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: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -3/1 -2/1 -9/5 -6/5 -1/1 -2/3 0/1 1/2 2/3 3/4 1/1 6/5 5/4 4/3 3/2 12/7 9/5 2/1 9/4 12/5 5/2 8/3 3/1 4/1 9/2 24/5 5/1 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 1/0 -6/1 -1/1 -5/1 -3/4 -14/3 -3/5 -9/2 -1/2 -4/1 -1/1 -3/1 -1/2 -8/3 -2/5 -13/5 -1/2 -18/7 -1/3 -5/2 -1/2 -12/5 -1/3 -7/3 -1/4 -16/7 0/1 -9/4 -1/3 0/1 -11/5 -1/4 -2/1 0/1 -13/7 -1/2 -24/13 0/1 -11/6 1/2 -9/5 1/0 -16/9 -2/1 -23/13 -3/2 -7/4 -2/1 -1/1 -12/7 -1/1 -5/3 -3/4 -13/8 -2/3 -3/5 -8/5 -2/3 -3/2 -1/2 -4/3 -1/3 -9/7 -1/4 -14/11 -1/5 -19/15 -1/8 -24/19 0/1 -5/4 -1/3 0/1 -11/9 -1/6 -6/5 0/1 -7/6 1/0 -8/7 0/1 -1/1 -1/2 -6/7 -1/3 -5/6 -1/4 -9/11 -1/2 -4/5 -1/3 -3/4 -1/3 0/1 -8/11 0/1 -5/7 -1/2 -12/17 -1/3 -7/10 -3/10 -16/23 -2/7 -9/13 -1/4 -2/3 -1/3 -9/14 -1/4 -16/25 0/1 -7/11 -1/4 -12/19 -1/3 -5/8 -1/3 -2/7 -8/13 -2/7 -3/5 -1/4 -4/7 -1/5 -9/16 -1/5 0/1 -5/9 -1/4 -6/11 0/1 -1/2 -1/4 0/1 0/1 1/2 1/2 3/5 1/2 5/8 2/3 1/1 7/11 1/2 16/25 0/1 9/14 1/2 2/3 1/1 9/13 1/2 7/10 3/4 5/7 1/0 8/11 0/1 3/4 0/1 1/1 4/5 1/1 5/6 1/2 6/7 1/1 1/1 1/0 7/6 -1/2 6/5 0/1 5/4 0/1 1/1 14/11 1/3 9/7 1/2 13/10 3/4 4/3 1/1 3/2 1/0 8/5 -2/1 13/8 -3/1 -2/1 18/11 -2/1 5/3 -3/2 17/10 -7/6 12/7 -1/1 7/4 -1/1 -2/3 16/9 -2/3 25/14 -5/8 9/5 -1/2 11/6 -1/4 2/1 0/1 13/6 -1/2 24/11 0/1 11/5 1/2 9/4 0/1 1/1 16/7 0/1 23/10 1/0 7/3 1/2 19/8 2/3 1/1 12/5 1/1 5/2 1/0 13/5 1/0 8/3 2/1 11/4 2/1 3/1 3/1 1/0 7/2 1/0 18/5 -2/1 11/3 1/0 15/4 -2/1 -1/1 4/1 -1/1 9/2 1/0 14/3 -3/1 19/4 -7/3 -2/1 24/5 -2/1 5/1 -3/2 11/2 -3/2 6/1 -1/1 7/1 -1/2 8/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,72,-4,-41) (-7/1,1/0) -> (-23/13,-7/4) Hyperbolic Matrix(7,48,8,55) (-7/1,-6/1) -> (6/7,1/1) Hyperbolic Matrix(31,168,-12,-65) (-6/1,-5/1) -> (-13/5,-18/7) Hyperbolic Matrix(71,336,-56,-265) (-5/1,-14/3) -> (-14/11,-19/15) Hyperbolic Matrix(31,144,48,223) (-14/3,-9/2) -> (9/14,2/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(17,48,-28,-79) (-3/1,-8/3) -> (-8/13,-3/5) Hyperbolic Matrix(55,144,76,199) (-8/3,-13/5) -> (5/7,8/11) Hyperbolic Matrix(47,120,56,143) (-18/7,-5/2) -> (5/6,6/7) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,-112,-265) (-12/5,-7/3) -> (-7/11,-12/19) Hyperbolic Matrix(31,72,-28,-65) (-7/3,-16/7) -> (-8/7,-1/1) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(119,264,32,71) (-9/4,-11/5) -> (11/3,15/4) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(233,432,48,89) (-13/7,-24/13) -> (24/5,5/1) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(119,216,92,167) (-11/6,-9/5) -> (9/7,13/10) Hyperbolic