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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -14/3 -4/1 -18/5 -10/3 -30/11 -8/3 -5/2 -12/5 -16/7 -2/1 -5/3 -30/19 -10/7 -5/4 0/1 1/1 5/4 10/7 3/2 30/19 5/3 2/1 20/9 40/17 12/5 5/2 100/39 8/3 30/11 20/7 3/1 10/3 7/2 18/5 40/11 11/3 4/1 9/2 14/3 5/1 16/3 40/7 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -3/8 -6/1 -1/3 -17/3 -5/18 -11/2 -1/4 -5/1 -1/4 -19/4 -3/16 -14/3 -1/3 -9/2 -1/4 -4/1 -2/9 0/1 -11/3 -1/4 -18/5 -1/5 -7/2 -1/6 -10/3 -1/5 -13/4 -3/16 -3/1 -1/6 -14/5 -3/19 -11/4 -1/6 -30/11 -1/6 -19/7 -9/56 -8/3 -2/13 -21/8 -11/74 -13/5 -1/6 -18/7 -1/7 -5/2 -1/7 -22/9 -1/7 -39/16 -11/78 -17/7 -5/36 -29/12 -3/22 -12/5 -4/29 -2/15 -31/13 -3/22 -19/8 -13/96 -26/11 -9/67 -7/3 -5/38 -16/7 -4/31 -41/18 -21/164 -25/11 -1/8 -9/4 -1/8 -2/1 -1/9 -11/6 -1/10 -20/11 -1/10 -9/5 -1/10 -16/9 -4/41 -39/22 -19/196 -23/13 -1/10 -7/4 -5/52 -40/23 -2/21 -33/19 -23/242 -26/15 -9/95 -19/11 -13/138 -50/29 -3/32 -31/18 -3/32 -12/7 -2/21 -4/43 -29/17 -3/32 -17/10 -5/54 -5/3 -1/11 -23/14 -1/12 -41/25 -5/54 -100/61 -1/11 -59/36 -11/122 -18/11 -1/11 -13/8 -1/12 -21/13 -11/124 -8/5 -2/23 -19/12 -9/106 -30/19 -1/12 -41/26 -3/38 -11/7 -1/12 -14/9 -3/35 -17/11 -1/12 -20/13 -1/12 -3/2 -1/12 -10/7 -1/13 -17/12 -1/14 -41/29 -7/92 -24/17 -2/27 -7/5 -1/12 -18/13 -1/13 -29/21 -3/40 -40/29 -2/27 -11/8 -1/14 -4/3 -2/27 0/1 -9/7 -1/14 -14/11 -1/15 -19/15 -3/38 -5/4 -1/14 -21/17 -5/74 -16/13 0/1 -11/9 -1/14 -17/14 -5/72 -40/33 -2/29 -23/19 -7/102 -6/5 -1/15 -13/11 -5/74 -20/17 -1/15 -7/6 -3/46 -1/1 -1/18 0/1 0/1 1/1 1/18 7/6 3/46 6/5 1/15 17/14 5/72 11/9 1/14 5/4 1/14 19/15 3/38 14/11 1/15 9/7 1/14 4/3 0/1 2/27 11/8 1/14 18/13 1/13 7/5 1/12 10/7 1/13 13/9 3/38 3/2 1/12 14/9 3/35 11/7 1/12 30/19 1/12 19/12 9/106 8/5 2/23 21/13 11/124 13/8 1/12 18/11 1/11 5/3 1/11 22/13 1/11 39/23 11/120 17/10 5/54 29/17 3/32 12/7 4/43 2/21 31/18 3/32 19/11 13/138 26/15 9/95 7/4 5/52 16/9 4/41 41/23 21/214 25/14 1/10 9/5 1/10 2/1 1/9 11/5 1/8 20/9 1/8 9/4 1/8 16/7 4/31 39/17 19/146 23/10 1/8 7/3 5/38 40/17 2/15 33/14 23/172 26/11 9/67 19/8 13/96 50/21 3/22 31/13 3/22 12/5 2/15 4/29 29/12 3/22 17/7 5/36 5/2 1/7 23/9 1/6 41/16 5/36 100/39 1/7 59/23 11/76 18/7 1/7 13/5 1/6 21/8 11/74 8/3 2/13 19/7 9/56 30/11 1/6 41/15 3/16 11/4 1/6 14/5 3/19 17/6 1/6 20/7 1/6 3/1 1/6 10/3 1/5 17/5 1/4 41/12 7/34 24/7 2/9 7/2 1/6 18/5 1/5 29/8 3/14 40/11 2/9 11/3 1/4 4/1 0/1 2/9 9/2 1/4 14/3 1/3 19/4 3/16 5/1 1/4 21/4 5/16 16/3 0/1 11/2 1/4 17/3 5/18 40/7 2/7 23/4 7/24 6/1 1/3 13/2 5/16 20/3 1/3 7/1 3/8 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,160,8,61) (-7/1,1/0) -> (13/5,21/8) Hyperbolic Matrix(19,120,-16,-101) (-7/1,-6/1) -> (-6/5,-13/11) Hyperbolic Matrix(41,240,-34,-199) (-6/1,-17/3) -> (-23/19,-6/5) Hyperbolic Matrix(121,680,50,281) (-17/3,-11/2) -> (29/12,17/7) Hyperbolic Matrix(59,320,-26,-141) (-11/2,-5/1) -> (-25/11,-9/4) Hyperbolic Matrix(141,680,-62,-299) (-5/1,-19/4) -> (-41/18,-25/11) Hyperbolic Matrix(161,760,68,321) (-19/4,-14/3) -> (26/11,19/8) Hyperbolic