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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -6/1 -4/1 -10/3 -8/3 -12/5 -2/1 -12/7 -8/5 -4/3 -8/7 0/1 1/1 8/7 4/3 3/2 8/5 12/7 2/1 16/7 12/5 5/2 8/3 14/5 48/17 3/1 16/5 10/3 64/19 7/2 4/1 32/7 14/3 5/1 16/3 6/1 32/5 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 1/2 -7/1 7/12 -6/1 1/1 -5/1 1/2 -4/1 0/1 2/3 -11/3 1/2 -18/5 7/11 -7/2 7/10 -24/7 3/4 -17/5 3/4 -10/3 1/1 -3/1 1/2 -8/3 1/1 -13/5 3/2 -31/12 1/0 -18/7 -1/1 -5/2 1/2 -22/9 5/7 -17/7 3/4 -12/5 2/3 4/5 -31/13 3/4 -19/8 11/14 -26/11 9/11 -7/3 7/8 -2/1 1/1 -9/5 5/6 -16/9 1/1 -7/4 7/6 -19/11 11/8 -12/7 4/3 2/1 -29/17 11/8 -17/10 3/2 -5/3 1/0 -8/5 1/1 -11/7 5/4 -14/9 7/5 -31/20 3/2 -48/31 3/2 -17/11 3/2 -3/2 1/0 -16/11 1/1 -13/9 5/4 -10/7 1/1 -37/26 13/10 -64/45 4/3 -27/19 11/8 -17/12 3/2 -24/17 3/2 -7/5 7/4 -4/3 0/1 2/1 -9/7 7/4 -32/25 2/1 -23/18 21/10 -14/11 7/3 -5/4 1/0 -16/13 1/0 -11/9 1/0 -6/5 1/1 -19/16 11/6 -32/27 2/1 -13/11 9/4 -7/6 7/2 -8/7 1/0 -1/1 1/0 0/1 0/1 1/1 1/4 8/7 1/4 7/6 7/26 6/5 1/3 5/4 1/4 4/3 0/1 2/7 11/8 1/4 18/13 7/25 7/5 7/24 24/17 3/10 17/12 3/10 10/7 1/3 3/2 1/4 8/5 1/3 13/8 3/8 31/19 1/2 18/11 1/1 5/3 1/4 22/13 5/17 17/10 3/10 12/7 2/7 4/13 31/18 3/10 19/11 11/36 26/15 9/29 7/4 7/22 2/1 1/3 9/4 5/16 16/7 1/3 7/3 7/20 19/8 11/30 12/5 4/11 2/5 29/12 11/30 17/7 3/8 5/2 1/2 8/3 1/3 11/4 5/14 14/5 7/19 31/11 3/8 48/17 3/8 17/6 3/8 3/1 1/2 16/5 1/3 13/4 5/14 10/3 1/3 37/11 13/36 64/19 4/11 27/8 11/30 17/5 3/8 24/7 3/8 7/2 7/18 4/1 0/1 2/5 9/2 7/18 32/7 2/5 23/5 21/52 14/3 7/17 5/1 1/2 16/3 1/2 11/2 1/2 6/1 1/3 19/3 11/28 32/5 2/5 13/2 9/22 7/1 7/16 8/1 1/2 1/0 1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,160,12,113) (-8/1,1/0) -> (24/17,17/12) Hyperbolic Matrix(31,224,22,159) (-8/1,-7/1) -> (7/5,24/17) Hyperbolic Matrix(33,224,-14,-95) (-7/1,-6/1) -> (-26/11,-7/3) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(79,288,48,175) (-11/3,-18/5) -> (18/11,5/3) Hyperbolic Matrix(143,512,-112,-401) (-18/5,-7/2) -> (-23/18,-14/11) Hyperbolic Matrix(65,224,56,193) (-7/2,-24/7) -> (8/7,7/6) Hyperbolic Matrix(47,160,42,143) (-24/7,-17/5) -> (1/1,8/7) Hyperbolic Matrix(161,544,-66,-223) (-17/5,-10/3) -> (-22/9,-17/7) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic Matrix(383,992,222,575) (-13/5,-31/12) -> (31/18,19/11) Hyperbolic Matrix(385,992,-248,-639) (-31/12,-18/7) -> (-14/9,-31/20) Hyperbolic Matrix(113,288,82,209) (-18/7,-5/2) -> (11/8,18/13) Hyperbolic Matrix(353,864,-248,-607) (-5/2,-22/9) -> (-10/7,-37/26) Hyperbolic Matrix(239,576,-100,-241) (-17/7,-12/5) -> (-12/5,-31/13) Parabolic Matrix(417,992,256,609) (-31/13,-19/8) -> (13/8,31/19) Hyperbolic