4,-1) Matrix(919,1156,190,239) -> Matrix(9,2,4,1) Matrix(307,374,252,307) -> Matrix(31,8,120,31) Matrix(169,204,140,169) -> Matrix(17,4,72,17) Matrix(1769,2108,1028,1225) -> Matrix(7,2,10,3) Matrix(1087,1292,228,271) -> Matrix(9,2,4,1) Matrix(373,442,200,237) -> Matrix(9,2,22,5) Matrix(985,1156,144,169) -> Matrix(39,8,34,7) Matrix(987,1156,146,171) -> Matrix(31,6,36,7) Matrix(237,272,88,101) -> Matrix(1,0,8,1) Matrix(239,272,210,239) -> Matrix(1,0,16,1) Matrix(271,306,240,271) -> Matrix(1,0,0,1) Matrix(579,646,458,511) -> Matrix(3,2,10,7) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(307,-340,214,-237) -> Matrix(1,0,0,1) Matrix(579,-680,86,-101) -> Matrix(19,-4,24,-5) Matrix(749,-884,172,-203) -> Matrix(9,-2,-4,1) Matrix(305,-374,84,-103) -> Matrix(15,-4,4,-1) Matrix(647,-816,134,-169) -> Matrix(15,-4,4,-1) Matrix(2279,-2890,884,-1121) -> Matrix(29,-8,40,-11) Matrix(1633,-2074,374,-475) -> Matrix(7,-2,-10,3) Matrix(373,-476,134,-171) -> Matrix(7,-2,18,-5) Matrix(815,-1054,474,-613) -> Matrix(27,-8,44,-13) Matrix(2073,-2686,1444,-1871) -> Matrix(13,-4,36,-11) Matrix(681,-884,104,-135) -> Matrix(13,-4,10,-3) Matrix(101,-136,26,-35) -> Matrix(1,0,-2,1) Matrix(271,-374,50,-69) -> Matrix(1,0,-4,1) Matrix(3433,-4930,782,-1123) -> Matrix(1,0,-2,1) Matrix(305,-442,118,-171) -> Matrix(5,-2,8,-3) Matrix(305,-476,66,-103) -> Matrix(5,-2,8,-3) Matrix(407,-646,172,-273) -> Matrix(1,0,0,1) Matrix(169,-272,64,-103) -> Matrix(1,0,-2,1) Matrix(271,-442,84,-137) -> Matrix(11,-4,14,-5) Matrix(103,-170,20,-33) -> Matrix(5,-2,-2,1) Matrix(3773,-6494,850,-1463) -> Matrix(7,-4,-12,7) Matrix(1223,-2108,474,-817) -> Matrix(13,-8,18,-11) Matrix(373,-646,138,-239) -> Matrix(3,-2,8,-5) Matrix(305,-544,134,-239) -> Matrix(1,0,4,1) Matrix(205,-374,74,-135) -> Matrix(1,0,0,1) Matrix(137,-306,30,-67) -> Matrix(1,0,0,1) Matrix(271,-646,86,-205) -> Matrix(5,-2,8,-3) Matrix(441,-1054,100,-239) -> Matrix(5,-2,-2,1) Matrix(239,-612,66,-169) -> Matrix(7,-4,2,-1) Matrix(407,-1292,86,-273) -> Matrix(3,-2,2,-1) Matrix(815,-2686,186,-613) -> Matrix(3,-2,-4,3) Matrix(103,-340,10,-33) -> Matrix(3,-2,-4,3) Matrix(239,-1122,36,-169) -> 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 Degree of the the map X: 32 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 17/14 17/13 17/12 3/2 17/11 17/10 2/1 17/8 34/15 17/7 5/2 17/6 3/1 17/5 7/2 15/4 34/9 4/1 17/4 9/2 85/18 5/1 11/2 17/3 6/1 13/2 7/1 8/1 17/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -2/1 1/0 -7/1 -2/1 -1/1 -6/1 -1/2 0/1 -17/3 0/1 -11/2 0/1 1/2 -5/1 0/1 1/0 -14/3 -1/1 