Matrix(161,288,-232,-415) (-9/5,-16/9) -> (-16/23,-9/13) Hyperbolic Matrix(271,480,424,751) (-16/9,-23/13) -> (7/11,16/25) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(103,168,-84,-137) (-5/3,-13/8) -> (-5/4,-11/9) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(55,72,-68,-89) (-4/3,-9/7) -> (-9/11,-4/5) Hyperbolic Matrix(113,144,164,209) (-9/7,-14/11) -> (2/3,9/13) Hyperbolic Matrix(569,720,260,329) (-19/15,-24/19) -> (24/11,11/5) 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(271,312,152,175) (-7/6,-8/7) -> (16/9,25/14) Hyperbolic Matrix(55,48,8,7) (-1/1,-6/7) -> (6/1,7/1) Hyperbolic Matrix(113,96,20,17) (-6/7,-5/6) -> (11/2,6/1) Hyperbolic Matrix(175,144,96,79) (-5/6,-9/11) -> (9/5,11/6) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(65,48,88,65) (-3/4,-8/11) -> (8/11,3/4) Hyperbolic Matrix(199,144,76,55) (-8/11,-5/7) -> (13/5,8/3) Hyperbolic Matrix(409,288,240,169) (-12/17,-7/10) -> (17/10,12/7) Hyperbolic Matrix(689,480,300,209) (-7/10,-16/23) -> (16/7,23/10) Hyperbolic Matrix(209,144,164,113) (-9/13,-2/3) -> (14/11,9/7) Hyperbolic Matrix(223,144,48,31) (-2/3,-9/14) -> (9/2,14/3) Hyperbolic Matrix(449,288,700,449) (-9/14,-16/25) -> (16/25,9/14) Hyperbolic Matrix(263,168,36,23) (-16/25,-7/11) -> (7/1,8/1) Hyperbolic Matrix(457,288,192,121) (-12/19,-5/8) -> (19/8,12/5) Hyperbolic Matrix(233,144,144,89) (-5/8,-8/13) -> (8/5,13/8) Hyperbolic Matrix(169,96,44,25) (-4/7,-9/16) -> (15/4,4/1) Hyperbolic Matrix(257,144,116,65) (-9/16,-5/9) -> (11/5,9/4) Hyperbolic Matrix(217,120,132,73) (-5/9,-6/11) -> (18/11,5/3) Hyperbolic Matrix(89,48,76,41) (-6/11,-1/2) -> (7/6,6/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(265,-168,112,-71) (5/8,7/11) -> (7/3,19/8) Hyperbolic Matrix(415,-288,232,-161) (9/13,7/10) -> (25/14,9/5) Hyperbolic Matrix(169,-120,100,-71) (7/10,5/7) -> (5/3,17/10) Hyperbolic Matrix(89,-72,68,-55) (4/5,5/6) -> (13/10,4/3) Hyperbolic Matrix(65,-72,28,-31) (1/1,7/6) -> (23/10,7/3) Hyperbolic