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 -4/5 -7/10 -19/30 -3/5 -1/2 -9/20 -17/40 -5/12 -2/5 -39/100 -3/8 -11/30 -7/20 -3/10 -5/18 -11/40 -1/4 -3/14 -1/5 -3/16 -7/40 -1/6 -3/20 0/1 1/7 1/6 1/5 3/14 2/9 1/4 5/18 2/7 3/10 1/3 11/30 3/8 2/5 5/12 7/16 1/2 5/9 3/5 19/30 2/3 7/10 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/18 -6/7 5/76 -5/6 1/15 2/29 -14/17 5/72 -9/11 1/14 -4/5 1/14 -15/19 11/150 -11/14 2/27 1/13 -7/9 1/14 -3/4 1/13 -8/11 1/12 -13/18 0/1 1/13 -5/7 3/38 -7/10 1/12 -9/13 5/58 -2/3 1/12 -9/14 1/11 2/21 -7/11 1/10 -19/30 1/10 -12/19 5/48 -5/8 0/1 -13/21 5/58 -8/13 3/32 -11/18 0/1 1/11 -3/5 1/10 -13/22 2/19 1/9 -23/39 7/62 -10/17 1/8 -17/29 1/10 -7/12 1/9 -18/31 1/8 -11/19 3/22 -15/26 0/1 1/5 -4/7 1/12 -9/16 0/1 -23/41 5/54 -14/25 1/10 -5/9 1/10 -1/2 0/1 1/9 -5/11 1/10 -9/20 1/9 -4/9 1/8 -7/16 0/1 -17/39 5/46 -10/23 1/8 -3/7 1/6 -17/40 0/1 -14/33 1/20 -11/26 0/1 1/13 -8/19 3/32 -21/50 1/10 -13/31 1/10 -5/12 1/9 -12/29 1/8 -7/17 1/10 -2/5 1/8 -9/23 3/22 -16/41 9/64 -39/100 1/7 -23/59 9/62 -7/18 0/1 1/7 -5/13 3/22 -8/21 5/32 -3/8 0/1 -7/19 5/42 -11/30 1/8 -15/41 11/86 -4/11 1/8 -5/14 2/15 1/7 -6/17 5/36 -7/20 1/7 -1/3 1/6 -3/10 1/6 -5/17 1/6 -12/41 17/96 -7/24 2/11 -2/7 3/16 -5/18 0/1 1/5 -8/29 1/4 -11/40 0/1 -3/11 1/6 -1/4 1/5 -2/9 1/4 -3/14 1/5 2/9 -4/19 11/48 -1/5 1/4 -4/21 13/48 -3/16 2/7 -2/11 1/4 -3/17 5/18 -7/40 2/7 -4/23 7/24 -1/6 2/7 1/3 -2/13 9/28 -3/20 1/3 -1/7 5/14 0/1 1/0 1/7 -5/14 1/6 -1/3 -2/7 3/17 -5/18 2/11 -1/4 1/5 -1/4 4/19 -11/48 3/14 -2/9 -1/5 2/9 -1/4 1/4 -1/5 3/11 -1/6 5/18 -1/5 0/1 2/7 -3/16 3/10 -1/6 4/13 -5/32 1/3 -1/6 5/14 -1/7 -2/15 4/11 -1/8 11/30 -1/8 7/19 -5/42 3/8 0/1 8/21 -5/32 5/13 -3/22 7/18 -1/7 0/1 2/5 -1/8 9/22 -2/17 -1/9 16/39 -7/64 7/17 -1/10 12/29 -1/8 5/12 -1/9 13/31 -1/10 8/19 -3/32 11/26 -1/13 0/1 3/7 -1/6 7/16 0/1 18/41 -5/36 11/25 -1/8 4/9 -1/8 1/2 -1/9 0/1 6/11 -1/8 11/20 -1/9 5/9 -1/10 9/16 0/1 22/39 -5/44 13/23 -1/10 4/7 -1/12 23/40 0/1 19/33 1/2 15/26 -1/5 0/1 11/19 -3/22 29/50 -1/8 18/31 -1/8 7/12 -1/9 17/29 -1/10 10/17 -1/8 3/5 -1/10 14/23 -3/32 25/41 -9/98 61/100 -1/11 36/59 -9/100 11/18 -1/11 0/1 8/13 -3/32 13/21 -5/58 5/8 0/1 12/19 -5/48 19/30 -1/10 26/41 -11/112 7/11 -1/10 9/14 -2/21 -1/11 11/17 -5/54 13/20 -1/11 2/3 -1/12 7/10 -1/12 12/17 -1/12 29/41 -17/210 17/24 -2/25 5/7 -3/38 13/18 -1/13 0/1 21/29 -1/14 29/40 0/1 8/11 -1/12 3/4 -1/13 7/9 -1/14 11/14 -1/13 -2/27 15/19 -11/150 4/5 -1/14 17/21 -13/186 13/16 -2/29 9/11 -1/14 14/17 -5/72 33/40 -2/29 