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 -5/3 -3/2 -4/3 -11/9 -1/1 -5/6 -3/4 -2/3 -5/9 -1/2 -11/24 -5/11 -3/7 -1/3 -3/11 -1/4 -1/5 -1/6 -1/7 0/1 1/6 1/5 1/4 1/3 2/5 5/12 1/2 5/9 7/12 2/3 3/4 4/5 5/6 1/1 11/9 4/3 3/2 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -9/5 -1/1 -2/3 -7/4 0/1 -5/3 1/0 -13/8 -2/1 -8/5 1/0 -11/7 -2/1 -5/3 -3/2 -1/1 -13/9 -1/2 -10/7 -1/2 -7/5 1/0 -11/8 0/1 -4/3 1/0 -9/7 -3/2 -14/11 -1/2 -5/4 -2/1 -11/9 -3/2 -6/5 -5/4 -1/1 -1/1 0/1 -6/7 -5/4 -5/6 -1/1 -9/11 -11/12 -4/5 -5/6 -7/9 -3/4 -10/13 -1/2 -3/4 -2/3 -11/15 -1/2 -19/26 -1/3 -8/11 1/0 -5/7 -3/4 -7/10 -1/1 -2/3 -1/2 -9/14 -1/3 -16/25 -1/6 -23/36 0/1 -7/11 -1/1 0/1 -12/19 -1/2 -5/8 0/1 -3/5 -1/2 -10/17 -1/4 -7/12 0/1 -4/7 1/0 -5/9 1/0 -6/11 -5/2 -1/2 -1/1 -6/13 -13/16 -11/24 -4/5 -5/11 -11/14 -4/9 -3/4 -3/7 -5/7 -2/3 -8/19 -11/16 -5/12 -2/3 -2/5 -1/2 -1/3 -1/2 -2/7 -1/2 -5/18 -1/3 -3/11 -1/3 0/1 -4/15 -1/2 -1/4 0/1 -1/5 -2/1 -1/1 -2/11 -5/4 -1/6 -1/1 -2/13 -9/10 -1/7 -5/6 0/1 -1/2 1/6 -1/1 1/5 -3/4 1/4 -2/3 1/3 -1/2 3/8 -2/5 5/13 -5/14 7/18 -1/3 2/5 -1/4 5/12 0/1 3/7 1/0 4/9 -1/2 5/11 -1/3 0/1 1/2 -1/1 7/13 -5/7 -2/3 13/24 -2/3 6/11 -5/8 5/9 -1/2 4/7 -3/4 7/12 -2/3 3/5 -2/3 -3/5 5/8 -4/7 7/11 -11/20 2/3 -1/2 9/13 -9/20 25/36 -4/9 16/23 -15/34 7/10 -3/7 5/7 -2/5 -1/3 13/18 -1/3 8/11 -1/4 11/15 -1/2 3/4 0/1 10/13 -3/4 7/9 -1/2 4/5 -1/2 9/11 -1/1 0/1 5/6 -1/1 1/1 -1/2 7/6 -1/3 6/5 -1/2 11/9 -1/2 5/4 0/1 14/11 1/0 9/7 -1/1 -2/3 4/3 -1/2 15/11 -11/24 11/8 -4/9 7/5 -3/7 -2/5 17/12 -2/5 10/7 -3/8 13/9 -1/2 16/11 -5/12 3/2 -1/3 11/7 -1/4 19/12 0/1 8/5 1/0 5/3 -1/2 12/7 -7/16 19/11 -11/26 7/4 -2/5 9/5 -3/8 11/6 -1/3 13/7 -1/3 -2/7 2/1 -1/2 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(11,20,-60,-109) (-2/1,-9/5) -> (-1/5,-2/11) Hyperbolic Matrix(83,148,60,107) (-9/5,-7/4) -> (11/8,7/5) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(121,196,-192,-311) (-13/8,-8/5) -> (-12/19,-5/8) Hyperbolic Matrix(71,112,-168,-265) (-8/5,-11/7) -> (-3/7,-8/19) Hyperbolic Matrix(59,92,84,131) (-11/7,-3/2) -> (7/10,5/7) Hyperbolic Matrix(193,280,-264,-383) (-3/2,-13/9) -> (-11/15,-19/26) Hyperbolic Matrix(131,188,108,155) (-13/9,-10/7) -> (6/5,11/9) Hyperbolic Matrix(71,100,-120,-169) (-10/7,-7/5) -> (-3/5,-10/17) Hyperbolic Matrix(107,148,60,83) (-7/5,-11/8) -> (7/4,9/5) Hyperbolic Matrix(35,48,-132,-181) (-11/8,-4/3) -> (-4/15,-1/4) Hyperbolic