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 -6/1 -9/2 -4/1 -10/3 -3/1 -8/3 -16/7 -2/1 -9/5 -3/2 -4/3 -6/5 0/1 1/1 6/5 4/3 3/2 36/23 12/7 9/5 2/1 24/11 12/5 5/2 8/3 3/1 36/11 10/3 24/7 7/2 11/3 4/1 9/2 24/5 5/1 11/2 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/1 -2/3 -6/1 -1/2 -5/1 -1/1 0/1 -14/3 -3/4 -23/5 -2/3 -7/11 -9/2 -1/2 -4/1 -1/2 1/0 -15/4 -1/2 -11/3 -1/1 0/1 -18/5 -1/2 -7/2 -1/3 0/1 -10/3 -1/2 -3/1 -1/1 0/1 -14/5 -1/2 -25/9 -1/7 0/1 -36/13 0/1 -11/4 0/1 1/1 -8/3 0/1 -13/5 1/1 2/1 -5/2 -1/1 0/1 -12/5 0/1 -7/3 0/1 1/1 -23/10 2/3 1/1 -16/7 1/1 -25/11 5/3 2/1 -9/4 1/0 -11/5 -1/1 0/1 -2/1 1/0 -13/7 -3/1 -2/1 -24/13 -2/1 -11/6 -2/1 -1/1 -9/5 -2/1 -1/1 -25/14 -1/1 0/1 -16/9 -1/1 -7/4 -2/1 -1/1 -12/7 -1/1 -5/3 -1/1 0/1 -18/11 -1/2 -13/8 -1/3 0/1 -8/5 0/1 -11/7 2/3 1/1 -3/2 1/0 -13/9 -10/3 -3/1 -36/25 -3/1 -23/16 -3/1 -14/5 -10/7 -5/2 -17/12 -7/3 -2/1 -24/17 -2/1 -7/5 -2/1 -1/1 -18/13 1/0 -11/8 -2/1 -1/1 -4/3 -3/2 1/0 -13/10 -2/1 -1/1 -9/7 -2/1 -1/1 -23/18 -4/3 -1/1 -14/11 1/0 -19/15 -4/3 -1/1 -24/19 -1/1 -5/4 -1/1 0/1 -11/9 1/1 2/1 -6/5 1/0 -7/6 -4/1 -3/1 -8/7 -3/1 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 -2/3 7/6 -3/5 -4/7 6/5 -1/2 5/4 -1/1 0/1 14/11 -1/2 23/18 -1/1 -4/5 9/7 -1/1 -2/3 4/3 -3/4 -1/2 15/11 -1/1 -2/3 11/8 -1/1 -2/3 18/13 -1/2 7/5 -1/1 -2/3 10/7 -5/8 3/2 -1/2 14/9 -3/8 25/16 -6/17 -1/3 36/23 -1/3 11/7 -1/3 -2/7 8/5 0/1 13/8 0/1 1/1 5/3 -1/1 0/1 12/7 -1/1 7/4 -1/1 -2/3 23/13 -1/3 0/1 16/9 -1/1 25/14 -1/1 0/1 9/5 -1/1 -2/3 11/6 -1/1 -2/3 2/1 -1/2 13/6 -1/3 0/1 24/11 0/1 11/5 -1/1 0/1 9/4 -1/2 25/11 -2/5 -5/13 16/7 -1/3 7/3 -1/3 0/1 12/5 0/1 5/2 -1/1 0/1 18/7 -1/2 13/5 -2/5 -1/3 8/3 0/1 11/4 -1/3 0/1 3/1 -1/1 0/1 13/4 -1/3 0/1 36/11 0/1 23/7 0/1 1/5 10/3 1/0 17/5 -1/3 0/1 24/7 0/1 7/2 0/1 1/1 18/5 1/0 11/3 -1/1 0/1 4/1 -1/2 1/0 13/3 -1/1 0/1 9/2 1/0 23/5 -7/3 -2/1 14/3 -3/2 19/4 -4/3 -1/1 24/5 -1/1 5/1 -1/1 0/1 11/2 0/1 1/1 6/1 1/0 7/1 -2/1 -1/1 8/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(23,168,-10,-73) (-7/1,1/0) -> (-7/3,-23/10) Hyperbolic Matrix(25,168,18,121) (-7/1,-6/1) -> (18/13,7/5) Hyperbolic Matrix(23,120,-14,-73) (-6/1,-5/1) -> (-5/3,-18/11) Hyperbolic Matrix(71,336,-56,-265) (-5/1,-14/3) -> (-14/11,-19/15) Hyperbolic Matrix(119,552,36,167) (-14/3,-23/5) -> (23/7,10/3) Hyperbolic Matrix(95,432,42,191) (-23/5,-9/2) -> (9/4,25/11) Hyperbolic Matrix(23,96,-6,-25) (-9/2,-4/1) -> (-4/1,-15/4) Parabolic Matrix(71,264,32,119) (-15/4,-11/3) -> (11/5,9/4) Hyperbolic Matrix(119,432,46,167) (-11/3,-18/5) -> (18/7,13/5) Hyperbolic Matrix(47,168,40,143) (-18/5,-7/2) -> (7/6,6/5) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(241,672,52,145) (-14/5,-25/9) -> (23/5,14/3) Hyperbolic Matrix(623,1728,190,527) (-25/9,-36/13) -> (36/11,23/7) Hyperbolic Matrix(313,864,96,265) (-36/13,-11/4) -> (13/4,36/11) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(47,120,-38,-97) (-13/5,-5/2) -> (-5/4,-11/9) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(335,768,188,431) (-23/10,-16/7) -> (16/9,25/14) Hyperbolic