1/0 -9/2 0/1 1/0 -4/1 -1/1 1/0 -15/4 -1/1 1/0 -11/3 0/1 1/0 -7/2 -3/2 -1/1 -17/5 -1/1 -10/3 -1/1 -4/5 -13/4 -1/1 -1/2 -3/1 -1/1 -1/2 -11/4 -1/2 0/1 -8/3 0/1 1/0 -5/2 -1/1 -1/2 -17/7 -1/2 -12/5 -1/2 -2/5 -19/8 -1/2 -1/3 -7/3 -1/3 0/1 -16/7 -1/4 0/1 -9/4 0/1 1/0 -11/5 -1/2 0/1 -13/6 -1/1 -1/2 -2/1 -1/2 0/1 -9/5 -1/4 0/1 -34/19 0/1 -25/14 0/1 1/4 -16/9 0/1 1/0 -7/4 -1/1 -1/2 -5/3 -1/2 -2/5 -18/11 -2/5 -3/8 -31/19 -3/8 -1/3 -13/8 -3/8 -1/3 -8/5 -1/2 0/1 -27/17 -1/3 0/1 -19/12 -1/2 -1/3 -11/7 -1/2 -2/5 -25/16 -2/5 -3/8 -14/9 -3/8 -1/3 -17/11 -1/3 -3/2 -1/3 0/1 -7/5 -1/3 0/1 -18/13 -1/2 0/1 -29/21 -1/4 0/1 -11/8 -1/2 0/1 -15/11 -1/2 0/1 -34/25 -1/2 -19/14 -1/2 -1/3 -4/3 -1/2 -1/3 -5/4 -1/3 -1/4 -6/5 -1/4 -2/9 -1/1 -1/4 0/1 0/1 0/1 1/1 0/1 1/4 7/6 1/6 1/5 6/5 2/9 1/4 17/14 1/4 11/9 1/4 4/15 5/4 1/4 1/3 9/7 2/7 3/10 13/10 5/16 1/3 17/13 1/3 4/3 1/3 1/2 15/11 0/1 1/2 11/8 0/1 1/2 18/13 0/1 1/2 7/5 0/1 1/3 17/12 1/4 1/2 10/7 0/1 1/3 3/2 0/1 1/3 17/11 1/3 14/9 1/3 3/8 11/7 2/5 1/2 19/12 1/3 1/2 8/5 0/1 1/2 13/8 1/3 3/8 18/11 3/8 2/5 5/3 2/5 1/2 17/10 1/2 12/7 1/2 4/7 7/4 1/2 1/1 16/9 0/1 1/0 9/5 0/1 1/4 11/6 0/1 1/2 2/1 0/1 1/2 17/8 1/2 15/7 1/2 2/3 13/6 1/2 1/1 11/5 0/1 1/2 9/4 0/1 1/0 34/15 0/1 25/11 0/1 1/8 16/7 0/1 1/4 7/3 0/1 1/3 26/11 0/1 1/2 19/8 1/3 1/2 31/13 1/3 1/2 12/5 2/5 1/2 17/7 1/2 5/2 1/2 1/1 13/5 3/4 1/1 34/13 1/1 21/8 1/1 3/2 8/3 0/1 1/0 19/7 0/1 1/2 11/4 0/1 1/2 14/5 1/3 1/2 17/6 1/2 3/1 1/2 1/1 19/6 1/2 1/1 16/5 1/2 2/3 13/4 1/2 1/1 23/7 1/2 2/3 10/3 4/5 1/1 17/5 1/1 7/2 1/1 3/2 18/5 2/1 1/0 11/3 0/1 1/0 15/4 1/1 1/0 34/9 1/0 19/5 0/1 1/0 4/1 1/1 1/0 17/4 1/0 13/3 -1/1 1/0 9/2 0/1 1/0 23/5 0/1 1/2 14/3 1/1 1/0 33/7 2/1 1/0 85/18 1/0 52/11 0/1 1/0 19/4 1/1 1/0 5/1 0/1 1/0 16/3 -1/2 0/1 11/2 -1/2 0/1 17/3 0/1 6/1 0/1 1/2 13/2 1/2 1/1 7/1 1/1 2/1 8/1 2/1 1/0 17/2 1/0 9/1 -2/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,17,0,1) (-8/1,1/0) -> (9/1,1/0) Parabolic Matrix(33,238,14,101) (-8/1,-7/1) -> (7/3,26/11) Hyperbolic Matrix(33,221,10,67) (-7/1,-6/1) -> (23/7,10/3) 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(67,323,28,135) (-5/1,-14/3) -> (31/13,12/5) Hyperbolic Matrix(103,476,-66,-305) (-14/3,-9/2) -> (-25/16,-14/9) Hyperbolic Matrix(35,153,8,35) (-9/2,-4/1) -> (13/3,9/2) 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(33,119,28,101) (-11/3,-7/2) -> (7/6,6/5) 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(67,221,10,33) (-10/3,-13/4) -> (13/2,7/1) Hyperbolic Matrix(137,442,-84,-271) (-13/4,-3/1) -> (-31/19,-13/8) Hyperbolic Matrix(67,187,24,67) (-3/1,-11/4) -> (11/4,14/5) Hyperbolic