Matrix(37,48,84,109) (-4/3,-9/7) -> (3/7,4/9) Hyperbolic Matrix(25,32,-168,-215) (-9/7,-14/11) -> (-2/13,-1/7) Hyperbolic Matrix(73,92,96,121) (-14/11,-5/4) -> (3/4,10/13) Hyperbolic Matrix(71,88,96,119) (-5/4,-11/9) -> (11/15,3/4) Hyperbolic Matrix(155,188,108,131) (-11/9,-6/5) -> (10/7,13/9) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(61,52,156,133) (-6/7,-5/6) -> (7/18,2/5) Hyperbolic Matrix(107,88,276,227) (-5/6,-9/11) -> (5/13,7/18) Hyperbolic Matrix(227,184,132,107) (-9/11,-4/5) -> (12/7,19/11) Hyperbolic Matrix(61,48,108,85) (-4/5,-7/9) -> (5/9,4/7) Hyperbolic Matrix(119,92,216,167) (-7/9,-10/13) -> (6/11,5/9) Hyperbolic Matrix(121,92,96,73) (-10/13,-3/4) -> (5/4,14/11) Hyperbolic Matrix(119,88,96,71) (-3/4,-11/15) -> (11/9,5/4) Hyperbolic Matrix(493,360,708,517) (-19/26,-8/11) -> (16/23,7/10) Hyperbolic Matrix(11,8,-84,-61) (-8/11,-5/7) -> (-1/7,0/1) Hyperbolic Matrix(131,92,84,59) (-5/7,-7/10) -> (3/2,11/7) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(299,192,204,131) (-9/14,-16/25) -> (16/11,3/2) Hyperbolic Matrix(1201,768,1728,1105) (-16/25,-23/36) -> (25/36,16/23) Hyperbolic Matrix(395,252,732,467) (-23/36,-7/11) -> (7/13,13/24) Hyperbolic Matrix(215,136,264,167) (-7/11,-12/19) -> (4/5,9/11) Hyperbolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(409,240,288,169) (-10/17,-7/12) -> (17/12,10/7) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(85,48,108,61) (-4/7,-5/9) -> (7/9,4/5) Hyperbolic Matrix(167,92,216,119) (-5/9,-6/11) -> (10/13,7/9) Hyperbolic Matrix(23,12,-48,-25) (-6/11,-1/2) -> (-1/2,-6/13) Parabolic Matrix(313,144,576,265) (-6/13,-11/24) -> (13/24,6/11) Hyperbolic Matrix(491,224,708,323) (-11/24,-5/11) -> (9/13,25/36) Hyperbolic Matrix(179,80,132,59) (-5/11,-4/9) -> (4/3,15/11) Hyperbolic Matrix(109,48,84,37) (-4/9,-3/7) -> (9/7,4/3) Hyperbolic Matrix(457,192,288,121) (-8/19,-5/12) -> (19/12,8/5) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(157,44,132,37) (-2/7,-5/18) -> (7/6,6/5) Hyperbolic Matrix(217,60,264,73) (-5/18,-3/11) -> (9/11,5/6) Hyperbolic Matrix(119,32,264,71) (-3/11,-4/15) -> (4/9,5/11) Hyperbolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(23,4,-144,-25) (-2/11,-1/6) -> (-1/6,-2/13) Parabolic Matrix(61,-8,84,-11) (0/1,1/6) -> (13/18,8/11) Hyperbolic