19/23 -7/102 5/6 -2/29 -1/15 11/13 -9/134 17/20 -1/15 6/7 -5/76 1/1 -1/18 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(99,86,160,139) (-1/1,-6/7) -> (8/13,13/21) Hyperbolic Matrix(19,16,-120,-101) (-6/7,-5/6) -> (-1/6,-2/13) Hyperbolic Matrix(41,34,-240,-199) (-5/6,-14/17) -> (-4/23,-1/6) Hyperbolic Matrix(399,328,680,559) (-14/17,-9/11) -> (17/29,10/17) Hyperbolic Matrix(179,146,-320,-261) (-9/11,-4/5) -> (-14/25,-5/9) Hyperbolic Matrix(381,302,-680,-539) (-4/5,-15/19) -> (-23/41,-14/25) Hyperbolic Matrix(439,346,760,599) (-15/19,-11/14) -> (15/26,11/19) Hyperbolic Matrix(179,140,280,219) (-11/14,-7/9) -> (7/11,9/14) Hyperbolic Matrix(21,16,80,61) (-7/9,-3/4) -> (1/4,3/11) Hyperbolic Matrix(19,14,80,59) (-3/4,-8/11) -> (2/9,1/4) Hyperbolic Matrix(199,144,-720,-521) (-8/11,-13/18) -> (-5/18,-8/29) Hyperbolic Matrix(139,100,360,259) (-13/18,-5/7) -> (5/13,7/18) Hyperbolic Matrix(139,98,-200,-141) (-5/7,-7/10) -> (-7/10,-9/13) Parabolic Matrix(139,96,-320,-221) (-9/13,-2/3) -> (-10/23,-3/7) Hyperbolic Matrix(99,64,-280,-181) (-2/3,-9/14) -> (-5/14,-6/17) Hyperbolic Matrix(219,140,280,179) (-9/14,-7/11) -> (7/9,11/14) Hyperbolic Matrix(621,394,-1480,-939) (-7/11,-19/30) -> (-21/50,-13/31) Hyperbolic Matrix(639,404,-1520,-961) (-19/30,-12/19) -> (-8/19,-21/50) Hyperbolic Matrix(121,76,320,201) (-12/19,-5/8) -> (3/8,8/21) Hyperbolic Matrix(119,74,320,199) (-5/8,-13/21) -> (7/19,3/8) Hyperbolic Matrix(139,86,160,99) (-13/21,-8/13) -> (6/7,1/1) Hyperbolic Matrix(101,62,360,221) (-8/13,-11/18) -> (5/18,2/7) Hyperbolic Matrix(119,72,-200,-121) (-11/18,-3/5) -> (-3/5,-13/22) Parabolic Matrix(779,460,-2000,-1181) (-13/22,-23/39) -> (-23/59,-7/18) Hyperbolic Matrix(961,566,1360,801) (-23/39,-10/17) -> (12/17,29/41) Hyperbolic Matrix(559,328,680,399) (-10/17,-17/29) -> (9/11,14/17) Hyperbolic Matrix(301,176,720,421) (-17/29,-7/12) -> (5/12,13/31) Hyperbolic Matrix(299,174,720,419) (-7/12,-18/31) -> (12/29,5/12) Hyperbolic Matrix(541,314,-1480,-859) (-18/31,-11/19) -> (-15/41,-4/11) Hyperbolic Matrix(599,346,760,439) (-11/19,-15/26) -> (11/14,15/19) Hyperbolic Matrix(441,254,-1040,-599) (-15/26,-4/7) -> (-14/33,-11/26) Hyperbolic Matrix(81,46,-280,-159) (-4/7,-9/16) -> (-7/24,-2/7) Hyperbolic Matrix(1161,652,1640,921) (-9/16,-23/41) -> (29/41,17/24) Hyperbolic Matrix(19,10,-40,-21) (-5/9,-1/2) -> (-1/2,-5/11) Parabolic Matrix(221,100,400,181) (-5/11,-9/20) -> (11/20,5/9) Hyperbolic Matrix(219,98,400,179) (-9/20,-4/9) -> (6/11,11/20) Hyperbolic