Matrix(61,280,22,101) (-14/3,-9/2) -> (11/4,14/5) Hyperbolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(199,720,-144,-521) (-11/3,-18/5) -> (-18/13,-29/21) Hyperbolic Matrix(101,360,62,221) (-18/5,-7/2) -> (13/8,18/11) Hyperbolic Matrix(59,200,-18,-61) (-7/2,-10/3) -> (-10/3,-13/4) Parabolic Matrix(99,320,-56,-181) (-13/4,-3/1) -> (-23/13,-7/4) Hyperbolic Matrix(99,280,-64,-181) (-3/1,-14/5) -> (-14/9,-17/11) Hyperbolic Matrix(101,280,22,61) (-14/5,-11/4) -> (9/2,14/3) Hyperbolic Matrix(541,1480,-314,-859) (-11/4,-30/11) -> (-50/29,-31/18) Hyperbolic Matrix(559,1520,-324,-881) (-30/11,-19/7) -> (-19/11,-50/29) Hyperbolic Matrix(119,320,74,199) (-19/7,-8/3) -> (8/5,21/13) Hyperbolic Matrix(121,320,76,201) (-8/3,-21/8) -> (19/12,8/5) Hyperbolic Matrix(61,160,8,21) (-21/8,-13/5) -> (7/1,1/0) Hyperbolic Matrix(139,360,100,259) (-13/5,-18/7) -> (18/13,7/5) Hyperbolic Matrix(79,200,-32,-81) (-18/7,-5/2) -> (-5/2,-22/9) Parabolic Matrix(819,2000,-500,-1221) (-22/9,-39/16) -> (-59/36,-18/11) Hyperbolic Matrix(559,1360,164,399) (-39/16,-17/7) -> (17/5,41/12) Hyperbolic Matrix(281,680,50,121) (-17/7,-29/12) -> (11/2,17/3) Hyperbolic Matrix(299,720,174,419) (-29/12,-12/5) -> (12/7,31/18) Hyperbolic Matrix(301,720,176,421) (-12/5,-31/13) -> (29/17,12/7) Hyperbolic Matrix(621,1480,-394,-939) (-31/13,-19/8) -> (-41/26,-11/7) Hyperbolic Matrix(321,760,68,161) (-19/8,-26/11) -> (14/3,19/4) Hyperbolic Matrix(441,1040,-254,-599) (-26/11,-7/3) -> (-33/19,-26/15) Hyperbolic Matrix(121,280,-86,-199) (-7/3,-16/7) -> (-24/17,-7/5) Hyperbolic Matrix(719,1640,210,479) (-16/7,-41/18) -> (41/12,24/7) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(219,400,98,179) (-11/6,-20/11) -> (20/9,9/4) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(179,320,-146,-261) (-9/5,-16/9) -> (-16/13,-11/9) Hyperbolic Matrix(541,960,102,181) (-16/9,-39/22) -> (21/4,16/3) Hyperbolic Matrix(1039,1840,406,719) (-39/22,-23/13) -> (23/9,41/16) Hyperbolic Matrix(919,1600,390,679) (-7/4,-40/23) -> (40/17,33/14) Hyperbolic Matrix(921,1600,392,681) (-40/23,-33/19) -> (7/3,40/17) Hyperbolic Matrix(439,760,346,599) (-26/15,-19/11) -> (19/15,14/11) Hyperbolic Matrix(419,720,174,299) (-31/18,-12/7) -> (12/5,29/12) Hyperbolic Matrix(421,720,176,301) (-12/7,-29/17) -> (31/13,12/5) Hyperbolic Matrix(399,680,328,559) (-29/17,-17/10) -> (17/14,11/9) Hyperbolic Matrix(119,200,-72,-121) (-17/10,-5/3) -> (-5/3,-23/14) Parabolic Matrix(1121,1840,488,801) (-23/14,-41/25) -> (39/17,23/10) Hyperbolic Matrix(6099,10000,2378,3899) (-41/25,-100/61) -> (100/39,59/23) Hyperbolic Matrix(6101,10000,2380,3901) (-100/61,-59/36) -> (41/16,100/39) Hyperbolic Matrix(221,360,62,101) (-18/11,-13/8) -> (7/2,18/5) Hyperbolic Matrix(99,160,86,139) (-13/8,-21/13) -> (1/1,7/6) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(1139,1800,-722,-1141) (-19/12,-30/19) -> (-30/19,-41/26) Parabolic Matrix(179,280,140,219) (-11/7,-14/9) -> (14/11,9/7) Hyperbolic Matrix(259,400,90,139) (-17/11,-20/13) -> (20/7,3/1) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) Hyperbolic