Matrix(137,-168,84,-103) (6/5,5/4) -> (13/8,18/11) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(41,-72,4,-7) (7/4,16/9) -> (8/1,1/0) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,72,-4,-41) -> Matrix(3,2,-2,-1) Matrix(7,48,8,55) -> Matrix(1,2,0,1) Matrix(31,168,-12,-65) -> Matrix(5,4,-14,-11) Matrix(71,336,-56,-265) -> Matrix(3,2,-20,-13) Matrix(31,144,48,223) -> Matrix(7,4,12,7) Matrix(17,72,4,17) -> Matrix(3,2,-2,-1) Matrix(7,24,-12,-41) -> Matrix(3,2,-14,-9) Matrix(17,48,-28,-79) -> Matrix(9,4,-34,-15) Matrix(55,144,76,199) -> Matrix(5,2,2,1) Matrix(47,120,56,143) -> Matrix(1,0,4,1) Matrix(49,120,20,49) -> Matrix(5,2,2,1) Matrix(71,168,-112,-265) -> Matrix(1,0,0,1) Matrix(31,72,-28,-65) -> Matrix(1,0,2,1) Matrix(127,288,56,127) -> Matrix(1,0,4,1) Matrix(119,264,32,71) -> Matrix(7,2,-4,-1) Matrix(23,48,-12,-25) -> Matrix(1,0,2,1) Matrix(233,432,48,89) -> Matrix(7,2,-4,-1) Matrix(313,576,144,265) -> Matrix(1,0,-4,1) Matrix(119,216,92,167) -> Matrix(1,-2,2,-3) Matrix(161,288,-232,-415) -> Matrix(1,4,-4,-15) Matrix(271,480,424,751) -> Matrix(1,2,0,1) Matrix(97,168,56,97) -> Matrix(3,4,-4,-5) Matrix(71,120,-100,-169) -> Matrix(5,4,-14,-11) Matrix(103,168,-84,-137) -> Matrix(3,2,-14,-9) Matrix(119,192,44,71) -> Matrix(1,0,2,1) Matrix(31,48,20,31) -> Matrix(7,4,-2,-1) Matrix(17,24,12,17) -> Matrix(5,2,2,1) Matrix(55,72,-68,-89) -> Matrix(7,2,-18,-5) Matrix(113,144,164,209) -> Matrix(1,0,6,1) Matrix(569,720,260,329) -> Matrix(1,0,10,1) Matrix(457,576,96,121) -> Matrix(1,-2,0,1) Matrix(217,264,60,73) -> Matrix(11,2,-6,-1) Matrix(143,168,40,47) -> Matrix(1,-2,0,1) Matrix(271,312,152,175) -> Matrix(5,2,-8,-3) Matrix(55,48,8,7) -> Matrix(5,2,-8,-3) Matrix(113,96,20,17) -> Matrix(13,4,-10,-3) Matrix(175,144,96,79) -> Matrix(1,0,0,1) Matrix(31,24,40,31) -> Matrix(1,0,4,1) Matrix(65,48,88,65) -> Matrix(1,0,4,1) Matrix(199,144,76,55) -> Matrix(5,2,2,1) Matrix(409,288,240,169) -> Matrix(31,10,-28,-9) Matrix(689,480,300,209) -> Matrix(7,2,10,3) Matrix(209,144,164,113) -> Matrix(1,0,6,1) Matrix(223,144,48,31) -> Matrix(15,4,-4,-1) Matrix(449,288,700,449) -> Matrix(1,0,6,1) Matrix(263,168,36,23) -> Matrix(1,0,2,1) Matrix(457,288,192,121) -> Matrix(13,4,16,5) Matrix(233,144,144,89) -> Matrix(15,4,-4,-1) Matrix(169,96,44,25) -> Matrix(11,2,-6,-1) Matrix(257,144,116,65) -> Matrix(1,0,6,1) Matrix(217,120,132,73) -> Matrix(11,2,-6,-1) Matrix(89,48,76,41) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,6,1) Matrix(41,-24,12,-7) -> Matrix(5,-2,-2,1) Matrix(79,-48,28,-17) -> Matrix(7,-4,2,-1) Matrix(265,-168,112,-71) -> Matrix(1,0,0,1) Matrix(415,-288,232,-161) -> Matrix(7,-4,-12,7) Matrix(169,-120,100,-71) -> Matrix(3,-4,-2,3) Matrix(89,-72,68,-55) -> Matrix(1,-2,2,-3) Matrix(65,-72,28,-31) -> Matrix(1,0,2,1) Matrix(137,-168,84,-103) -> Matrix(5,-2,-2,1) Matrix(265,-336,56,-71) -> Matrix(9,-2,-4,1) Matrix(41,-72,4,-7) -> Matrix(3,2,-2,-1) Matrix(25,-48,12,-23) -> Matrix(1,0,2,1) Matrix(65,-168,12,-31) -> Matrix(3,-4,-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): 