Matrix(257,608,-216,-511) (-19/8,-26/11) -> (-6/5,-19/16) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(129,224,-110,-191) (-7/4,-19/11) -> (-13/11,-7/6) Hyperbolic Matrix(335,576,-196,-337) (-19/11,-12/7) -> (-12/7,-29/17) Parabolic Matrix(319,544,112,191) (-29/17,-17/10) -> (17/6,3/1) Hyperbolic Matrix(321,544,-226,-383) (-17/10,-5/3) -> (-27/19,-17/12) Hyperbolic Matrix(79,128,-50,-81) (-5/3,-8/5) -> (-8/5,-11/7) Parabolic Matrix(143,224,30,47) (-11/7,-14/9) -> (14/3,5/1) Hyperbolic Matrix(1487,2304,526,815) (-31/20,-48/31) -> (48/17,17/6) Hyperbolic Matrix(1489,2304,528,817) (-48/31,-17/11) -> (31/11,48/17) Hyperbolic Matrix(353,544,146,225) (-17/11,-3/2) -> (29/12,17/7) Hyperbolic Matrix(175,256,54,79) (-3/2,-16/11) -> (16/5,13/4) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(2879,4096,854,1215) (-37/26,-64/45) -> (64/19,27/8) Hyperbolic Matrix(2881,4096,856,1217) (-64/45,-27/19) -> (37/11,64/19) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(799,1024,174,223) (-9/7,-32/25) -> (32/7,23/5) Hyperbolic Matrix(801,1024,176,225) (-32/25,-23/18) -> (9/2,32/7) Hyperbolic Matrix(177,224,64,81) (-14/11,-5/4) -> (11/4,14/5) Hyperbolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(863,1024,134,159) (-19/16,-32/27) -> (32/5,13/2) Hyperbolic Matrix(865,1024,136,161) (-32/27,-13/11) -> (19/3,32/5) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(143,160,42,47) (-8/7,-1/1) -> (17/5,24/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(191,-224,110,-129) (7/6,6/5) -> (26/15,7/4) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(81,-128,50,-79) (3/2,8/5) -> (8/5,13/8) Parabolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(511,-864,152,-257) (5/3,22/13) -> (10/3,37/11) Hyperbolic Matrix(337,-576,196,-335) (17/10,12/7) -> (12/7,31/18) Parabolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(241,-576,100,-239) (19/8,12/5) -> (12/5,29/12) Parabolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,160,12,113) -> Matrix(1,-2,4,-7) Matrix(31,224,22,159) -> Matrix(25,-14,84,-47) Matrix(33,224,-14,-95) -> Matrix(23,-14,28,-17) Matrix(17,96,-14,-79) -> Matrix(3,-2,2,-1) Matrix(15,64,-4,-17) -> Matrix(1,0,0,1) Matrix(79,288,48,175) -> Matrix(3,-2,14,-9) Matrix(143,512,-112,-401) -> Matrix(43,-28,20,-13) Matrix(65,224,56,193) -> Matrix(19,-14,72,-53) Matrix(47,160,42,143) -> Matrix(3,-2,8,-5) Matrix(161,544,-66,-223) -> Matrix(23,-18,32,-25) Matrix(49,160,-34,-111) -> Matrix(7,-6,6,-5) Matrix(47,128,-18,-49) -> Matrix(5,-4,4,-3) Matrix(383,992,222,575) -> Matrix(3,-10,10,-33) Matrix(385,992,-248,-639) -> Matrix(3,10,2,7) Matrix(113,288,82,209) -> Matrix(5,-2,18,-7) Matrix(353,864,