Matrix(139,200,-98,-141) (-3/2,-10/7) -> (-10/7,-17/12) Parabolic Matrix(961,1360,566,801) (-17/12,-41/29) -> (39/23,17/10) Hyperbolic Matrix(1161,1640,652,921) (-41/29,-24/17) -> (16/9,41/23) Hyperbolic Matrix(259,360,100,139) (-7/5,-18/13) -> (18/7,13/5) Hyperbolic Matrix(1159,1600,318,439) (-29/21,-40/29) -> (40/11,11/3) Hyperbolic Matrix(1161,1600,320,441) (-40/29,-11/8) -> (29/8,40/11) Hyperbolic Matrix(59,80,14,19) (-11/8,-4/3) -> (4/1,9/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(219,280,140,179) (-9/7,-14/11) -> (14/9,11/7) Hyperbolic Matrix(599,760,346,439) (-14/11,-19/15) -> (19/11,26/15) Hyperbolic Matrix(159,200,-128,-161) (-19/15,-5/4) -> (-5/4,-21/17) Parabolic Matrix(779,960,340,419) (-21/17,-16/13) -> (16/7,39/17) Hyperbolic Matrix(559,680,328,399) (-11/9,-17/14) -> (17/10,29/17) Hyperbolic Matrix(1319,1600,230,279) (-17/14,-40/33) -> (40/7,23/4) Hyperbolic Matrix(1321,1600,232,281) (-40/33,-23/19) -> (17/3,40/7) Hyperbolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(139,160,86,99) (-7/6,-1/1) -> (21/13,13/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(199,-240,34,-41) (6/5,17/14) -> (23/4,6/1) Hyperbolic Matrix(261,-320,146,-179) (11/9,5/4) -> (25/14,9/5) Hyperbolic Matrix(539,-680,302,-381) (5/4,19/15) -> (41/23,25/14) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(221,-320,96,-139) (13/9,3/2) -> (23/10,7/3) Hyperbolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(939,-1480,394,-621) (11/7,30/19) -> (50/21,31/13) Hyperbolic Matrix(961,-1520,404,-639) (30/19,19/12) -> (19/8,50/21) Hyperbolic Matrix(121,-200,72,-119) (18/11,5/3) -> (5/3,22/13) Parabolic Matrix(1181,-2000,460,-779) (22/13,39/23) -> (59/23,18/7) Hyperbolic Matrix(859,-1480,314,-541) (31/18,19/11) -> (41/15,11/4) Hyperbolic Matrix(599,-1040,254,-441) (26/15,7/4) -> (33/14,26/11) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(41,-200,8,-39) (19/4,5/1) -> (5/1,21/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,160,8,61) -> Matrix(11,4,74,27) Matrix(19,120,-16,-101) -> Matrix(7,2,-102,-29) Matrix(41,240,-34,-199) -> Matrix(13,4,-192,-59) Matrix(121,680,50,281) -> Matrix(37,10,270,73) Matrix(59,320,-26,-141) -> Matrix(17,4,-132,-31) Matrix(141,680,-62,-299) -> Matrix(25,6,-196,-47) Matrix(161,760,68,321) -> Matrix(15,2,112,15) Matrix(61,280,22,101) -> Matrix(9,2,58,13) Matrix(19,80,14,59) -> Matrix(1,0,18,1) Matrix(21,80,16,61) -> Matrix(1,0,18,1) Matrix(199,720,-144,-521) -> Matrix(19,4,-252,-53) Matrix(101,360,62,221) -> Matrix(11,2,126,23) Matrix(59,200,-18,-61) -> Matrix(9,2,-50,-11) Matrix(99,320,-56,-181) -> Matrix(23,4,-236,-41) Matrix(99,280,-64,-181) -> Matrix(37,6,-438,-71) Matrix(101,280,22,61) -> Matrix(13,2,58,9) Matrix(541,1480,-314,-859) -> Matrix(15,2,-158,-21) Matrix(559,1520,-324,-881) -> Matrix(123,20,-1310,-213) Matrix(119,320,74,199) -> Matrix(51,8,580,91) Matrix(121,320,76,201) -> Matrix(53,8,616,93) Matrix(61,160,8,21) -> Matrix(27,4,74,11) Matrix(139,360,100,259) -> Matrix(13,2,162,25) Matrix(79,200,-32,-81) -> Matrix(41,6,-294,-43) Matrix(819,2000,-500