Matrix(59,26,-320,-141) (-4/9,-7/16) -> (-3/16,-2/11) Hyperbolic Matrix(779,340,960,419) (-7/16,-17/39) -> (17/21,13/16) Hyperbolic Matrix(1121,488,1840,801) (-17/39,-10/23) -> (14/23,25/41) Hyperbolic Matrix(921,392,1600,681) (-3/7,-17/40) -> (23/40,19/33) Hyperbolic Matrix(919,390,1600,679) (-17/40,-14/33) -> (4/7,23/40) Hyperbolic Matrix(161,68,760,321) (-11/26,-8/19) -> (4/19,3/14) Hyperbolic Matrix(421,176,720,301) (-13/31,-5/12) -> (7/12,17/29) Hyperbolic Matrix(419,174,720,299) (-5/12,-12/29) -> (18/31,7/12) Hyperbolic Matrix(121,50,680,281) (-12/29,-7/17) -> (3/17,2/11) Hyperbolic Matrix(79,32,-200,-81) (-7/17,-2/5) -> (-2/5,-9/23) Parabolic Matrix(1039,406,1840,719) (-9/23,-16/41) -> (22/39,13/23) Hyperbolic Matrix(6101,2380,10000,3901) (-16/41,-39/100) -> (61/100,36/59) Hyperbolic Matrix(6099,2378,10000,3899) (-39/100,-23/59) -> (25/41,61/100) Hyperbolic Matrix(259,100,360,139) (-7/18,-5/13) -> (5/7,13/18) Hyperbolic Matrix(21,8,160,61) (-5/13,-8/21) -> (0/1,1/7) Hyperbolic Matrix(201,76,320,121) (-8/21,-3/8) -> (5/8,12/19) Hyperbolic Matrix(199,74,320,119) (-3/8,-7/19) -> (13/21,5/8) Hyperbolic Matrix(659,242,-1800,-661) (-7/19,-11/30) -> (-11/30,-15/41) Parabolic Matrix(61,22,280,101) (-4/11,-5/14) -> (3/14,2/9) Hyperbolic Matrix(261,92,400,141) (-6/17,-7/20) -> (13/20,2/3) Hyperbolic Matrix(259,90,400,139) (-7/20,-1/3) -> (11/17,13/20) Hyperbolic Matrix(59,18,-200,-61) (-1/3,-3/10) -> (-3/10,-5/17) Parabolic Matrix(559,164,1360,399) (-5/17,-12/41) -> (16/39,7/17) Hyperbolic Matrix(719,210,1640,479) (-12/41,-7/24) -> (7/16,18/41) Hyperbolic Matrix(221,62,360,101) (-2/7,-5/18) -> (11/18,8/13) Hyperbolic Matrix(1161,320,1600,441) (-8/29,-11/40) -> (29/40,8/11) Hyperbolic Matrix(1159,318,1600,439) (-11/40,-3/11) -> (21/29,29/40) Hyperbolic Matrix(61,16,80,21) (-3/11,-1/4) -> (3/4,7/9) Hyperbolic Matrix(59,14,80,19) (-1/4,-2/9) -> (8/11,3/4) Hyperbolic Matrix(101,22,280,61) (-2/9,-3/14) -> (5/14,4/11) Hyperbolic Matrix(321,68,760,161) (-3/14,-4/19) -> (8/19,11/26) Hyperbolic Matrix(39,8,-200,-41) (-4/19,-1/5) -> (-1/5,-4/21) Parabolic Matrix(541,102,960,181) (-4/21,-3/16) -> (9/16,22/39) Hyperbolic Matrix(281,50,680,121) (-2/11,-3/17) -> (7/17,12/29) Hyperbolic Matrix(1321,232,1600,281) (-3/17,-7/40) -> (33/40,19/23) Hyperbolic Matrix(1319,230,1600,279) (-7/40,-4/23) -> (14/17,33/40) Hyperbolic Matrix(341,52,400,61) (-2/13,-3/20) -> (17/20,6/7) Hyperbolic Matrix(339,50,400,59) (-3/20,-1/7) -> (11/13,17/20) Hyperbolic