-248,-607) -> Matrix(31,-22,24,-17) Matrix(239,576,-100,-241) -> Matrix(1,0,0,1) Matrix(417,992,256,609) -> Matrix(13,-10,30,-23) Matrix(257,608,-216,-511) -> Matrix(27,-22,16,-13) Matrix(15,32,-8,-17) -> Matrix(3,-2,2,-1) Matrix(143,256,62,111) -> Matrix(13,-12,38,-35) Matrix(145,256,64,113) -> Matrix(11,-12,34,-37) Matrix(129,224,-110,-191) -> Matrix(11,-14,4,-5) Matrix(335,576,-196,-337) -> Matrix(1,0,0,1) Matrix(319,544,112,191) -> Matrix(13,-18,34,-47) Matrix(321,544,-226,-383) -> Matrix(11,-18,8,-13) Matrix(79,128,-50,-81) -> Matrix(5,-4,4,-3) Matrix(143,224,30,47) -> Matrix(11,-14,26,-33) Matrix(1487,2304,526,815) -> Matrix(73,-108,194,-287) Matrix(1489,2304,528,817) -> Matrix(71,-108,190,-289) Matrix(353,544,146,225) -> Matrix(11,-18,30,-49) Matrix(175,256,54,79) -> Matrix(5,-4,14,-11) Matrix(177,256,56,81) -> Matrix(3,-4,10,-13) Matrix(2879,4096,854,1215) -> Matrix(73,-96,200,-263) Matrix(2881,4096,856,1217) -> Matrix(71,-96,196,-265) Matrix(113,160,12,17) -> Matrix(1,-2,4,-7) Matrix(159,224,22,31) -> Matrix(9,-14,20,-31) Matrix(47,64,-36,-49) -> Matrix(1,0,0,1) Matrix(799,1024,174,223) -> Matrix(29,-56,72,-139) Matrix(801,1024,176,225) -> Matrix(27,-56,68,-141) Matrix(177,224,64,81) -> Matrix(5,-14,14,-39) Matrix(207,256,38,47) -> Matrix(1,-4,2,-7) Matrix(209,256,40,49) -> Matrix(1,4,2,9) Matrix(863,1024,134,159) -> Matrix(21,-40,52,-99) Matrix(865,1024,136,161) -> Matrix(19,-40,48,-101) Matrix(193,224,56,65) -> Matrix(3,-14,8,-37) Matrix(143,160,42,47) -> Matrix(3,-2,8,-5) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(191,-224,110,-129) -> Matrix(51,-14,164,-45) Matrix(79,-96,14,-17) -> Matrix(7,-2,18,-5) Matrix(49,-64,36,-47) -> Matrix(1,0,0,1) Matrix(369,-512,80,-111) -> Matrix(99,-28,244,-69) Matrix(383,-544,226,-321) -> Matrix(59,-18,200,-61) Matrix(111,-160,34,-49) -> Matrix(19,-6,54,-17) Matrix(81,-128,50,-79) -> Matrix(13,-4,36,-11) Matrix(607,-992,216,-353) -> Matrix(17,-10,46,-27) Matrix(511,-864,152,-257) -> Matrix(75,-22,208,-61) Matrix(337,-576,196,-335) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(71,-22,184,-57) Matrix(17,-32,8,-15) -> Matrix(7,-2,18,-5) Matrix(95,-224,14,-33) -> Matrix(39,-14,92,-33) Matrix(241,-576,100,-239) -> Matrix(1,0,0,1) Matrix(223,-544,66,-161) -> Matrix(47,-18,128,-49) Matrix(49,-128,18,-47) -> Matrix(13,-4,36,-11) Matrix(17,-64,4,-15) -> 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): 32 Degree of the the map X: 32 Degree of the the map Y: 64 Permutation triple for Y: ((2,6,22,7)(3,12,13,4)(5,18)(8,26)(9,10)(15,16)(17,44,52,27)(19,48,54,25)(21,51)(23,53)(28,58,42,29)(30,50,55,41)(31,60,40,32)(33,61,39,46)(34,35)(37,38); (1,4,16,42,56,54,53,61,64,50,34,31,43,17,5,2)(3,10,32,49,25,8,7,24,55,37,58,62,44,51,33,11)(6,20,30,9,29,45,27,26,39,14,13,38,60,63,48,21)(12,35,28,47,19,18,46,59,41,15,40,57,52,23,22,36); (1,2,8,27,57,40,38,55,64,61,51,48,47,28,9,3)(4,14,39,53,52,62,58,35,50,20,6,5,19,49,32,15)(7,23,54,63,60,34,12,11,33,18,17,45,29,16,41,24)(10,30,59,46,26,25,56,42,37,13,36,22,21,44,43,31)) ----------------------------------------------------------------------- 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): 192 Minimal number of generators: 33 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: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 8/7 4/3 8/5 12/7 2/1 16/7 12/5 8/3 48/17 3/1 16/5 10/3 64/19 4/1 32/7 14/3 5/1 16/3 6/1 32/5 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/4 8/7 1/4 7/6 7/26 6/5 1/3 5/4 1/4 4/3 0/1 2/7 11/8 1/4 18/13 7/25 7/5 7/24 24/17 3/10 17/12 3/10 10/7 1/3 3/2 1/4 8/5 1/3 13/8 3/8 31/19 1/2 18/11 1/1 5/3 1/4 22/13 5/17 17/10 3/10 12/7 2/7 4/13 31/18 3/10 19/11 11/36 26/15 9/29 7/4 7/22 2/1 1/3 9/4 5/16 16/7 1/3 7/3 7/20 19/8 11/30 12/5 4/11 2/5 29/12 11/30 17/7 3/8 5/2 1/2 8/3 1/3 11/4 5/14 14/5 7/19 31/11 3/8 48/17 3/8 17/6 3/8 3/1 1/2 16/5 1/3 13/4 5/14 10/3 1/3 37/11 13/36 64/19 4/11 27/8 11/30 17/5 3/8 24/7 3/8 7/2 7/18 4/1 0/1 2/5 9/2 7/18 32/7 2/5 23/5 21/52 14/3 7/17 5/1 1/2 16/3 1/2 11/2 1/2 6/1 1/3 19/3 11/28 32/5 2/5 13/2 9/22 7/1 7/16 8/1 1/2 1/0 1/2 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(143,-160,101,-113) (1/1,8/7) -> (24/17,17/12) Hyperbolic Matrix(193,-224,137,-159) (8/7,7/6) -> (7/5,24/17) Hyperbolic Matrix(191,-224,110,-129) (7/6,6/5) -> (26/15,7/4) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(209,-288,127,-175) (11/8,18/13) -> (18/11,5/3) Hyperbolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(81,-128,50,-79) (3/2,8/5) -> (8/5,13/8) Parabolic Matrix(609,-992,353,-575) (13/8,31/19) -> (31/18,19/11) Hyperbolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(511,-864,152,-257) (5/3,22/13) -> (10/3,37/11) Hyperbolic Matrix(337,-576,196,-335) (17/10,12/7) -> (12/7,31/18) Parabolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(113,-256,49,-111) (9/4,16/7) -> (16/7,7/3) Parabolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(241,-576,100,-239) (19/8,12/5) -> (12/5,29/12) Parabolic Matrix(225,-544,79,-191) (29/12,17/7) -> (17/6,3/1) Hyperbolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(81,-224,17,-47) (11/4,14/5) -> (14/3,5/1) Hyperbolic Matrix(817,-2304,289,-815) (31/11,48/17) -> (48/17,17/6) Parabolic Matrix(81,-256,25,-79) (3/1,16/5) -> (16/5,13/4) Parabolic Matrix(1217,-4096,361,-1215) (37/11,64/19) -> (64/19,27/8) Parabolic