,-1221) -> Matrix(1,0,-4,1) Matrix(559,1360,164,399) -> Matrix(29,4,152,21) Matrix(281,680,50,121) -> Matrix(73,10,270,37) Matrix(299,720,174,419) -> Matrix(1,0,18,1) Matrix(301,720,176,421) -> Matrix(1,0,18,1) Matrix(621,1480,-394,-939) -> Matrix(15,2,-158,-21) Matrix(321,760,68,161) -> Matrix(15,2,112,15) Matrix(441,1040,-254,-599) -> Matrix(269,36,-2832,-379) Matrix(121,280,-86,-199) -> Matrix(15,2,-218,-29) Matrix(719,1640,210,479) -> Matrix(109,14,506,65) Matrix(19,40,-10,-21) -> Matrix(17,2,-162,-19) Matrix(219,400,98,179) -> Matrix(159,16,1262,127) Matrix(221,400,100,181) -> Matrix(161,16,1298,129) Matrix(179,320,-146,-261) -> Matrix(41,4,-564,-55) Matrix(541,960,102,181) -> Matrix(41,4,92,9) Matrix(1039,1840,406,719) -> Matrix(41,4,256,25) Matrix(919,1600,390,679) -> Matrix(587,56,4392,419) Matrix(921,1600,392,681) -> Matrix(589,56,4428,421) Matrix(439,760,346,599) -> Matrix(21,2,220,21) Matrix(419,720,174,299) -> Matrix(1,0,18,1) Matrix(421,720,176,301) -> Matrix(1,0,18,1) Matrix(399,680,328,559) -> Matrix(107,10,1530,143) Matrix(119,200,-72,-121) -> Matrix(65,6,-726,-67) Matrix(1121,1840,488,801) -> Matrix(47,4,364,31) Matrix(6099,10000,2378,3899) -> Matrix(175,16,1214,111) Matrix(6101,10000,2380,3901) -> Matrix(177,16,1250,113) Matrix(221,360,62,101) -> Matrix(23,2,126,11) Matrix(99,160,86,139) -> Matrix(45,4,686,61) Matrix(199,320,74,119) -> Matrix(91,8,580,51) Matrix(201,320,76,121) -> Matrix(93,8,616,53) Matrix(1139,1800,-722,-1141) -> Matrix(71,6,-864,-73) Matrix(179,280,140,219) -> Matrix(23,2,310,27) Matrix(259,400,90,139) -> Matrix(95,8,558,47) Matrix(261,400,92,141) -> Matrix(97,8,594,49) Matrix(139,200,-98,-141) -> Matrix(25,2,-338,-27) Matrix(961,1360,566,801) -> Matrix(51,4,548,43) Matrix(1161,1640,652,921) -> Matrix(187,14,1910,143) Matrix(259,360,100,139) -> Matrix(25,2,162,13) Matrix(1159,1600,318,439) -> Matrix(107,8,468,35) Matrix(1161,1600,320,441) -> Matrix(109,8,504,37) Matrix(59,80,14,19) -> Matrix(1,0,18,1) Matrix(61,80,16,21) -> Matrix(1,0,18,1) Matrix(219,280,140,179) -> Matrix(27,2,310,23) Matrix(599,760,346,439) -> Matrix(21,2,220,21) Matrix(159,200,-128,-161) -> Matrix(27,2,-392,-29) Matrix(779,960,340,419) -> Matrix(63,4,488,31) Matrix(559,680,328,399) -> Matrix(143,10,1530,107) Matrix(1319,1600,230,279) -> Matrix(347,24,1200,83) Matrix(1321,1600,232,281) -> Matrix(349,24,1236,85) Matrix(339,400,50,59) -> Matrix(119,8,342,23) Matrix(341,400,52,61) -> Matrix(121,8,378,25) Matrix(139,160,86,99) -> Matrix(61,4,686,45) Matrix(1,0,2,1) -> Matrix(1,0,36,1) Matrix(101,-120,16,-19) -> Matrix(29,-2,102,-7) Matrix(199,-240,34,-41) -> Matrix(59,-4,192,-13) Matrix(261,-320,146,-179) -> Matrix(55,-4,564,-41) Matrix(539,-680,302,-381) -> Matrix(83,-6,844,-61) Matrix(521,-720,144,-199) -> Matrix(53,-4,252,-19) Matrix(141,-200,98,-139) -> Matrix(27,-2,338,-25) Matrix(221,-320,96,-139) -> Matrix(49,-4,380,-31) Matrix(181,-280,64,-99) -> Matrix(71,-6,438,-37) Matrix(939,-1480,394,-621) -> Matrix(21,-2,158,-15) Matrix(961,-1520,404,-639) -> Matrix(237,-20,1742,-147) Matrix