Matrix(337,768,190,433) (-16/7,-25/11) -> (23/13,16/9) Hyperbolic Matrix(191,432,42,95) (-25/11,-9/4) -> (9/2,23/5) Hyperbolic Matrix(97,216,22,49) (-9/4,-11/5) -> (13/3,9/2) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(311,576,142,263) (-13/7,-24/13) -> (24/11,11/5) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(145,264,106,193) (-11/6,-9/5) -> (15/11,11/8) Hyperbolic Matrix(241,432,188,337) (-9/5,-25/14) -> (23/18,9/7) Hyperbolic Matrix(121,216,14,25) (-25/14,-16/9) -> (8/1,1/0) Hyperbolic Matrix(95,168,-82,-145) (-16/9,-7/4) -> (-7/6,-8/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(265,432,192,313) (-18/11,-13/8) -> (11/8,18/13) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(599,864,382,551) (-13/9,-36/25) -> (36/23,11/7) Hyperbolic Matrix(1201,1728,768,1105) (-36/25,-23/16) -> (25/16,36/23) Hyperbolic Matrix(385,552,302,433) (-23/16,-10/7) -> (14/11,23/18) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(121,168,18,25) (-7/5,-18/13) -> (6/1,7/1) Hyperbolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(71,96,-54,-73) (-11/8,-4/3) -> (-4/3,-13/10) Parabolic Matrix(167,216,92,119) (-13/10,-9/7) -> (9/5,11/6) Hyperbolic Matrix(337,432,188,241) (-9/7,-23/18) -> (25/14,9/5) Hyperbolic Matrix(527,672,338,431) (-23/18,-14/11) -> (14/9,25/16) Hyperbolic Matrix(455,576,94,119) (-19/15,-24/19) -> (24/5,5/1) Hyperbolic Matrix(457,576,96,121) (-24/19,-5/4) -> (19/4,24/5) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(191,216,84,95) (-8/7,-1/1) -> (25/11,16/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(145,-168,82,-95) (1/1,7/6) -> (7/4,23/13) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(73,-96,54,-71) (9/7,4/3) -> (4/3,15/11) Parabolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(73,-168,10,-23) (16/7,7/3) -> (7/1,8/1) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(23,168,-10,-73) -> Matrix(3,2,4,3) Matrix(25,168,18,121) -> Matrix(1,0,0,1) Matrix(23,120,-14,-73) -> Matrix(1,0,0,1) Matrix(71,336,-56,-265) -> Matrix(5,4,-4,-3) Matrix(119,552,36,167) -> Matrix(3,2,4,3) Matrix(95,432,42,191) -> Matrix(7,4,-16,-9) Matrix(23,96,-6,-25) -> Matrix(1,0,0,1) Matrix(71,264,32,119) -> Matrix(1,0,0,1) Matrix(119,432,46,167) -> Matrix(3,2,-8,-5) Matrix(47,168,40,143) -> Matrix(9,4,-16,-7) Matrix(71,240,-50,-169) -> Matrix(1,-2,0,1) Matrix(23,72,-8,-25) -> Matrix(1,0,0,1) Matrix(241,672,52,145) -> Matrix(7,2,-4,-1) Matrix(623,1728,190,527) -> Matrix(1,0,12,1) Matrix(313,864,96,265) -> Matrix(1,0,-4,1) Matrix(71,192,44,119) -> Matrix(1,0,0,1) Matrix(73,192,46,121) -> Matrix(1,0,-4,1) Matrix(47,120,-38,-97) -> Matrix(1,0,0,1) Matrix(49,120,20,49) -> Matrix(1,0,0,1) Matrix(71,168,30,71) -> Matrix(1,0,-4,1) Matrix(335,768,188,431) -> Matrix(3,-2,-4,3) Matrix(337,768,190,433) -> Matrix(1,-2,0,