Matrix(69,187,38,103) (-11/4,-8/3) -> (9/5,11/6) Hyperbolic Matrix(33,85,26,67) (-8/3,-5/2) -> (5/4,9/7) 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(135,323,28,67) (-12/5,-19/8) -> (19/4,5/1) 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(103,221,48,103) (-13/6,-2/1) -> (15/7,13/6) Hyperbolic Matrix(103,187,38,69) (-2/1,-9/5) -> (8/3,19/7) 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(69,119,40,69) (-7/4,-5/3) -> (12/7,7/4) Hyperbolic Matrix(373,612,-270,-443) (-5/3,-18/11) -> (-18/13,-29/21) Hyperbolic Matrix(781,1275,166,271) (-18/11,-31/19) -> (14/3,33/7) Hyperbolic Matrix(137,221,106,171) (-13/8,-8/5) -> (9/7,13/10) 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(35,51,24,35) (-3/2,-7/5) -> (10/7,3/2) 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(67,85,26,33) (-4/3,-5/4) -> (5/2,13/5) Hyperbolic Matrix(69,85,56,69) (-5/4,-6/5) -> (11/9,5/4) Hyperbolic Matrix(101,119,28,33) (-6/5,-1/1) -> (18/5,11/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(103,-119,58,-67) (1/1,7/6) -> (7/4,16/9) Hyperbolic Matrix(239,-289,196,-237) (6/5,17/14) -> (17/14,11/9) Parabolic Matrix(613,-799,234,-305) (13/10,17/13) -> (34/13,21/8) Hyperbolic Matrix(271,-357,104,-137) (17/13,4/3) -> (13/5,34/13) Hyperbolic Matrix(101,-136,26,-35) (4/3,15/11) -> (19/5,4/1) Hyperbolic Matrix(137,-187,74,-101) (15/11,11/8) -> (11/6,2/1) Hyperbolic Matrix(271,-374,50,-69) (11/8,18/13) -> (16/3,11/2) Hyperbolic Matrix(205,-289,144,-203) (7/5,17/12) -> (17/12,10/7) Parabolic 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(171,-289,100,-169) (5/3,17/10) -> (17/10,12/7) Parabolic Matrix(305,-544,134,-239) (16/9,9/5) -> (25/11,16/7) Hyperbolic Matrix(137,-289,64,-135) (2/1,17/8) -> (17/8,15/7) Parabolic Matrix(101,-221,16,-35) (13/6,11/5) -> (6/1,13/2) 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(103,-289,36,-101) (14/5,17/6) -> (17/6,3/1) Parabolic Matrix(407,-1292,86,-273) (19/6,16/5) -> (52/11,19/4) Hyperbolic Matrix(69,-289,16,-67) (4/1,17/4) -> (17/4,13/3) Parabolic Matrix(1531,-7225,324,-1529) (33/7,85/18) -> (85/18,52/11) Parabolic Matrix(35,-289,4,-33) (8/1,17/2) -> (17/2,9/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,17,0,1) -> Matrix(1,0,0,1) Matrix(33,238,14,101) -> Matrix(1,2,2,5) Matrix(33,221,10,67) -> Matrix(3,2,4,3) Matrix(35,204,6,35) -> Matrix(1,0,4,1) Matrix(67,374,12,67) -> Matrix(1,0,-4,1) Matrix(69,374,-50,-271) -> Matrix(1,0,-4,1) Matrix(67,323,28,135) -> Matrix(1,2,2,5) Matrix(103,476,-66,-305) -> Matrix(3,2,-8,-5) Matrix(35,153,8,35