(121,-200,72,-119) -> Matrix(67,-6,726,-65) Matrix(1181,-2000,460,-779) -> Matrix(1,0,-4,1) Matrix(859,-1480,314,-541) -> Matrix(21,-2,158,-15) Matrix(599,-1040,254,-441) -> Matrix(379,-36,2832,-269) Matrix(159,-280,46,-81) -> Matrix(21,-2,74,-7) Matrix(21,-40,10,-19) -> Matrix(19,-2,162,-17) Matrix(141,-320,26,-59) -> Matrix(31,-4,132,-17) Matrix(81,-200,32,-79) -> Matrix(43,-6,294,-41) Matrix(661,-1800,242,-659) -> Matrix(37,-6,216,-35) Matrix(61,-200,18,-59) -> Matrix(11,-2,50,-9) Matrix(41,-200,8,-39) -> Matrix(9,-2,32,-7) 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): 48 Degree of the the map X: 48 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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): 288 Minimal number of generators: 49 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: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 5/4 4/3 10/7 30/19 8/5 5/3 12/7 2/1 20/9 40/17 12/5 5/2 100/39 8/3 30/11 20/7 3/1 10/3 24/7 18/5 40/11 4/1 9/2 14/3 5/1 16/3 40/7 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/18 7/6 3/46 6/5 1/15 17/14 5/72 11/9 1/14 5/4 1/14 19/15 3/38 14/11 1/15 9/7 1/14 4/3 0/1 2/27 11/8 1/14 18/13 1/13 7/5 1/12 10/7 1/13 13/9 3/38 3/2 1/12 14/9 3/35 11/7 1/12 30/19 1/12 19/12 9/106 8/5 2/23 21/13 11/124 13/8 1/12 18/11 1/11 5/3 1/11 22/13 1/11 39/23 11/120 17/10 5/54 29/17 3/32 12/7 4/43 2/21 31/18 3/32 19/11 13/138 26/15 9/95 7/4 5/52 16/9 4/41 41/23 21/214 25/14 1/10 9/5 1/10 2/1 1/9 11/5 1/8 20/9 1/8 9/4 1/8 16/7 4/31 39/17 19/146 23/10 1/8 7/3 5/38 40/17 2/15 33/14 23/172 26/11 9/67 19/8 13/96 50/21 3/22 31/13 3/22 12/5 2/15 4/29 29/12 3/22 17/7 5/36 5/2 1/7 23/9 1/6 41/16 5/36 100/39 1/7 59/23 11/76 18/7 1/7 13/5 1/6 21/8 11/74 8/3 2/13 19/7 9/56 30/11 1/6 41/15 3/16 11/4 1/6 14/5 3/19 17/6 1/6 20/7 1/6 3/1 1/6 10/3 1/5 17/5 1/4 41/12 7/34 24/7 2/9 7/2 1/6 18/5 1/5 29/8 3/14 40/11 2/9 11/3 1/4 4/1 0/1 2/9 9/2 1/4 14/3 1/3 19/4 3/16 5/1 1/4 21/4 5/16 16/3 0/1 11/2 1/4 17/3 5/18 40/7 2/7 23/4 7/24 6/1 1/3 13/2 5/16 20/3 1/3 7/1 3/8 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(139,-160,53,-61) (1/1,7/6) -> (13/5,21/8) Hyperbolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(199,-240,34,-41) (6/5,17/14) -> (23/4,6/1) Hyperbolic Matrix(559,-680,231,-281) (17/14,11/9) -> (29/12,17/7) Hyperbolic Matrix(261,-320,146,-179) (11/9,5/4) -> (25/14,9/5) Hyperbolic Matrix(539,-680,302,-381) (5/4,19/15) -> (41/23,25/14) Hyperbolic Matrix(599,-760,253,-321) (19/15,14/11) -> (26/11,19/8) Hyperbolic Matrix(219,-280,79,-101) (14/11,9/7) -> (11/4,14/5) Hyperbolic Matrix(61,-80,45,-59) (9/7,4/3) -> (4/3,11/8) Parabolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(259,-360,159,-221) (18/13,7/5) -> (13/8,18/11) Hyperbolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(221,-320,96,-139) (13/9,3/2) -> (23/10,7/3) Hyperbolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(179,-280,39,-61) (14/9,11/7) -> (9/2,14/3) Hyperbolic Matrix(939,-1480,394,-621) (11/7,30/19) -> (50/21,31/13) Hyperbolic Matrix(961,-1520,404,-639) (30/19,19/12) -> (19/8,50/21) Hyperbolic