1) Matrix(191,432,42,95) -> Matrix(1,-4,0,1) Matrix(97,216,22,49) -> Matrix(1,0,0,1) Matrix(23,48,-12,-25) -> Matrix(1,-2,0,1) Matrix(311,576,142,263) -> Matrix(1,2,0,1) Matrix(313,576,144,265) -> Matrix(1,2,-4,-7) Matrix(145,264,106,193) -> Matrix(3,4,-4,-5) Matrix(241,432,188,337) -> Matrix(3,4,-4,-5) Matrix(121,216,14,25) -> Matrix(1,0,0,1) Matrix(95,168,-82,-145) -> Matrix(1,-2,0,1) Matrix(97,168,56,97) -> Matrix(3,4,-4,-5) Matrix(71,120,42,71) -> Matrix(1,0,0,1) Matrix(265,432,192,313) -> Matrix(5,2,-8,-3) Matrix(119,192,44,71) -> Matrix(1,0,0,1) Matrix(121,192,46,73) -> Matrix(1,0,-4,1) Matrix(47,72,-32,-49) -> Matrix(1,-4,0,1) Matrix(599,864,382,551) -> Matrix(5,16,-16,-51) Matrix(1201,1728,768,1105) -> Matrix(11,32,-32,-93) Matrix(385,552,302,433) -> Matrix(1,2,0,1) Matrix(407,576,118,167) -> Matrix(1,2,4,9) Matrix(409,576,120,169) -> Matrix(1,2,-4,-7) Matrix(121,168,18,25) -> Matrix(1,0,0,1) Matrix(191,264,34,47) -> Matrix(1,2,0,1) Matrix(71,96,-54,-73) -> Matrix(1,0,0,1) Matrix(167,216,92,119) -> Matrix(3,4,-4,-5) Matrix(337,432,188,241) -> Matrix(3,4,-4,-5) Matrix(527,672,338,431) -> Matrix(3,2,-8,-5) Matrix(455,576,94,119) -> Matrix(3,4,-4,-5) Matrix(457,576,96,121) -> Matrix(5,4,-4,-3) Matrix(217,264,60,73) -> Matrix(1,-2,0,1) Matrix(143,168,40,47) -> Matrix(1,4,0,1) Matrix(191,216,84,95) -> Matrix(3,8,-8,-21) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(145,-168,82,-95) -> Matrix(3,2,-8,-5) Matrix(97,-120,38,-47) -> Matrix(1,0,0,1) Matrix(265,-336,56,-71) -> Matrix(5,4,-4,-3) Matrix(73,-96,54,-71) -> Matrix(1,0,0,1) Matrix(169,-240,50,-71) -> Matrix(3,2,-8,-5) Matrix(49,-72,32,-47) -> Matrix(7,4,-16,-9) Matrix(73,-120,14,-23) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(3,2,-8,-5) Matrix(73,-168,10,-23) -> Matrix(7,2,-4,-1) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(25,-96,6,-23) -> Matrix(1,0,0,1) 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): 10 Degree of the the map X: 20 Degree of the the map Y: 64 Permutation triple for Y: ((2,6,24,56,25,7)(3,12,42,43,13,4)(5,18,19)(8,30,31)(9,36,10)(11,22,21)(14,28,27)(15,47,16)(17,37)(20,29,59,35,48,54)(32,46)(33,34)(38,45,52,51,60,39)(40,49)(41,55,50)(44,58,57); (1,4,16,49,57,59,62,60,50,17,5,2)(3,10,39,11)(6,22,32,31,51,61,35,9,34,14,13,23)(7,28,29,8)(12,26,25,58,46,15,38,53,20,19,33,41)(18,52,44,43)(21,54,64,45,27,37,36,56,63,42,30,40)(24,55,48,47); (1,2,8,32,58,52,64,54,55,33,9,3)(4,14,45,15)(5,20,21,6)(7,26,12,11,40,16,48,61,51,18,17,27)(10,37,50,24,23,13,44,49,30,29,53,38)(19,43,63,56,47,46,22,39,62,59,28,34)(25,36,35,57)(31,42,41,60)) ----------------------------------------------------------------------- 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): 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 -6/1 -4/1 -10/3 -3/1 -8/3 -2/1 -3/2 -6/5 0/1 1/1 6/5 3/2 2/1 12/5 5/2 8/3 3/1 36/11 10/3 24/7 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 -5/1 -1/1 0/1 -4/1 -1/2 1/0 -7/2 -1/3 0/1 -10/3 -1/2 -3/1 -1/1 0/1 -14/5 -1/2 -11/4 0/1 1/1 -8/3 0/1 -13/5 1/1 2/1 -5/2 -1/1 0/1 -2/1 1/0 -7/4 -2/1 -1/1 -12/7 -1/1 -5/3 -1/1 0/1 -18/11 -1/2 -13/8 -1/3 0/1 -8/5 0/1 -11/7 2/3 1/1 -3/2 1/0 -13/9 -10/3 -3/1 -36/25 -3/1 -23/16 -3/1 -14/5 -10/7 -5/2 -17/12 -7/3 -2/1 -24/17 -2/1 -7/5 -2/1 -1/1 -4/3 -3/2 1/0 -5/4 -1/1 0/1 -6/5 1/0 -7/6 -4/1 -3/1 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 -2/3 6/5 -1/2 5/4 -1/1 0/1 4/3 -3/4 -1/2 7/5 -1/1 -2/3 10/7 -5/8 3/2 -1/2 14/9 -3/8 11/7 -1/3 -2/7 8/5 0/1 13/8 0/1 1/1 5/3 -1/1 0/1 2/1 -1/2 7/3 -1/3 0/1 12/5 0/1 5/2 -1/1 0/1 18/7 -1/2 13/5 -2/5 -1/3 8/3 0/1 11/4 -1/3 0/1 3/1 -1/1 0/1 13/4 -1/3 0/1 36/11 0/1 23/7 0/1 1/5 10/3 1/0 17/5 -1/3 0/1 24/7 0/1 7/2 0/1 1/1 4/1 -1/2 1/0 5/1 -1/1 0/1 6/1 1/0 7/1 -2/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,96,-8,-59) (-6/1,1/0) -> (-18/11,-13/8) Hyperbolic Matrix(23,120,-14,-73) (-6/1,-5/1) -> (-5/3,-18/11) Hyperbolic Matrix(11,48,8,35) (-5/1,-4/1) -> (4/3,7/5) Hyperbolic Matrix(13,48,10,37) (-4/1,-7/2) -> (5/4,4/3) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(155,432,-108,-301) (-14/5,-11/4) -> (-23/16,-10/7) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(37,96,-32,-83) (-13/5,-5/2) -> (-7/6,-1/1) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,34,59) (-7/4,-12/7) -> (12/5,5/2) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(59,72,-50,-61) (-5/4,-6/5) -> (-6/5,-7/6) Parabolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(83,-96,32,-37) (1/1,6/5) -> (18/7,13/5) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(277,-432,84,-131) (14/9,11/7) -> (23/7,10/3) Hyperbolic Matrix(59,-96,8,-13) (13/8,5/3) -> (7/1,1/0) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,96,-8,-59) -> Matrix(1,1,-4,-3) Matrix(23,120,-14,-73) -> Matrix(1,0,0,1) Matrix(11,48,8,35) -> Matrix(3,1,-4,-1) Matrix(13,48,10,37) -> Matrix(3,1,-4,-1) Matrix(71,240,-50,-169) -> Matrix(1,-2,0,1) Matrix(23,72,-8,-25) -> Matrix(1,0,0,1) Matrix(155,432,-108,-301) -> Matrix(11,3,-4,-1) Matrix(71,192,44,119) -> Matrix(1,0,0,1) Matrix(73,192,46,121) -> Matrix(1,0,-4,1) Matrix(37,96,-32,-83) -> Matrix(1,-3,0,1) Matrix(11,24,-6,-13) -> Matrix(1,-1,0,1) Matrix(83,144,34,59) -> Matrix(1,1,0,1) Matrix(85,144,36,61) -> Matrix(1,1,-4,-3) Matrix(119,192,44,71) -> Matrix(1,0,0,1) Matrix(121,192,46,73) -> Matrix(1,0,-4,1) Matrix(47,72,-32,-49) -> Matrix(1,-4,0,1) Matrix(899,1296,274,395) -> Matrix(1,3,8,25) Matrix(901,1296,276,397) -> Matrix(1,3,-8,-23) Matrix(407,576,118,167) -> Matrix(1,2,4,9) Matrix(409,576,120,169) -> Matrix(1,2,-4,-7) Matrix(35,48,8,11) -> Matrix(1,1,0,1) Matrix(37,48,10,13) -> Matrix(1,1,0,1) Matrix(59,72,-50,-61) -> Matrix(1,-3,0,1) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(83,-96,32,-37) -> Matrix(5,3,-12,-7) Matrix(97,-120,38,-47) -> Matrix(1,0,0,1) Matrix(169,-240,50,-71) -> Matrix(3,2,-8,-5) Matrix(49,-72,32,-47) -> Matrix(7,4,-16,-9) Matrix(277,-432,84,-131) -> Matrix(3,1,8,3) Matrix(59,-96,8,-13) -> Matrix(1,-1,0,1) Matrix(13,-24,6,-11) -> Matrix(1,1,-4,-3) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(13,-72,2,-11) -> Matrix(1,-1,0,1) 