) -> Matrix(1,0,0,1) Matrix(35,136,-26,-101) -> Matrix(1,0,-2,1) Matrix(101,374,64,237) -> Matrix(1,2,2,5) Matrix(33,119,28,101) -> Matrix(1,2,4,9) Matrix(69,238,20,69) -> Matrix(5,6,4,5) Matrix(101,340,30,101) -> Matrix(9,8,10,9) Matrix(67,221,10,33) -> Matrix(3,2,4,3) Matrix(137,442,-84,-271) -> Matrix(5,4,-14,-11) Matrix(67,187,24,67) -> Matrix(1,0,4,1) Matrix(69,187,38,103) -> Matrix(1,0,4,1) Matrix(33,85,26,67) -> Matrix(3,2,10,7) Matrix(69,170,28,69) -> Matrix(3,2,4,3) Matrix(169,408,70,169) -> Matrix(9,4,20,9) Matrix(135,323,28,67) -> Matrix(5,2,2,1) Matrix(273,646,-172,-407) -> Matrix(1,0,0,1) Matrix(103,238,74,171) -> Matrix(1,0,6,1) Matrix(239,544,-134,-305) -> Matrix(1,0,4,1) Matrix(169,374,-108,-239) -> Matrix(3,2,-8,-5) Matrix(203,442,62,135) -> Matrix(3,2,4,3) Matrix(103,221,48,103) -> Matrix(3,2,4,3) Matrix(103,187,38,69) -> Matrix(1,0,4,1) Matrix(645,1156,284,509) -> Matrix(1,0,12,1) Matrix(647,1156,286,511) -> Matrix(1,0,-4,1) Matrix(135,238,38,67) -> Matrix(1,2,0,1) Matrix(69,119,40,69) -> Matrix(3,2,4,3) Matrix(373,612,-270,-443) -> Matrix(5,2,-18,-7) Matrix(781,1275,166,271) -> Matrix(11,4,8,3) Matrix(137,221,106,171) -> Matrix(7,2,24,7) Matrix(171,272,22,35) -> Matrix(5,2,2,1) Matrix(237,374,64,101) -> Matrix(5,2,2,1) Matrix(307,476,198,307) -> Matrix(17,6,48,17) Matrix(67,102,44,67) -> Matrix(1,0,6,1) Matrix(35,51,24,35) -> Matrix(1,0,6,1) Matrix(171,238,74,103) -> Matrix(1,0,6,1) Matrix(273,374,100,137) -> Matrix(1,0,4,1) Matrix(849,1156,224,305) -> Matrix(1,0,2,1) Matrix(851,1156,226,307) -> Matrix(5,2,2,1) Matrix(67,85,26,33) -> Matrix(7,2,10,3) Matrix(69,85,56,69) -> Matrix(7,2,24,7) Matrix(101,119,28,33) -> Matrix(9,2,4,1) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(103,-119,58,-67) -> Matrix(1,0,-4,1) Matrix(239,-289,196,-237) -> Matrix(25,-6,96,-23) Matrix(613,-799,234,-305) -> Matrix(25,-8,22,-7) Matrix(271,-357,104,-137) -> Matrix(11,-4,14,-5) Matrix(101,-136,26,-35) -> Matrix(1,0,-2,1) Matrix(137,-187,74,-101) -> Matrix(1,0,0,1) Matrix(271,-374,50,-69) -> Matrix(1,0,-4,1) Matrix(205,-289,144,-203) -> Matrix(1,0,0,1) Matrix(305,-476,66,-103) -> Matrix(5,-2,8,-3) Matrix(407,-646,172,-273) -> Matrix(1,0,0,1) Matrix(169,-272,64,-103) -> Matrix(1,0,-2,1) Matrix(271,-442,84,-137) -> Matrix(11,-4,14,-5) Matrix(103,-170,20,-33) -> Matrix(5,-2,-2,1) Matrix(171,-289,100,-169) -> Matrix(13,-6,24,-11) Matrix(305,-544,134,-239) -> Matrix(1,0,4,1) Matrix(137,-289,64,-135) -> Matrix(5,-2,8,-3) Matrix(101,-221,16,-35) -> Matrix(1,0,0,1) Matrix(137,-306,30,-67) -> Matrix(1,0,0,1) Matrix(271,-646,86,-205) -> Matrix(5,-2,8,-3) Matrix(103,-289,36,-101) -> Matrix(5,-2,8,-3) Matrix(407,-1292,86,-273) -> Matrix(3,-2,2,-1) Matrix(69,-289,16,-67) -> Matrix(1,-2,0,1) Matrix(1531,-7225,324,-1529) -> Matrix(1,-2,0,1) Matrix(35,-289,4,-33) -> Matrix(1,-4,0,1) 