24 Degree of the the map X: 24 Degree of the the map Y: 64 Permutation triple for Y: ((1,2)(3,10,33,62,34,11)(4,16,46,47,17,5)(6,21,55,60,56,22)(7,25,41,52,26,8)(9,30,28,20,51,19)(12,23,42,14,13,37)(15,31,18,24,36,27)(29,43)(32,49)(35,44)(38,54)(39,40)(45,58,61,57,48,59)(50,53)(63,64); (1,5,19,53,45,16,44,56,61,54,20,6)(2,8,28,39,57,25,35,34,59,29,9,3)(4,14,22,15)(7,12,11,24)(10,31,43,42,62,63,41,13,40,27,26,32)(17,49,21,36,38,37,60,64,46,23,50,18)(30,55,48,47)(33,58,52,51); (1,3,12,38,61,33,32,17,48,39,13,4)(2,6,14,43,59,55,49,26,58,53,23,7)(5,18,10,9)(8,27,21,20)(11,35,16,15,40,28,47,64,62,51,54,36)(19,52,63,60,30,29,31,22,44,25,24,50)(34,42,46,45)(37,41,57,56)) ----------------------------------------------------------------------- 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: 20 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -3/1 -2/1 -9/5 -1/1 0/1 1/2 3/4 1/1 4/3 3/2 12/7 9/5 2/1 3/1 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 -1/1 -5/1 -3/4 -4/1 -1/1 -3/1 -1/2 -2/1 0/1 -9/5 1/0 -7/4 -2/1 -1/1 -12/7 -1/1 -5/3 -3/4 -8/5 -2/3 -3/2 -1/2 -4/3 -1/3 -9/7 -1/4 -5/4 -1/3 0/1 -6/5 0/1 -1/1 -1/2 -6/7 -1/3 -5/6 -1/4 -9/11 -1/2 -4/5 -1/3 -3/4 -1/3 0/1 -8/11 0/1 -5/7 -1/2 -12/17 -1/3 -7/10 -3/10 -9/13 -1/4 -2/3 -1/3 -3/5 -1/4 -4/7 -1/5 -5/9 -1/4 -6/11 0/1 -1/2 -1/4 0/1 0/1 1/2 1/2 2/3 1/1 7/10 3/4 5/7 1/0 8/11 0/1 3/4 0/1 1/1 4/5 1/1 5/6 1/2 1/1 1/0 7/6 -1/2 6/5 0/1 5/4 0/1 1/1 9/7 1/2 13/10 3/4 4/3 1/1 3/2 1/0 8/5 -2/1 5/3 -3/2 17/10 -7/6 12/7 -1/1 7/4 -1/1 -2/3 9/5 -1/2 11/6 -1/4 2/1 0/1 5/2 1/0 3/1 1/0 7/2 1/0 11/3 1/0 4/1 -1/1 5/1 -3/2 11/2 -3/2 6/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,36,4,29) (-6/1,1/0) -> (6/5,5/4) Hyperbolic Matrix(11,60,-20,-109) (-6/1,-5/1) -> (-5/9,-6/11) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(19,36,-28,-53) (-2/1,-9/5) -> (-9/13,-2/3) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(55,72,-68,-89) (-4/3,-9/7) -> (-9/11,-4/5) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(113,96,20,17) (-6/7,-5/6) -> (11/2,6/1) Hyperbolic Matrix(175,144,96,79) (-5/6,-9/11) -> (9/5,11/6) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(65,48,88,65) (-3/4,-8/11) -> (8/11,3/4) Hyperbolic