Matrix(61,8,160,21) (-1/7,0/1) -> (8/21,5/13) Hyperbolic Matrix(101,-16,120,-19) (1/7,1/6) -> (5/6,11/13) Hyperbolic Matrix(199,-34,240,-41) (1/6,3/17) -> (19/23,5/6) Hyperbolic Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(299,-62,680,-141) (1/5,4/19) -> (18/41,11/25) Hyperbolic Matrix(521,-144,720,-199) (3/11,5/18) -> (13/18,21/29) Hyperbolic Matrix(61,-18,200,-59) (2/7,3/10) -> (3/10,4/13) Parabolic Matrix(181,-56,320,-99) (4/13,1/3) -> (13/23,4/7) Hyperbolic Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(859,-314,1480,-541) (4/11,11/30) -> (29/50,18/31) Hyperbolic Matrix(881,-324,1520,-559) (11/30,7/19) -> (11/19,29/50) Hyperbolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(1221,-500,2000,-819) (9/22,16/39) -> (36/59,11/18) Hyperbolic Matrix(939,-394,1480,-621) (13/31,8/19) -> (26/41,7/11) Hyperbolic Matrix(599,-254,1040,-441) (11/26,3/7) -> (19/33,15/26) Hyperbolic Matrix(199,-86,280,-121) (3/7,7/16) -> (17/24,5/7) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(261,-146,320,-179) (5/9,9/16) -> (13/16,9/11) Hyperbolic Matrix(121,-72,200,-119) (10/17,3/5) -> (3/5,14/23) Parabolic Matrix(1141,-722,1800,-1139) (12/19,19/30) -> (19/30,26/41) Parabolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(161,-128,200,-159) (15/19,4/5) -> (4/5,17/21) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-36,1) Matrix(99,86,160,139) -> Matrix(31,-2,-356,23) Matrix(19,16,-120,-101) -> Matrix(59,-4,192,-13) Matrix(41,34,-240,-199) -> Matrix(59,-4,192,-13) Matrix(399,328,680,559) -> Matrix(29,-2,-304,21) Matrix(179,146,-320,-261) -> Matrix(29,-2,276,-19) Matrix(381,302,-680,-539) -> Matrix(55,-4,564,-41) Matrix(439,346,760,599) -> Matrix(27,-2,-148,11) Matrix(179,140,280,219) -> Matrix(1,0,-24,1) Matrix(21,16,80,61) -> Matrix(27,-2,-148,11) Matrix(19,14,80,59) -> Matrix(25,-2,-112,9) Matrix(199,144,-720,-521) -> Matrix(1,0,-8,1) Matrix(139,100,360,259) -> Matrix(1,0,-20,1) Matrix(139,98,-200,-141) -> Matrix(49,-4,576,-47) Matrix(139,96,-320,-221) -> Matrix(23,-2,196,-17) Matrix(99,64,-280,-181) -> Matrix(43,-4,312,-29) Matrix(219,140,280,179) -> Matrix(1,0,-24,1) Matrix(621,394,-1480,-939) -> Matrix(41,-4,400,-39) Matrix(639,404,-1520,-961) -> Matrix(39,-4,400,-41) Matrix(121,76,320,201) -> Matrix(1,0,-16,1) Matrix(119,74,320,199) -> Matrix(1,0,-20,1) Matrix(139,86,160,99) -> Matrix(23,-2,-356,31) Matrix(101,62,360,221) -> Matrix(1,0,-16,1) Matrix(119,72,-200,-121) -> Matrix(21,-2,200,-19) Matrix(779,460,-