Matrix(47,-160,5,-17) (17/5,24/7) -> (8/1,1/0) Hyperbolic Matrix(65,-224,9,-31) (24/7,7/2) -> (7/1,8/1) Hyperbolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(225,-1024,49,-223) (9/2,32/7) -> (32/7,23/5) Parabolic Matrix(49,-256,9,-47) (5/1,16/3) -> (16/3,11/2) Parabolic Matrix(161,-1024,25,-159) (19/3,32/5) -> (32/5,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(143,-160,101,-113) -> Matrix(5,-2,18,-7) Matrix(193,-224,137,-159) -> Matrix(53,-14,178,-47) Matrix(191,-224,110,-129) -> Matrix(51,-14,164,-45) Matrix(79,-96,14,-17) -> Matrix(7,-2,18,-5) Matrix(49,-64,36,-47) -> Matrix(1,0,0,1) Matrix(209,-288,127,-175) -> Matrix(7,-2,32,-9) Matrix(369,-512,80,-111) -> Matrix(99,-28,244,-69) Matrix(383,-544,226,-321) -> Matrix(59,-18,200,-61) Matrix(111,-160,34,-49) -> Matrix(19,-6,54,-17) Matrix(81,-128,50,-79) -> Matrix(13,-4,36,-11) Matrix(609,-992,353,-575) -> Matrix(23,-10,76,-33) Matrix(607,-992,216,-353) -> Matrix(17,-10,46,-27) Matrix(511,-864,152,-257) -> Matrix(75,-22,208,-61) Matrix(337,-576,196,-335) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(71,-22,184,-57) Matrix(17,-32,8,-15) -> Matrix(7,-2,18,-5) Matrix(113,-256,49,-111) -> Matrix(37,-12,108,-35) Matrix(95,-224,14,-33) -> Matrix(39,-14,92,-33) Matrix(241,-576,100,-239) -> Matrix(1,0,0,1) Matrix(225,-544,79,-191) -> Matrix(49,-18,128,-47) Matrix(223,-544,66,-161) -> Matrix(47,-18,128,-49) Matrix(49,-128,18,-47) -> Matrix(13,-4,36,-11) Matrix(81,-224,17,-47) -> Matrix(39,-14,92,-33) Matrix(817,-2304,289,-815) -> Matrix(289,-108,768,-287) Matrix(81,-256,25,-79) -> Matrix(13,-4,36,-11) Matrix(1217,-4096,361,-1215) -> Matrix(265,-96,726,-263) Matrix(47,-160,5,-17) -> Matrix(5,-2,18,-7) Matrix(65,-224,9,-31) -> Matrix(37,-14,82,-31) Matrix(17,-64,4,-15) -> Matrix(1,0,0,1) Matrix(225,-1024,49,-223) -> Matrix(141,-56,350,-139) Matrix(49,-256,9,-47) -> Matrix(9,-4,16,-7) Matrix(161,-1024,25,-159) -> Matrix(101,-40,250,-99) 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 2 1 2/1 1/3 2 8 16/7 1/3 12 1 7/3 7/20 1 16 19/8 11/30 1 16 12/5 0 4 29/12 11/30 1 16 17/7 3/8 1 16 5/2 1/2 1 16 8/3 1/3 4 2 11/4 5/14 1 16 14/5 7/19 2 8 48/17 3/8 12 1 17/6 3/8 1 16 3/1 1/2 1 16 16/5 1/3 4 1 10/3 1/3 2 8 64/19 4/11 6 1 27/8 11/30 1 16 17/5 3/8 1 16 24/7 3/8 4 2 7/2 7/18 1 16 4/1 0 4 9/2 7/18 1 16 32/7 2/5 14 1 14/3 7/17 2 8 5/1 1/2 1 16 16/3 1/2 4 1 6/1 1/3 2 8 32/5 2/5 10 1 13/2 9/22 1 16 7/1 7/16 1 16 8/1 1/2 4 2 1/0 1/2 1 16 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(15,-32,7,-15) (2/1,16/7) -> (2/1,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(241,-576,100,-239) (19/8,12/5) -> (12/5,29/12) Parabolic