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 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): 16 ----------------------------------------------------------------------- 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 4 1 1/1 (0/1,1/4) 0 17 6/5 (2/9,1/4) 0 17 17/14 1/4 6 1 5/4 (1/4,1/3) 0 17 4/3 (1/3,1/2) 0 17 15/11 (0/1,1/2) 0 17 11/8 (0/1,1/2) 0 17 18/13 (0/1,1/2) 0 17 7/5 (0/1,1/3) 0 17 17/12 (0/1,1/3) 0 1 3/2 (0/1,1/3) 0 17 17/11 1/3 3 1 14/9 (1/3,3/8) 0 17 11/7 (2/5,1/2) 0 17 19/12 (1/3,1/2) 0 17 8/5 (0/1,1/2) 0 17 13/8 (1/3,3/8) 0 17 18/11 (3/8,2/5) 0 17 5/3 (2/5,1/2) 0 17 17/10 1/2 6 1 7/4 (1/2,1/1) 0 17 16/9 (0/1,1/0) 0 17 9/5 (0/1,1/4) 0 17 2/1 (0/1,1/2) 0 17 17/8 1/2 2 1 13/6 (1/2,1/1) 0 17 11/5 (0/1,1/2) 0 17 9/4 (0/1,1/0) 0 17 34/15 0/1 8 1 25/11 (0/1,1/8) 0 17 16/7 (0/1,1/4) 0 17 7/3 (0/1,1/3) 0 17 26/11 (0/1,1/2) 0 17 19/8 (1/3,1/2) 0 17 12/5 (2/5,1/2) 0 17 17/7 1/2 3 1 5/2 (1/2,1/1) 0 17 13/5 (3/4,1/1) 0 17 34/13 1/1 6 1 21/8 (1/1,3/2) 0 17 8/3 (0/1,1/0) 0 17 19/7 (0/1,1/2) 0 17 11/4 (0/1,1/2) 0 17 17/6 1/2 2 1 3/1 (1/2,1/1) 0 17 16/5 (1/2,2/3) 0 17 13/4 (1/2,1/1) 0 17 10/3 (4/5,1/1) 0 17 17/5 1/1 7 1 7/2 (1/1,3/2) 0 17 18/5 (2/1,1/0) 0 17 11/3 (0/1,1/0) 0 17 15/4 (1/1,1/0) 0 17 34/9 1/0 1 1 19/5 (0/1,1/0) 0 17 4/1 (1/1,1/0) 0 17 17/4 1/0 2 1 9/2 (0/1,1/0) 0 17 23/5 (0/1,1/2) 0 17 14/3 (1/1,1/0) 0 17 33/7 (2/1,1/0) 0 17 85/18 1/0 2 1 19/4 (1/1,1/0) 0 17 5/1 (0/1,1/0) 0 17 16/3 (-1/2,0/1) 0 17 11/2 (-1/2,0/1) 0 17 17/3 0/1 4 1 6/1 (0/1,1/2) 0 17 13/2 (1/2,1/1) 0 17 7/1 (1/1,2/1) 0 17 8/1 (2/1,1/0) 0 17 17/2 1/0 4 1 1/0 (0/1,1/0) 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(101,-119,28,-33) (1/1,6/5) -> (18/5,11/3) Glide Reflection Matrix(169,-204,140,-169) (6/5,17/14) -> (6/5,17/14) Reflection Matrix(69,-85,56,-69) (17/14,5/4) -> (17/14,5/4) Reflection Matrix(67,-85,26,-33) (5/4,4/3) -> (5/2,13/5) Glide 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(271,-374,50,-69) (11/8,18/13) -> (16/3,11/2) Hyperbolic Matrix(171,-238,74,-103) (18/13,7/5) -> (16/7,7/3) Glide Reflection Matrix(169,-238,120,-169) (7/5,17/12) -> (7/5,17/12) Reflection Matrix(35,-51,24,-35) (17/12,3/2) -> (17/12,3/2) Reflection Matrix(67,-102,44,-67) (3/2,17/11) -> (3/2,17/11) Reflection