Matrix(201,-320,125,-199) (19/12,8/5) -> (8/5,21/13) Parabolic Matrix(99,-160,13,-21) (21/13,13/8) -> (7/1,1/0) Hyperbolic Matrix(121,-200,72,-119) (18/11,5/3) -> (5/3,22/13) Parabolic Matrix(1181,-2000,460,-779) (22/13,39/23) -> (59/23,18/7) Hyperbolic Matrix(801,-1360,235,-399) (39/23,17/10) -> (17/5,41/12) Hyperbolic Matrix(399,-680,71,-121) (17/10,29/17) -> (11/2,17/3) Hyperbolic Matrix(421,-720,245,-419) (29/17,12/7) -> (12/7,31/18) Parabolic Matrix(859,-1480,314,-541) (31/18,19/11) -> (41/15,11/4) Hyperbolic Matrix(439,-760,93,-161) (19/11,26/15) -> (14/3,19/4) Hyperbolic Matrix(599,-1040,254,-441) (26/15,7/4) -> (33/14,26/11) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(921,-1640,269,-479) (16/9,41/23) -> (41/12,24/7) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(181,-400,81,-179) (11/5,20/9) -> (20/9,9/4) Parabolic Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(419,-960,79,-181) (16/7,39/17) -> (21/4,16/3) Hyperbolic Matrix(801,-1840,313,-719) (39/17,23/10) -> (23/9,41/16) Hyperbolic Matrix(681,-1600,289,-679) (7/3,40/17) -> (40/17,33/14) Parabolic Matrix(301,-720,125,-299) (31/13,12/5) -> (12/5,29/12) Parabolic Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(3901,-10000,1521,-3899) (41/16,100/39) -> (100/39,59/23) Parabolic Matrix(139,-360,39,-101) (18/7,13/5) -> (7/2,18/5) Hyperbolic Matrix(121,-320,45,-119) (21/8,8/3) -> (8/3,19/7) Parabolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(141,-400,49,-139) (17/6,20/7) -> (20/7,3/1) Parabolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(441,-1600,121,-439) (29/8,40/11) -> (40/11,11/3) Parabolic Matrix(21,-80,5,-19) (11/3,4/1) -> (4/1,9/2) Parabolic Matrix(41,-200,8,-39) (19/4,5/1) -> (5/1,21/4) Parabolic Matrix(281,-1600,49,-279) (17/3,40/7) -> (40/7,23/4) Parabolic Matrix(61,-400,9,-59) (13/2,20/3) -> (20/3,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,18,1) Matrix(139,-160,53,-61) -> Matrix(61,-4,412,-27) Matrix(101,-120,16,-19) -> Matrix(29,-2,102,-7) Matrix(199,-240,34,-41) -> Matrix(59,-4,192,-13) Matrix(559,-680,231,-281) -> Matrix(143,-10,1044,-73) Matrix(261,-320,146,-179) -> Matrix(55,-4,564,-41) Matrix(539,-680,302,-381) -> Matrix(83,-6,844,-61) Matrix(599,-760,253,-321) -> Matrix(21,-2,158,-15) Matrix(219,-280,79,-101) -> Matrix(27,-2,176,-13) Matrix(61,-80,45,-59) -> Matrix(1,0,0,1) Matrix(521,-720,144,-199) -> Matrix(53,-4,252,-19) Matrix(259,-360,159,-221) -> Matrix(25,-2,288,-23) Matrix(141,-200,98,-139) -> Matrix(27,-2,338,-25) Matrix(221,-320,96,-139) -> Matrix(49,-4,380,-31) Matrix(181,-280,64,-99) -> Matrix(71,-6,438,-37) Matrix(179,-280,39,-61) -> Matrix(23,-2,104,-9) Matrix(939,-1480,394,-621) -> Matrix(21,-2,158,-15) Matrix(961,-1520,404,-639) -> Matrix(237,-20,1742,-147) Matrix(201,-320,125,-199) -> Matrix(93,-8,1058,-91) Matrix(99,-160,13,-21) -> Matrix(45,-4,124,-11) Matrix(121,-200,72,-119) -> Matrix(67,-6,726,-65) Matrix(1181,-2000,460,-779) -> Matrix(1,0,-4,1) Matrix(801,-1360,235,-399) -> Matrix(43,-4,226,-21) Matrix(399,-680,71,-121