Matrix(109,-20,60,-11) (1/6,1/5) -> (9/5,11/6) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(263,-100,192,-73) (3/8,5/13) -> (15/11,11/8) Hyperbolic Matrix(265,-112,168,-71) (5/12,3/7) -> (11/7,19/12) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(169,-100,120,-71) (7/12,3/5) -> (7/5,17/12) Hyperbolic Matrix(229,-144,132,-83) (5/8,7/11) -> (19/11,7/4) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(311,-224,168,-121) (5/7,13/18) -> (11/6,13/7) Hyperbolic Matrix(383,-280,264,-193) (8/11,11/15) -> (13/9,16/11) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(157,-200,84,-107) (14/11,9/7) -> (13/7,2/1) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(11,20,-60,-109) -> Matrix(5,4,-4,-3) Matrix(83,148,60,107) -> Matrix(7,4,-16,-9) Matrix(59,100,-36,-61) -> Matrix(1,-2,0,1) Matrix(121,196,-192,-311) -> Matrix(1,2,-2,-3) Matrix(71,112,-168,-265) -> Matrix(11,20,-16,-29) Matrix(59,92,84,131) -> Matrix(5,8,-12,-19) Matrix(193,280,-264,-383) -> Matrix(3,2,-8,-5) Matrix(131,188,108,155) -> Matrix(1,0,0,1) Matrix(71,100,-120,-169) -> Matrix(1,0,-2,1) Matrix(107,148,60,83) -> Matrix(3,2,-8,-5) Matrix(35,48,-132,-181) -> Matrix(1,0,-2,1) Matrix(37,48,84,109) -> Matrix(1,2,-2,-3) Matrix(25,32,-168,-215) -> Matrix(7,8,-8,-9) Matrix(73,92,96,121) -> Matrix(1,2,-2,-3) Matrix(71,88,96,119) -> Matrix(1,2,-4,-7) Matrix(155,188,108,131) -> Matrix(1,2,-4,-7) Matrix(11,12,-12,-13) -> Matrix(1,0,0,1) Matrix(61,52,156,133) -> Matrix(5,6,-16,-19) Matrix(107,88,276,227) -> Matrix(17,16,-50,-47) Matrix(227,184,132,107) -> Matrix(25,22,-58,-51) Matrix(61,48,108,85) -> Matrix(3,2,-2,-1) Matrix(119,92,216,167) -> Matrix(11,8,-18,-13) Matrix(121,92,96,73) -> Matrix(3,2,-2,-1) Matrix(119,88,96,71) -> Matrix(3,2,-8,-5) Matrix(493,360,708,517) -> Matrix(15,4,-34,-9) Matrix(11,8,-84,-61) -> Matrix(1,2,-2,-3) Matrix(131,92,84,59) -> Matrix(3,2,-8,-5) Matrix(47,32,-72,-49) -> Matrix(3,2,-8,-5) Matrix(299,192,204,131) -> Matrix(7,2,-18,-5) Matrix(1201,768,1728,1105) -> Matrix(39,4,-88,-9) Matrix(395,252,732,467) -> Matrix(3,-2,-4,3) Matrix(215,136,264,167) -> Matrix(1,0,0,1) Matrix(13,8,60,37) -> Matrix(1,2,-2,-3) Matrix(409,240,288,169) -> Matrix(11,2,-28,-5) Matrix(97,56,168,97) -> Matrix(3,-2,-4,3) Matrix(85,48,108,61) -> Matrix(1,0,-2,1) Matrix(167,92,216,119) -> Matrix(1,4,-2,-7) Matrix(23,12,-48,-25) -> Matrix(5,6,-6,-7) Matrix(313,144,576,265) -> Matrix(57,46,-88,-71) Matrix