Matrix(149,108,40,29) (-8/11,-5/7) -> (11/3,4/1) Hyperbolic Matrix(409,288,240,169) (-12/17,-7/10) -> (17/10,12/7) Hyperbolic Matrix(259,180,200,139) (-7/10,-9/13) -> (9/7,13/10) Hyperbolic Matrix(107,60,148,83) (-4/7,-5/9) -> (5/7,8/11) Hyperbolic Matrix(89,48,76,41) (-6/11,-1/2) -> (7/6,6/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(19,-12,8,-5) (1/2,2/3) -> (2/1,5/2) Hyperbolic Matrix(53,-36,28,-19) (2/3,7/10) -> (11/6,2/1) Hyperbolic Matrix(169,-120,100,-71) (7/10,5/7) -> (5/3,17/10) Hyperbolic Matrix(89,-72,68,-55) (4/5,5/6) -> (13/10,4/3) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(43,-156,8,-29) (7/2,11/3) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,36,4,29) -> Matrix(1,1,0,1) Matrix(11,60,-20,-109) -> Matrix(1,1,-8,-7) Matrix(13,60,8,37) -> Matrix(5,3,-2,-1) Matrix(7,24,-12,-41) -> Matrix(3,2,-14,-9) Matrix(5,12,-8,-19) -> Matrix(3,1,-10,-3) Matrix(19,36,-28,-53) -> Matrix(1,1,-4,-3) Matrix(61,108,48,85) -> Matrix(1,1,2,3) Matrix(97,168,56,97) -> Matrix(3,4,-4,-5) Matrix(71,120,-100,-169) -> Matrix(5,4,-14,-11) Matrix(37,60,8,13) -> Matrix(5,3,-2,-1) Matrix(31,48,20,31) -> Matrix(7,4,-2,-1) Matrix(17,24,12,17) -> Matrix(5,2,2,1) Matrix(55,72,-68,-89) -> Matrix(7,2,-18,-5) Matrix(85,108,48,61) -> Matrix(5,1,-6,-1) Matrix(29,36,4,5) -> Matrix(3,1,-4,-1) Matrix(11,12,-12,-13) -> Matrix(1,1,-4,-3) Matrix(113,96,20,17) -> Matrix(13,4,-10,-3) Matrix(175,144,96,79) -> Matrix(1,0,0,1) Matrix(31,24,40,31) -> Matrix(1,0,4,1) Matrix(65,48,88,65) -> Matrix(1,0,4,1) Matrix(149,108,40,29) -> Matrix(1,1,-2,-1) Matrix(409,288,240,169) -> Matrix(31,10,-28,-9) Matrix(259,180,200,139) -> Matrix(11,3,18,5) Matrix(107,60,148,83) -> Matrix(5,1,4,1) Matrix(89,48,76,41) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,6,1) Matrix(19,-12,8,-5) -> Matrix(1,-1,2,-1) Matrix(53,-36,28,-19) -> Matrix(1,-1,0,1) Matrix(169,-120,100,-71) -> Matrix(3,-4,-2,3) Matrix(89,-72,68,-55) -> Matrix(1,-2,2,-3) Matrix(13,-12,12,-11) -> Matrix(1,-1,0,1) Matrix(13,-36,4,-11) -> Matrix(1,-3,0,1) Matrix(43,-156,8,-29) -> Matrix(3,5,-2,-3) 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 0/1 3 2 1/2 1/2 1 12 2/3 1/1 1 6 7/10 3/4 1 12 5/7 1/0 1 12 8/11 0/1 1 6 3/4 (0/1,1/1) 0 4 4/5 1/1 3 6 5/6 1/2 1 12 1/1 1/0 1 12 7/6 -1/2 1 12 6/5 0/1 3 2 5/4 (0/1,1/1) 0 12 9/7 1/2 3 4 13/10 3/4 1 12 4/3 1/1 3 6 3/2 1/0 3 4 8/5 -2/1 1 6 5/3 -3/2 1 