) -> Matrix(107,-10,396,-37) Matrix(421,-720,245,-419) -> Matrix(1,0,0,1) Matrix(859,-1480,314,-541) -> Matrix(21,-2,158,-15) Matrix(439,-760,93,-161) -> Matrix(21,-2,158,-15) Matrix(599,-1040,254,-441) -> Matrix(379,-36,2832,-269) Matrix(159,-280,46,-81) -> Matrix(21,-2,74,-7) Matrix(921,-1640,269,-479) -> Matrix(143,-14,664,-65) Matrix(21,-40,10,-19) -> Matrix(19,-2,162,-17) Matrix(181,-400,81,-179) -> Matrix(129,-16,1024,-127) Matrix(141,-320,26,-59) -> Matrix(31,-4,132,-17) Matrix(419,-960,79,-181) -> Matrix(31,-4,70,-9) Matrix(801,-1840,313,-719) -> Matrix(31,-4,194,-25) Matrix(681,-1600,289,-679) -> Matrix(421,-56,3150,-419) Matrix(301,-720,125,-299) -> Matrix(1,0,0,1) Matrix(81,-200,32,-79) -> Matrix(43,-6,294,-41) Matrix(3901,-10000,1521,-3899) -> Matrix(113,-16,784,-111) Matrix(139,-360,39,-101) -> Matrix(13,-2,72,-11) Matrix(121,-320,45,-119) -> Matrix(53,-8,338,-51) Matrix(661,-1800,242,-659) -> Matrix(37,-6,216,-35) Matrix(141,-400,49,-139) -> Matrix(49,-8,288,-47) Matrix(61,-200,18,-59) -> Matrix(11,-2,50,-9) Matrix(441,-1600,121,-439) -> Matrix(37,-8,162,-35) Matrix(21,-80,5,-19) -> Matrix(1,0,0,1) Matrix(41,-200,8,-39) -> Matrix(9,-2,32,-7) Matrix(281,-1600,49,-279) -> Matrix(85,-24,294,-83) Matrix(61,-400,9,-59) -> Matrix(25,-8,72,-23) 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): 24 ----------------------------------------------------------------------- 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 18 1 2/1 1/9 2 10 20/9 1/8 16 1 9/4 1/8 1 20 16/7 4/31 2 5 7/3 5/38 1 20 40/17 2/15 14 1 26/11 9/67 2 10 19/8 13/96 1 20 31/13 3/22 1 20 12/5 0 5 29/12 3/22 1 20 17/7 5/36 1 20 5/2 1/7 3 4 23/9 1/6 1 20 41/16 5/36 1 20 100/39 1/7 16 1 18/7 1/7 2 10 13/5 1/6 1 20 21/8 11/74 1 20 8/3 2/13 2 5 19/7 9/56 1 20 30/11 1/6 6 2 41/15 3/16 1 20 11/4 1/6 1 20 14/5 3/19 2 10 20/7 1/6 8 1 3/1 1/6 1 20 10/3 1/5 2 2 17/5 1/4 1 20 41/12 7/34 1 20 24/7 2/9 2 5 7/2 1/6 1 20 18/5 1/5 2 10 40/11 2/9 2 1 11/3 1/4 1 20 4/1 0 5 9/2 1/4 1 20 14/3 1/3 2 10 19/4 3/16 1 20 5/1 1/4 1 4 21/4 5/16 1 20 16/3 0/1 2 5 11/2 1/4 1 20 17/3 5/18 1 20 40/7 2/7 6 1 6/1 1/3 2 10 20/3 1/3 8 1 7/1 3/8 1 20 1/0 1/0 1 20 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(19,-40,9,-19) (2/1,20/9) -> (2/1,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(121,-280,35,-81) (16/7,7/3) -> (24/7,7/2) Glide Reflection Matrix(239,-560,102,-239) (7/3,40/17) -> (7/3,40/17) Reflection Matrix(441,-1040,187,-441) (40/17,26/11) -> (40/17,26/11) Reflection Matrix(321,-760,68,-161) (26/11,19/8) -> (14/3,19/4) Glide Reflection Matrix(621,-1480,227,-541) (19/8,31/13) -> (41/15,11/4) Glide Reflection Matrix(301,-720,125,-299) (31/13,12/5) -> (12/5,29/12) Parabolic Matrix(281,-680,50,-121) (29/12,17/7) -> (11/2,17/3) Glide Reflection Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(641,-1640,188,-481) (23/9,41/16) -> (17/5,41/12) Glide Reflection Matrix(3199,-8200,1248,-3199) (41/16,100/39) -> (41/16,100/39) Reflection Matrix(701,-1800,273,-701) (100/39,18/7) -> (100/39,18/7) Reflection