2000,-1181) -> Matrix(19,-2,124,-13) Matrix(961,566,1360,801) -> Matrix(33,-4,-404,49) Matrix(559,328,680,399) -> Matrix(21,-2,-304,29) Matrix(301,176,720,421) -> Matrix(19,-2,-180,19) Matrix(299,174,720,419) -> Matrix(17,-2,-144,17) Matrix(541,314,-1480,-859) -> Matrix(33,-4,256,-31) Matrix(599,346,760,439) -> Matrix(11,-2,-148,27) Matrix(441,254,-1040,-599) -> Matrix(1,0,8,1) Matrix(81,46,-280,-159) -> Matrix(21,-2,116,-11) Matrix(1161,652,1640,921) -> Matrix(25,-2,-312,25) Matrix(19,10,-40,-21) -> Matrix(1,0,0,1) Matrix(221,100,400,181) -> Matrix(19,-2,-180,19) Matrix(219,98,400,179) -> Matrix(17,-2,-144,17) Matrix(59,26,-320,-141) -> Matrix(17,-2,60,-7) Matrix(779,340,960,419) -> Matrix(21,-2,-304,29) Matrix(1121,488,1840,801) -> Matrix(35,-4,-376,43) Matrix(921,392,1600,681) -> Matrix(1,0,-4,1) Matrix(919,390,1600,679) -> Matrix(1,0,-32,1) Matrix(161,68,760,321) -> Matrix(25,-2,-112,9) Matrix(421,176,720,301) -> Matrix(19,-2,-180,19) Matrix(419,174,720,299) -> Matrix(17,-2,-144,17) Matrix(121,50,680,281) -> Matrix(15,-2,-52,7) Matrix(79,32,-200,-81) -> Matrix(17,-2,128,-15) Matrix(1039,406,1840,719) -> Matrix(29,-4,-268,37) Matrix(6101,2380,10000,3901) -> Matrix(127,-18,-1404,199) Matrix(6099,2378,10000,3899) -> Matrix(125,-18,-1368,197) Matrix(259,100,360,139) -> Matrix(1,0,-20,1) Matrix(21,8,160,61) -> Matrix(13,-2,-32,5) Matrix(201,76,320,121) -> Matrix(1,0,-16,1) Matrix(199,74,320,119) -> Matrix(1,0,-20,1) Matrix(659,242,-1800,-661) -> Matrix(65,-8,512,-63) Matrix(61,22,280,101) -> Matrix(1,0,-12,1) Matrix(261,92,400,141) -> Matrix(43,-6,-480,67) Matrix(259,90,400,139) -> Matrix(41,-6,-444,65) Matrix(59,18,-200,-61) -> Matrix(25,-4,144,-23) Matrix(559,164,1360,399) -> Matrix(23,-4,-224,39) Matrix(719,210,1640,479) -> Matrix(11,-2,-60,11) Matrix(221,62,360,101) -> Matrix(1,0,-16,1) Matrix(1161,320,1600,441) -> Matrix(1,0,-16,1) Matrix(1159,318,1600,439) -> Matrix(1,0,-20,1) Matrix(61,16,80,21) -> Matrix(11,-2,-148,27) Matrix(59,14,80,19) -> Matrix(9,-2,-112,25) Matrix(101,22,280,61) -> Matrix(1,0,-12,1) Matrix(321,68,760,161) -> Matrix(9,-2,-112,25) Matrix(39,8,-200,-41) -> Matrix(25,-6,96,-23) Matrix(541,102,960,181) -> Matrix(7,-2,-52,15) Matrix(281,50,680,121) -> Matrix(7,-2,-52,15) Matrix(1321,232,1600,281) -> Matrix(85,-24,-1236,349) Matrix(1319,230,1600,279) -> Matrix(83,-24,-1200,347) Matrix(341,52,400,61) -> Matrix(43,-14,-648,211) Matrix(339,50,400,59) -> Matrix(41,-14,-612,209) Matrix(61,8,160,21) -> Matrix(5,-