Matrix(307,-476,198,-307) (17/11,14/9) -> (17/11,14/9) Reflection Matrix(305,-476,66,-103) (14/9,11/7) -> (23/5,14/3) Hyperbolic Matrix(237,-374,64,-101) (11/7,19/12) -> (11/3,15/4) Glide Reflection 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,-170,60,-101) (5/3,17/10) -> (5/3,17/10) Reflection Matrix(69,-119,40,-69) (17/10,7/4) -> (17/10,7/4) Reflection 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(103,-187,38,-69) (9/5,2/1) -> (8/3,19/7) Glide Reflection Matrix(33,-68,16,-33) (2/1,17/8) -> (2/1,17/8) Reflection Matrix(103,-221,48,-103) (17/8,13/6) -> (17/8,13/6) Reflection Matrix(101,-221,16,-35) (13/6,11/5) -> (6/1,13/2) Hyperbolic 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(749,-1700,330,-749) (34/15,25/11) -> (34/15,25/11) Reflection Matrix(101,-238,14,-33) (7/3,26/11) -> (7/1,8/1) Glide Reflection Matrix(135,-323,28,-67) (19/8,12/5) -> (19/4,5/1) Glide Reflection 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(339,-884,130,-339) (13/5,34/13) -> (13/5,34/13) Reflection Matrix(545,-1428,208,-545) (34/13,21/8) -> (34/13,21/8) Reflection Matrix(67,-187,24,-67) (11/4,17/6) -> (11/4,17/6) Reflection Matrix(35,-102,12,-35) (17/6,3/1) -> (17/6,3/1) Reflection Matrix(103,-323,22,-69) (3/1,16/5) -> (14/3,33/7) Glide Reflection Matrix(67,-221,10,-33) (13/4,10/3) -> (13/2,7/1) Glide Reflection 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(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(35,-153,8,-35) (17/4,9/2) -> (17/4,9/2) Reflection Matrix(1189,-5610,252,-1189) (33/7,85/18) -> (33/7,85/18) Reflection Matrix(341,-1615,72,-341) (85/18,19/4) -> (85/18,19/4) 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(33,-272,4,-33) (8/1,17/2) -> (8/1,17/2) Reflection Matrix(-1,17,0,1) (17/2,1/0) -> (17/2,1/0) 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(1,0,2,-1) -> Matrix(1,0,8,-1) (0/1,1/1) -> (0/1,1/4) Matrix(101,-119,28,-33) -> Matrix(9,-2,4,-1) Matrix(169,-204,140,-169) -> Matrix(17,-4,72,-17) (6/5,17/14) -> (2/9,1/4) Matrix(69,-85,56,-69) -> Matrix(7,-2,24,-7) (17/14,5/4) -> (1/4,1/3) Matrix(67,-85,26,-33) -> Matrix(7,-2,10,-3) Matrix(101,-136,26,-35) -> Matrix(1,0,-2,1) 0/1 Matrix(273,-374,100,-137) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(271,-374,50,-69) -> Matrix(1,0,-4,1) 0/1 Matrix(171,-238,74,-103) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(169,-238,120,-169) -> Matrix(1,0,6,-1) (7/5,17/12) -> (0/1,1/3) Matrix(35,-51,24,-35) -> Matrix(1,0,6,-1) (17/12,3/2) -> (0/1,1/3) Matrix(67,-102,44,-67) -> Matrix(1,0,6,-1) (3/2,17/11) -> (0/1,1/3) Matrix(307,-476,198,-307) -> Matrix(17,-6,48,-17) (17/11,14/9) -> (1/3,3/8) Matrix(305,-476,66,-103) -> Matrix(5,-2,8,-3) 