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): 10 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 -1/1 2 1 1/1 (-1/1,-2/3) 0 12 6/5 -1/2 3 2 5/4 (-1/1,0/1) 0 12 4/3 0 3 7/5 (-1/1,-2/3) 0 12 10/7 -5/8 1 6 3/2 -1/2 2 4 14/9 -3/8 1 6 11/7 (-1/3,-2/7) 0 12 8/5 0/1 2 3 13/8 (0/1,1/1) 0 12 5/3 (-1/1,0/1) 0 12 2/1 -1/2 1 6 7/3 (-1/3,0/1) 0 12 12/5 0/1 2 1 5/2 (-1/1,0/1) 0 12 18/7 -1/2 3 2 13/5 (-2/5,-1/3) 0 12 8/3 0/1 2 3 11/4 (-1/3,0/1) 0 12 3/1 0 4 13/4 (-1/3,0/1) 0 12 36/11 0/1 8 1 23/7 (0/1,1/5) 0 12 10/3 1/0 1 6 17/5 (-1/3,0/1) 0 12 24/7 0/1 4 1 7/2 (0/1,1/1) 0 12 4/1 0 3 5/1 (-1/1,0/1) 0 12 6/1 1/0 1 2 7/1 (-2/1,-1/1) 0 12 1/0 (-1/1,0/1) 0 12 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(83,-96,32,-37) (1/1,6/5) -> (18/7,13/5) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(37,-48,10,-13) (5/4,4/3) -> (7/2,4/1) Glide Reflection Matrix(35,-48,8,-11) (4/3,7/5) -> (4/1,5/1) Glide Reflection Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(277,-432,84,-131) (14/9,11/7) -> (23/7,10/3) Hyperbolic Matrix(121,-192,46,-73) (11/7,8/5) -> (13/5,8/3) Glide Reflection Matrix(119,-192,44,-71) (8/5,13/8) -> (8/3,11/4) Glide Reflection Matrix(59,-96,8,-13) (13/8,5/3) -> (7/1,1/0) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(287,-936,88,-287) (13/4,36/11) -> (13/4,36/11) Reflection Matrix(505,-1656,154,-505) (36/11,23/7) -> (36/11,23/7) Reflection Matrix(239,-816,70,-239) (17/5,24/7) -> (17/5,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(5,4,-6,-5) (0/1,1/1) -> (-1/1,-2/3) Matrix(83,-96,32,-37) -> Matrix(5,3,-12,-7) -1/2 Matrix(97,-120,38,-47) -> Matrix(1,0,0,1) Matrix(37,-48,10,-13) -> Matrix(1,1,2,1) Matrix(35,-48,8,-11) -> Matrix(1,1,2,1) Matrix(169,-240,50,-71) -> Matrix(3,2,-8,-5) -1/2 Matrix(49,-72,32,-47) -> Matrix(7,4,-16,-9) -1/2 Matrix(277,-432,84,-131) -> Matrix(3,1,8,3) Matrix(121,-192,46,-73) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(119,-192,44,-71) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(59,-96,8,-13) -> Matrix(1,-1,0,1) 1/0 Matrix(13,-24,6,-11) -> Matrix(1,1,-4,-3) -1/2 Matrix(71,-168,30,-71) -> Matrix(-1,0,6,1) (7/3,12/5) -> (-1/3,0/1) Matrix(49,-120,20,-49) -> Matrix(-1,0,2,1) (12/5,5/2) -> (-1/1,0/1) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(287,-936,88,-287) -> Matrix(-1,0,6,1) (13/4,36/11) -> (-1/3,0/1) Matrix(505,-1656,154,-505) -> Matrix(1,0,10,-1) (36/11,23/7) -> (0/1,1/5) Matrix(239,-816,70,-239) -> Matrix(-1,0,6,1) (17/5,24/7) -> (-1/3,0/1) Matrix(97,-336,28,-97) -> Matrix(1,0,2,-1) (24/7,7/2) -> (0/1,1/1) Matrix(13,-72,2,-11) -> Matrix(1,-1,0,1) 1/0 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.