2,-32,13) Matrix(101,-16,120,-19) -> Matrix(13,4,-192,-59) Matrix(199,-34,240,-41) -> Matrix(13,4,-192,-59) Matrix(141,-26,320,-59) -> Matrix(7,2,-60,-17) Matrix(299,-62,680,-141) -> Matrix(17,4,-132,-31) Matrix(521,-144,720,-199) -> Matrix(1,0,-8,1) Matrix(61,-18,200,-59) -> Matrix(23,4,-144,-25) Matrix(181,-56,320,-99) -> Matrix(13,2,-124,-19) Matrix(181,-64,280,-99) -> Matrix(29,4,-312,-43) Matrix(859,-314,1480,-541) -> Matrix(31,4,-256,-33) Matrix(881,-324,1520,-559) -> Matrix(33,4,-256,-31) Matrix(81,-32,200,-79) -> Matrix(15,2,-128,-17) Matrix(1221,-500,2000,-819) -> Matrix(17,2,-196,-23) Matrix(939,-394,1480,-621) -> Matrix(39,4,-400,-41) Matrix(599,-254,1040,-441) -> Matrix(1,0,8,1) Matrix(199,-86,280,-121) -> Matrix(15,2,-188,-25) Matrix(21,-10,40,-19) -> Matrix(1,0,0,1) Matrix(261,-146,320,-179) -> Matrix(19,2,-276,-29) Matrix(121,-72,200,-119) -> Matrix(19,2,-200,-21) Matrix(1141,-722,1800,-1139) -> Matrix(79,8,-800,-81) Matrix(141,-98,200,-139) -> Matrix(47,4,-576,-49) Matrix(161,-128,200,-159) -> Matrix(83,6,-1176,-85) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 24 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 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): 144 Minimal number of generators: 25 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: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/7 1/6 1/5 2/9 1/4 3/10 1/3 11/30 3/8 2/5 1/2 11/20 3/5 5/8 13/20 3/4 4/5 17/20 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/7 -5/14 1/6 -1/3 -2/7 1/5 -1/4 3/14 -2/9 -1/5 2/9 -1/4 1/4 -1/5 3/11 -1/6 2/7 -3/16 3/10 -1/6 1/3 -1/6 5/14 -1/7 -2/15 4/11 -1/8 11/30 -1/8 7/19 -5/42 3/8 0/1 2/5 -1/8 5/12 -1/9 8/19 -3/32 3/7 -1/6 4/9 -1/8 1/2 -1/9 0/1 6/11 -1/8 11/20 -1/9 5/9 -1/10 4/7 -1/12 3/5 -1/10 8/13 -3/32 13/21 -5/58 5/8 0/1 12/19 -5/48 19/30 -1/10 7/11 -1/10 9/14 -2/21 -1/11 11/17 -5/54 13/20 -1/11 2/3 -1/12 7/10 -1/12 5/7 -3/38 8/11 -1/12 3/4 -1/13 7/9 -1/14 4/5 -1/14 9/11 -1/14 5/6 -2/29 -1/15 11/13 -9/134 17/20 -1/15 6/7 -5/76 1/1 -1/18 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(99,-13,160,-21) (0/1,1/7) -> (8/13,13/21) Hyperbolic Matrix(101,-16,120,-19) (1/7,1/6) -> (5/6,11/13) Hyperbolic Matrix(21,-4,100,-19) (1/6,1/5) -> (1/5,3/14) Parabolic Matrix(179,-39,280,-61) (3/14,2/9) -> (7/11,9/14) Hyperbolic Matrix(21,-5,80,-19) (2/9,1/4) -> (1/4,3/11) Parabolic Matrix(61,-17,140,-39) (3/11,2/7) -> (3/7,4/9) Hyperbolic Matrix(99,-29,140,-41) (2/7,3/10) -> (7/10,5/7) Hyperbolic