12 17/10 -7/6 1 12 12/7 -1/1 9 2 7/4 (-1/1,-2/3) 0 12 9/5 -1/2 3 4 11/6 -1/4 1 12 2/1 0/1 1 6 5/2 1/0 1 12 3/1 1/0 3 4 7/2 1/0 1 12 11/3 1/0 1 12 4/1 -1/1 1 6 5/1 -3/2 1 12 11/2 -3/2 1 12 6/1 -1/1 3 2 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(19,-12,8,-5) (1/2,2/3) -> (2/1,5/2) Hyperbolic Matrix(53,-36,28,-19) (2/3,7/10) -> (11/6,2/1) Hyperbolic Matrix(169,-120,100,-71) (7/10,5/7) -> (5/3,17/10) Hyperbolic Matrix(149,-108,40,-29) (5/7,8/11) -> (11/3,4/1) Glide Reflection Matrix(65,-48,88,-65) (8/11,3/4) -> (8/11,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(89,-72,68,-55) (4/5,5/6) -> (13/10,4/3) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(91,-108,16,-19) (7/6,6/5) -> (11/2,6/1) Glide Reflection Matrix(29,-36,4,-5) (6/5,5/4) -> (6/1,1/0) Glide Reflection Matrix(85,-108,48,-61) (5/4,9/7) -> (7/4,9/5) Glide Reflection Matrix(167,-216,92,-119) (9/7,13/10) -> (9/5,11/6) Glide Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(37,-60,8,-13) (8/5,5/3) -> (4/1,5/1) Glide Reflection Matrix(239,-408,140,-239) (17/10,12/7) -> (17/10,12/7) Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(43,-156,8,-29) (7/2,11/3) -> (5/1,11/2) Hyperbolic 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(1,0,4,-1) -> Matrix(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Matrix(19,-12,8,-5) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(53,-36,28,-19) -> Matrix(1,-1,0,1) 1/0 Matrix(169,-120,100,-71) -> Matrix(3,-4,-2,3) Matrix(149,-108,40,-29) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(65,-48,88,-65) -> Matrix(1,0,2,-1) (8/11,3/4) -> (0/1,1/1) Matrix(31,-24,40,-31) -> Matrix(1,0,2,-1) (3/4,4/5) -> (0/1,1/1) Matrix(89,-72,68,-55) -> Matrix(1,-2,2,-3) 1/1 Matrix(13,-12,12,-11) -> Matrix(1,-1,0,1) 1/0 Matrix(91,-108,16,-19) -> Matrix(5,1,-4,-1) Matrix(29,-36,4,-5) -> Matrix(1,-1,-2,1) Matrix(85,-108,48,-61) -> Matrix(3,-1,-4,1) Matrix(167,-216,92,-119) -> Matrix(3,-2,-8,5) Matrix(17,-24,12,-17) -> Matrix(-1,2,0,1) (4/3,3/2) -> (1/1,1/0) Matrix(31,-48,20,-31) -> Matrix(1,4,0,-1) (3/2,8/5) -> (-2/1,1/0) Matrix(37,-60,8,-13) -> Matrix(1,3,0,-1) *** -> (-3/2,1/0) Matrix(239,-408,140,-239) -> Matrix(13,14,-12,-13) (17/10,12/7) -> (-7/6,-1/1) Matrix(97,-168,56,-97) -> Matrix(5,4,-6,-5) (12/7,7/4) -> (-1/1,-2/3) Matrix(13,-36,4,-11) -> Matrix(1,-3,0,1) 1/0 Matrix(43,-156,8,-29) -> Matrix(3,5,-2,-3) (-2/1,-1/1).(-3/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.