1/2 Matrix(237,-374,64,-101) -> Matrix(5,-2,2,-1) Matrix(407,-646,172,-273) -> Matrix(1,0,0,1) Matrix(169,-272,64,-103) -> Matrix(1,0,-2,1) 0/1 Matrix(271,-442,84,-137) -> Matrix(11,-4,14,-5) Matrix(103,-170,20,-33) -> Matrix(5,-2,-2,1) Matrix(101,-170,60,-101) -> Matrix(9,-4,20,-9) (5/3,17/10) -> (2/5,1/2) Matrix(69,-119,40,-69) -> Matrix(3,-2,4,-3) (17/10,7/4) -> (1/2,1/1) Matrix(135,-238,38,-67) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(305,-544,134,-239) -> Matrix(1,0,4,1) 0/1 Matrix(103,-187,38,-69) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(33,-68,16,-33) -> Matrix(1,0,4,-1) (2/1,17/8) -> (0/1,1/2) Matrix(103,-221,48,-103) -> Matrix(3,-2,4,-3) (17/8,13/6) -> (1/2,1/1) Matrix(101,-221,16,-35) -> Matrix(1,0,0,1) Matrix(137,-306,30,-67) -> Matrix(1,0,0,1) Matrix(271,-612,120,-271) -> Matrix(1,0,0,-1) (9/4,34/15) -> (0/1,1/0) Matrix(749,-1700,330,-749) -> Matrix(1,0,16,-1) (34/15,25/11) -> (0/1,1/8) Matrix(101,-238,14,-33) -> Matrix(5,-2,2,-1) Matrix(135,-323,28,-67) -> Matrix(5,-2,2,-1) Matrix(169,-408,70,-169) -> Matrix(9,-4,20,-9) (12/5,17/7) -> (2/5,1/2) Matrix(69,-170,28,-69) -> Matrix(3,-2,4,-3) (17/7,5/2) -> (1/2,1/1) Matrix(339,-884,130,-339) -> Matrix(7,-6,8,-7) (13/5,34/13) -> (3/4,1/1) Matrix(545,-1428,208,-545) -> Matrix(5,-6,4,-5) (34/13,21/8) -> (1/1,3/2) Matrix(67,-187,24,-67) -> Matrix(1,0,4,-1) (11/4,17/6) -> (0/1,1/2) Matrix(35,-102,12,-35) -> Matrix(3,-2,4,-3) (17/6,3/1) -> (1/2,1/1) Matrix(103,-323,22,-69) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(67,-221,10,-33) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(101,-340,30,-101) -> Matrix(9,-8,10,-9) (10/3,17/5) -> (4/5,1/1) Matrix(69,-238,20,-69) -> Matrix(5,-6,4,-5) (17/5,7/2) -> (1/1,3/2) Matrix(271,-1020,72,-271) -> Matrix(-1,2,0,1) (15/4,34/9) -> (1/1,1/0) Matrix(341,-1292,90,-341) -> Matrix(1,0,0,-1) (34/9,19/5) -> (0/1,1/0) Matrix(33,-136,8,-33) -> Matrix(-1,2,0,1) (4/1,17/4) -> (1/1,1/0) Matrix(35,-153,8,-35) -> Matrix(1,0,0,-1) (17/4,9/2) -> (0/1,1/0) Matrix(1189,-5610,252,-1189) -> Matrix(-1,4,0,1) (33/7,85/18) -> (2/1,1/0) Matrix(341,-1615,72,-341) -> Matrix(-1,2,0,1) (85/18,19/4) -> (1/1,1/0) Matrix(67,-374,12,-67) -> Matrix(-1,0,4,1) (11/2,17/3) -> (-1/2,0/1) Matrix(35,-204,6,-35) -> Matrix(1,0,4,-1) (17/3,6/1) -> (0/1,1/2) Matrix(33,-272,4,-33) -> Matrix(-1,4,0,1) (8/1,17/2) -> (2/1,1/0) Matrix(-1,17,0,1) -> Matrix(1,0,0,-1) (17/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.