(491,224,708,323) -> Matrix(101,80,-226,-179) Matrix(179,80,132,59) -> Matrix(29,22,-62,-47) Matrix(109,48,84,37) -> Matrix(11,8,-18,-13) Matrix(457,192,288,121) -> Matrix(3,2,16,11) Matrix(49,20,120,49) -> Matrix(3,2,-14,-9) Matrix(11,4,-36,-13) -> Matrix(3,2,-8,-5) Matrix(157,44,132,37) -> Matrix(1,0,0,1) Matrix(217,60,264,73) -> Matrix(1,0,2,1) Matrix(119,32,264,71) -> Matrix(1,0,0,1) Matrix(37,8,60,13) -> Matrix(1,4,-2,-7) Matrix(23,4,-144,-25) -> Matrix(13,14,-14,-15) Matrix(61,-8,84,-11) -> Matrix(1,0,-2,1) Matrix(109,-20,60,-11) -> Matrix(7,6,-20,-17) Matrix(13,-4,36,-11) -> Matrix(7,4,-16,-9) Matrix(263,-100,192,-73) -> Matrix(37,14,-82,-31) Matrix(265,-112,168,-71) -> Matrix(1,0,-4,1) Matrix(25,-12,48,-23) -> Matrix(1,2,-2,-3) Matrix(169,-100,120,-71) -> Matrix(19,12,-46,-29) Matrix(229,-144,132,-83) -> Matrix(39,22,-94,-53) Matrix(49,-32,72,-47) -> Matrix(19,10,-40,-21) Matrix(311,-224,168,-121) -> Matrix(11,4,-36,-13) Matrix(383,-280,264,-193) -> Matrix(3,2,-8,-5) Matrix(13,-12,12,-11) -> Matrix(3,2,-8,-5) Matrix(157,-200,84,-107) -> Matrix(1,0,-2,1) Matrix(61,-100,36,-59) -> Matrix(7,4,-16,-9) 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: ((1,6,23,54,53,61,63,62,32,24,7,2)(3,12,41,59,52,60,64,56,48,42,13,4)(5,18,55,19)(8,25,50,49,38,46,45,35,58,22,21,26)(9,30,31,10)(11,36,57,37)(14,43,34,33,20,29,28,17,51,40,39,44)(15,47,27,16); (1,4,16,50,17,5)(2,10,34,35,11,3)(6,22)(7,8)(9,29,38,37,42,23)(12,40)(13,14)(15,46,20,19,24,41)(18,54,48,27,26,39)(21,36,59,32,31,44)(25,57,64,61,30,51)(28,52)(33,56)(43,55,63,60,47,58)(45,53)(49,62); (1,3)(2,8,27,60,28,9)(4,14,31,61,45,15)(5,20,56,57,21,6)(7,19,43,13,37,25)(10,32,49,16,48,33)(11,38,62,55,39,12)(17,52,36,35,53,18)(22,47,41,40,30,23)(24,59)(26,44)(29,46)(34,58)(42,54)(50,51)(63,64)) ----------------------------------------------------------------------- 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 -5/3 -3/2 -11/9 -1/1 -5/6 -2/3 -1/2 -1/3 -1/5 -1/6 0/1 1/6 1/5 1/3 3/7 1/2 5/9 7/11 2/3 1/1 3/2 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -5/3 1/0 -8/5 1/0 -11/7 -2/1 -5/3 -3/2 -1/1 -4/3 1/0 -5/4 -2/1 -11/9 -3/2 -6/5 -5/4 -1/1 -1/1 0/1 -6/7 -5/4 -5/6 -1/1 -9/11 -11/12 -4/5 -5/6 -7/9 -3/4 -3/4 -2/3 -5/7 -3/4 -7/10 -1/1 -2/3 -1/2 -5/8 0/1 -3/5 -1/2 -7/12 0/1 -4/7 1/0 -1/2 -1/1 -1/3 -1/2 -1/4 0/1 -1/5 -2/1 -1/1 -2/11 -5/4 -1/6 -1/1 0/1 -1/2 1/6 -1/1 1/5 -3/4 1/4 -2/3 1/3 -1/2 3/8 -2/5 5/13 -5/14 7/18 -1/3 2/5 -1/4 5/12 0/1 3/7 1/0 1/2 -1/1 5/9 -1/2 4/7 -3/4 7/12 -2/3 3/5 -2/3 -3/5 5/8 -4/7 7/11 -11/20 2/3 -1/2 1/1 -1/2 4/3 -1/2 15/11 -11/24 11/8 -4/9 7/5 -3/7 -2/5 17/12 -2/5 10/7 -3/8 13/9 -1/2 3/2 -1/3 11/7 -1/4 19/12 0/1 8/5 1/0 5/3 -1/2 12/7 -7/16 19/11 -11/26 7/4 -2/5 2/1 -1/2 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(19,30,-102,-161) (-8/5,-11/7) -> (-1/5,-2/11) Hyperbolic Matrix(91,142,66,103) (-11/7,-3/2) -> (11/8,7/5) Hyperbolic Matrix(19,26,-30,-41) (-3/2,-4/3) -> (-2/3,-5/8) Hyperbolic Matrix(29,38,-42,-55) (-4/3,-5/4) -> (-7/10,-2/3) Hyperbolic Matrix(79,98,54,67) (-5/4,-11/9) -> (13/9,3/2) Hyperbolic Matrix(155,188,108,131) (-11/9,-6/5) -> (10/7,13/9) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(61,52,156,133) (-6/7,-5/6) -> (7/18,2/5) Hyperbolic Matrix(107,88,276,227) (-5/6,-9/11) -> (5/13,7/18) Hyperbolic Matrix(227,184,132,107) (-9/11,-4/5) -> (12/7,19/11) Hyperbolic Matrix(61,48,108,85) (-4/5,-7/9) -> (5/9,4/7) Hyperbolic Matrix(29,22,54,41) (-7/9,-3/4) -> (1/2,5/9) Hyperbolic Matrix(19,14,42,31) (-3/4,-5/7) -> (3/7,1/2) Hyperbolic Matrix(131,92,84,59) (-5/7,-7/10) -> (3/2,11/7) Hyperbolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(53,30,30,17) (-4/7,-1/2) -> (7/4,2/1) Hyperbolic Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(257,46,162,29) (-2/11,-1/6) -> (19/12,8/5) Hyperbolic Matrix(17,2,42,5) (-1/6,0/1) -> (2/5,5/12) Hyperbolic Matrix(77,-10,54,-7) (0/1,1/6) -> (17/12,10/7) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(263,-100,192,-73) (3/8,5/13) -> (15/11,11/8) Hyperbolic Matrix(265,-112,168,-71) (5/12,3/7) -> (11/7,19/12) Hyperbolic Matrix(169,-100,120,-71) (7/12,3/5) -> (7/5,17/12) Hyperbolic Matrix(229,-144,132,-83) (5/8,7/11) -> (19/11,7/4) Hyperbolic Matrix(89,-58,66,-43) (7/11,2/3) -> (4/3,15/11) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(29,50,-18,-31) -> Matrix(1,-1,0,1) Matrix(19,30,-102,-161) -> Matrix(5,9,-4,-7) Matrix(91,142,66,103) -> Matrix(7,11,-16,-25) Matrix(19,26,-30,-41) -> Matrix(1,1,-2,-1) Matrix(29,38,-42,-55) -> Matrix(1,3,-2,-5) Matrix(79,98,54,67) -> Matrix(3,5,-8,-13) Matrix(155,188,108,131) -> Matrix(1,2,-4,-7) Matrix(11,12,-12,-13) -> Matrix(1,0,0,1) Matrix(61,52,156,133) -> Matrix(5,6,-16,-19) Matrix(107,88,276,227) -> Matrix(17,16,-50,-47