Matrix(139,-360,39,-101) (18/7,13/5) -> (7/2,18/5) Hyperbolic Matrix(61,-160,8,-21) (13/5,21/8) -> (7/1,1/0) Glide Reflection Matrix(121,-320,45,-119) (21/8,8/3) -> (8/3,19/7) Parabolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(101,-280,22,-61) (11/4,14/5) -> (9/2,14/3) Glide Reflection Matrix(99,-280,35,-99) (14/5,20/7) -> (14/5,20/7) Reflection Matrix(41,-120,14,-41) (20/7,3/1) -> (20/7,3/1) Reflection Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(339,-1160,64,-219) (41/12,24/7) -> (21/4,16/3) Glide Reflection Matrix(199,-720,55,-199) (18/5,40/11) -> (18/5,40/11) Reflection Matrix(241,-880,66,-241) (40/11,11/3) -> (40/11,11/3) Reflection Matrix(21,-80,5,-19) (11/3,4/1) -> (4/1,9/2) Parabolic Matrix(41,-200,8,-39) (19/4,5/1) -> (5/1,21/4) Parabolic Matrix(239,-1360,42,-239) (17/3,40/7) -> (17/3,40/7) Reflection Matrix(41,-240,7,-41) (40/7,6/1) -> (40/7,6/1) Reflection Matrix(19,-120,3,-19) (6/1,20/3) -> (6/1,20/3) Reflection Matrix(41,-280,6,-41) (20/3,7/1) -> (20/3,7/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,18,-1) (0/1,2/1) -> (0/1,1/9) Matrix(19,-40,9,-19) -> Matrix(17,-2,144,-17) (2/1,20/9) -> (1/9,1/8) Matrix(161,-360,72,-161) -> Matrix(111,-14,880,-111) (20/9,9/4) -> (1/8,7/55) Matrix(141,-320,26,-59) -> Matrix(31,-4,132,-17) Matrix(121,-280,35,-81) -> Matrix(15,-2,52,-7) Matrix(239,-560,102,-239) -> Matrix(151,-20,1140,-151) (7/3,40/17) -> (5/38,2/15) Matrix(441,-1040,187,-441) -> Matrix(269,-36,2010,-269) (40/17,26/11) -> (2/15,9/67) Matrix(321,-760,68,-161) -> Matrix(15,-2,112,-15) *** -> (1/8,1/7) Matrix(621,-1480,227,-541) -> Matrix(15,-2,112,-15) *** -> (1/8,1/7) Matrix(301,-720,125,-299) -> Matrix(1,0,0,1) Matrix(281,-680,50,-121) -> Matrix(73,-10,270,-37) Matrix(81,-200,32,-79) -> Matrix(43,-6,294,-41) 1/7 Matrix(641,-1640,188,-481) -> Matrix(13,-2,58,-9) Matrix(3199,-8200,1248,-3199) -> Matrix(71,-10,504,-71) (41/16,100/39) -> (5/36,1/7) Matrix(701,-1800,273,-701) -> Matrix(41,-6,280,-41) (100/39,18/7) -> (1/7,3/20) Matrix(139,-360,39,-101) -> Matrix(13,-2,72,-11) 1/6 Matrix(61,-160,8,-21) -> Matrix(27,-4,74,-11) Matrix(121,-320,45,-119) -> Matrix(53,-8,338,-51) 2/13 Matrix(661,-1800,242,-659) -> Matrix(37,-6,216,-35) 1/6 Matrix(101,-280,22,-61) -> Matrix(13,-2,58,-9) Matrix(99,-280,35,-99) -> Matrix(37,-6,228,-37) (14/5,20/7) -> (3/19,1/6) Matrix(41,-120,14,-41) -> Matrix(11,-2,60,-11) (20/7,3/1) -> (1/6,1/5) Matrix(61,-200,18,-59) -> Matrix(11,-2,50,-9) 1/5 Matrix(339,-1160,64,-219) -> Matrix(9,-2,22,-5) Matrix(199,-720,55,-199) -> Matrix(19,-4,90,-19) (18/5,40/11) -> (1/5,2/9) Matrix(241,-880,66,-241) -> Matrix(17,-4,72,-17) (40/11,11/3) -> (2/9,1/4) Matrix(21,-80,5,-19) -> Matrix(1,0,0,1) Matrix(41,-200,8,-39) -> Matrix(9,-2,32,-7) 1/4 Matrix(239,-1360,42,-239) -> Matrix(71,-20,252,-71) (17/3,40/7) -> (5/18,2/7) Matrix(41,-240,7,-41) -> Matrix(13,-4,42,-13) (40/7,6/1) -> (2/7,1/3) Matrix(19,-120,3,-19) -> Matrix(7,-2,24,-7) (6/1,20/3) -> (1/4,1/3) Matrix(41,-280,6,-41) -> Matrix(17,-6,48,-17) (20/3,7/1) -> (1/3,3/8) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.