Matrix(41,-13,60,-19) (3/10,1/3) -> (2/3,7/10) Hyperbolic Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(181,-65,220,-79) (5/14,4/11) -> (9/11,5/6) Hyperbolic Matrix(419,-153,660,-241) (4/11,11/30) -> (19/30,7/11) Hyperbolic Matrix(721,-265,1140,-419) (11/30,7/19) -> (12/19,19/30) Hyperbolic Matrix(159,-59,380,-141) (7/19,3/8) -> (5/12,8/19) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(121,-51,140,-59) (8/19,3/7) -> (6/7,1/1) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(221,-121,400,-219) (6/11,11/20) -> (11/20,5/9) Parabolic Matrix(101,-57,140,-79) (5/9,4/7) -> (5/7,8/11) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(201,-125,320,-199) (13/21,5/8) -> (5/8,12/19) Parabolic Matrix(261,-169,400,-259) (11/17,13/20) -> (13/20,2/3) Parabolic Matrix(61,-45,80,-59) (8/11,3/4) -> (3/4,7/9) Parabolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(341,-289,400,-339) (11/13,17/20) -> (17/20,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-18,1) Matrix(99,-13,160,-21) -> Matrix(5,2,-58,-23) Matrix(101,-16,120,-19) -> Matrix(13,4,-192,-59) Matrix(21,-4,100,-19) -> Matrix(11,3,-48,-13) Matrix(179,-39,280,-61) -> Matrix(1,0,-6,1) Matrix(21,-5,80,-19) -> Matrix(9,2,-50,-11) Matrix(61,-17,140,-39) -> Matrix(5,1,-46,-9) Matrix(99,-29,140,-41) -> Matrix(17,3,-210,-37) Matrix(41,-13,60,-19) -> Matrix(7,1,-78,-11) Matrix(181,-64,280,-99) -> Matrix(29,4,-312,-43) Matrix(181,-65,220,-79) -> Matrix(23,3,-330,-43) Matrix(419,-153,660,-241) -> Matrix(23,3,-238,-31) Matrix(721,-265,1140,-419) -> Matrix(41,5,-402,-49) Matrix(159,-59,380,-141) -> Matrix(9,1,-82,-9) Matrix(41,-16,100,-39) -> Matrix(7,1,-64,-9) Matrix(121,-51,140,-59) -> Matrix(11,1,-166,-15) Matrix(21,-10,40,-19) -> Matrix(1,0,0,1) Matrix(221,-121,400,-219) -> Matrix(17,2,-162,-19) Matrix(101,-57,140,-79) -> Matrix(9,1,-118,-13) Matrix(61,-36,100,-59) -> Matrix(9,1,-100,-11) Matrix(201,-125,320,-199) -> Matrix(1,0,2,1) Matrix(261,-169,400,-259) -> Matrix(65,6,-726,-67) Matrix(61,-45,80,-59) -> Matrix(25,2,-338,-27) Matrix(81,-64,100,-79) -> Matrix(41,3,-588,-43) Matrix(341,-289,400,-339) -> Matrix(209,14,-3150,-211) 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): 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 1/0 1 20 1/7 -5/14 1 20 3/20 -1/3 14 1 1/6 (-1/3,-2/7) 0 10 1/5 -1/4 3 4 3/14 (-2/9,-1/5) 0 10 2/9 -1/4 1 20 1/4 -1/5 2 5 3/11 -1/6 1 20 2/7 -3/16 1 20 3/10 -1/6 4 2 1/3 -1/6 1 20 7/20 -1/7 6 1 5/14 (-1/7,-2/15) 0 10 4/11 -1/8 1 20 11/30 -1/8 8 2 7/19 -5/42 1 20 3/8 0/1 2 5 2/5 -1/8 1 4 5/12 -1/9 2 5 8/19 -3/32 1 20 3/7 -1/6 1 20 4/9 -1/8 1 20 9/20 -1/9 2 1 1/2 (-1/9,0/1) 0 10 1/0 0/1 18 1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(59,-8,140,-19) (0/1,1/7) -> (8/19,3/7) Glide Reflection Matrix(41,-6,280,-41) (1/7,3/20) -> (1/7,3/20) Reflection Matrix(19,-3,120,-19) (3/20,1/6) -> (3/20,1/6) Reflection Matrix(21,-4,100,-19) (1/6,1/5) -> (1/5,3/14) Parabolic Matrix(101,-22,280,-61) (3/14,2/9) -> (5/14,4/11) Glide Reflection Matrix(21,-5,80,-19) (2/9,1/4) -> (1/4,3/11) Parabolic Matrix(61,-17,140,-39) (3/11,2/7) -> (3/7,4/9) Hyperbolic Matrix(41,-12,140,-41) (2/7,3/10) -> (2/7,3/10) Reflection Matrix(19,-6,60,-19) (3/10,1/3) -> (3/10,1/3) Reflection Matrix(41,-14,120,-41) (1/3,7/20) -> (1/3,7/20) Reflection Matrix(99,-35,280,-99) (7/20,5/14) -> (7/20,5/14) Reflection Matrix(241,-88,660,-241) (4/11,11/30) -> (4/11,11/30) Reflection Matrix(419,-154,1140,-419) (11/30,7/19) -> (11/30,7/19) Reflection Matrix(159,-59,380,-141) (7/19,3/8) -> (5/12,8/19) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(161,-72,360,-161) (4/9,9/20) -> (4/9,9/20) Reflection Matrix(19,-9,40,-19) (9/20,1/2) -> (9/20,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(59,-8,140,-19) -> Matrix(3,1,-32,-11) Matrix(41,-6,280,-41) -> Matrix(29,10,-84,-29) (1/7,3/20) -> (-5/14,-1/3) Matrix(19,-3,120,-19) -> Matrix(13,4,-42,-13) (3/20,1/6) -> (-1/3,-2/7) Matrix(21,-4,100,-19) -> Matrix(11,3,-48,-13) -1/4 Matrix(101,-22,280,-61) -> Matrix(-1,0,12,1) *** -> (-1/6,0/1) Matrix(21,-5,80,-19) -> Matrix(9,2,-50,-11) -1/5 Matrix(61,-17,140,-39) -> Matrix(5,1,-46,-9) Matrix(41,-12,140,-41) -> Matrix(17,3,-96,-17) (2/7,3/10) -> (-3/16,-1/6) Matrix(19,-6,60,-19) -> Matrix(7,1,-48,-7) (3/10,1/3) -> (-1/6,-1/8) Matrix(41,-14,120,-41) -> Matrix(13,2,-84,-13) (1/3,7/20) -> (-1/6,-1/7) Matrix(99,-35,280,-99) -> Matrix(29,4,-210,-29) (7/20,5/14) -> (-1/7,-2/15) Matrix(241,-88,660,-241) -> Matrix(23,3,-176,-23) (4/11,11/30) -> (-3/22,-1/8) Matrix(419,-154,1140,-419) -> Matrix(41,5,-336,-41) (11/30,7/19) -> (-1/8,-5/42) Matrix(159,-59,380,-141) -> Matrix(9,1,-82,-9) (-1/8,-1/10).(-1/9,0/1) Matrix(41,-16,100,-39) -> Matrix(7,1,-64,-9) -1/8 Matrix(161,-72,360,-161) -> Matrix(17,2,-144,-17) (4/9,9/20) -> (-1/8,-1/9) Matrix(19,-9,40,-19) -> Matrix(-1,0,18,1) (9/20,1/2) -> (-1/9,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,18,1) (1/2,1/0) -> (-1/9,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.