Matrix(225,-544,79,-191) (29/12,17/7) -> (17/6,3/1) Hyperbolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(81,-224,17,-47) (11/4,14/5) -> (14/3,5/1) Hyperbolic Matrix(239,-672,85,-239) (14/5,48/17) -> (14/5,48/17) Reflection Matrix(577,-1632,204,-577) (48/17,17/6) -> (48/17,17/6) Reflection Matrix(31,-96,10,-31) (3/1,16/5) -> (3/1,16/5) Reflection Matrix(49,-160,15,-49) (16/5,10/3) -> (16/5,10/3) Reflection Matrix(191,-640,57,-191) (10/3,64/19) -> (10/3,64/19) Reflection Matrix(1025,-3456,304,-1025) (64/19,27/8) -> (64/19,27/8) Reflection Matrix(47,-160,5,-17) (17/5,24/7) -> (8/1,1/0) Hyperbolic Matrix(65,-224,9,-31) (24/7,7/2) -> (7/1,8/1) Hyperbolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(127,-576,28,-127) (9/2,32/7) -> (9/2,32/7) Reflection Matrix(97,-448,21,-97) (32/7,14/3) -> (32/7,14/3) Reflection Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(31,-192,5,-31) (6/1,32/5) -> (6/1,32/5) Reflection Matrix(129,-832,20,-129) (32/5,13/2) -> (32/5,13/2) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,4,-1) (0/1,1/0) -> (0/1,1/2) Matrix(1,0,1,-1) -> Matrix(1,0,6,-1) (0/1,2/1) -> (0/1,1/3) Matrix(15,-32,7,-15) -> Matrix(5,-2,12,-5) (2/1,16/7) -> (1/3,1/2) Matrix(97,-224,42,-97) -> Matrix(41,-14,120,-41) (16/7,7/3) -> (1/3,7/20) Matrix(95,-224,14,-33) -> Matrix(39,-14,92,-33) Matrix(241,-576,100,-239) -> Matrix(1,0,0,1) Matrix(225,-544,79,-191) -> Matrix(49,-18,128,-47) 3/8 Matrix(223,-544,66,-161) -> Matrix(47,-18,128,-49) 3/8 Matrix(49,-128,18,-47) -> Matrix(13,-4,36,-11) 1/3 Matrix(81,-224,17,-47) -> Matrix(39,-14,92,-33) Matrix(239,-672,85,-239) -> Matrix(113,-42,304,-113) (14/5,48/17) -> (7/19,3/8) Matrix(577,-1632,204,-577) -> Matrix(175,-66,464,-175) (48/17,17/6) -> (3/8,11/29) Matrix(31,-96,10,-31) -> Matrix(5,-2,12,-5) (3/1,16/5) -> (1/3,1/2) Matrix(49,-160,15,-49) -> Matrix(17,-6,48,-17) (16/5,10/3) -> (1/3,3/8) Matrix(191,-640,57,-191) -> Matrix(23,-8,66,-23) (10/3,64/19) -> (1/3,4/11) Matrix(1025,-3456,304,-1025) -> Matrix(241,-88,660,-241) (64/19,27/8) -> (4/11,11/30) Matrix(47,-160,5,-17) -> Matrix(5,-2,18,-7) 1/3 Matrix(65,-224,9,-31) -> Matrix(37,-14,82,-31) Matrix(17,-64,4,-15) -> Matrix(1,0,0,1) Matrix(127,-576,28,-127) -> Matrix(71,-28,180,-71) (9/2,32/7) -> (7/18,2/5) Matrix(97,-448,21,-97) -> Matrix(69,-28,170,-69) (32/7,14/3) -> (2/5,7/17) Matrix(31,-160,6,-31) -> Matrix(13,-6,28,-13) (5/1,16/3) -> (3/7,1/2) Matrix(17,-96,3,-17) -> Matrix(5,-2,12,-5) (16/3,6/1) -> (1/3,1/2) Matrix(31,-192,5,-31) -> Matrix(11,-4,30,-11) (6/1,32/5) -> (1/3,2/5) Matrix(129,-832,20,-129) -> Matrix(89,-36,220,-89) (32/5,13/2) -> (2/5,9/22) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.