) Matrix(227,184,132,107) -> Matrix(25,22,-58,-51) Matrix(61,48,108,85) -> Matrix(3,2,-2,-1) Matrix(29,22,54,41) -> Matrix(7,5,-10,-7) Matrix(19,14,42,31) -> Matrix(1,1,-4,-3) Matrix(131,92,84,59) -> Matrix(3,2,-8,-5) Matrix(13,8,60,37) -> Matrix(1,2,-2,-3) Matrix(17,10,90,53) -> Matrix(5,1,-6,-1) Matrix(97,56,168,97) -> Matrix(3,-2,-4,3) Matrix(53,30,30,17) -> Matrix(1,-1,-2,3) Matrix(5,2,-18,-7) -> Matrix(1,1,-4,-3) Matrix(37,8,60,13) -> Matrix(1,4,-2,-7) Matrix(257,46,162,29) -> Matrix(1,1,4,5) Matrix(17,2,42,5) -> Matrix(1,1,-6,-5) Matrix(77,-10,54,-7) -> Matrix(1,-1,-2,3) Matrix(13,-4,36,-11) -> Matrix(7,4,-16,-9) Matrix(263,-100,192,-73) -> Matrix(37,14,-82,-31) Matrix(265,-112,168,-71) -> Matrix(1,0,-4,1) Matrix(169,-100,120,-71) -> Matrix(19,12,-46,-29) Matrix(229,-144,132,-83) -> Matrix(39,22,-94,-53) Matrix(89,-58,66,-43) -> Matrix(21,11,-44,-23) Matrix(7,-6,6,-5) -> Matrix(1,1,-4,-3) Matrix(61,-100,36,-59) -> Matrix(7,4,-16,-9) 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): 32 ----------------------------------------------------------------------- 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 -1/1 (-1/1,0/1) 0 12 -5/6 -1/1 11 2 -4/5 -5/6 1 12 -7/9 -3/4 2 4 -3/4 -2/3 1 6 -5/7 -3/4 2 12 -2/3 -1/2 1 4 -3/5 -1/2 2 12 -1/2 -1/1 3 6 -1/3 -1/2 2 4 -1/4 0/1 3 6 -1/5 (-2/1,-1/1) 0 12 -1/6 -1/1 7 2 0/1 -1/2 1 12 1/6 -1/1 5 2 1/5 -3/4 2 12 1/4 -2/3 3 6 1/3 -1/2 4 4 3/8 -2/5 3 6 5/13 -5/14 2 12 7/18 -1/3 11 2 2/5 -1/4 1 12 5/12 0/1 7 2 3/7 1/0 2 12 1/2 -1/1 1 6 5/9 -1/2 2 4 4/7 -3/4 1 12 7/12 -2/3 5 2 3/5 (-2/3,-3/5) 0 12 5/8 -4/7 3 6 7/11 -11/20 2 12 2/3 -1/2 5 4 1/1 -1/2 2 12 1/0 0/1 1 2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(11,10,-12,-11) (-1/1,-5/6) -> (-1/1,-5/6) Reflection Matrix(47,38,120,97) (-5/6,-4/5) -> (7/18,2/5) Glide Reflection Matrix(61,48,108,85) (-4/5,-7/9) -> (5/9,4/7) Hyperbolic Matrix(29,22,54,41) (-7/9,-3/4) -> (1/2,5/9) Hyperbolic Matrix(19,14,42,31) (-3/4,-5/7) -> (3/7,1/2) Hyperbolic Matrix(29,20,-42,-29) (-5/7,-2/3) -> (-5/7,-2/3) Reflection Matrix(19,12,-30,-19) (-2/3,-3/5) -> (-2/3,-3/5) Reflection Matrix(7,4,30,17) (-3/5,-1/2) -> (1/5,1/4) Glide Reflection Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(11,2,-60,-11) (-1/5,-1/6) -> (-1/5,-1/6) Reflection Matrix(17,2,42,5) (-1/6,0/1) -> (2/5,5/12) Hyperbolic Matrix(31,-4,54,-7) (0/1,1/6) -> (4/7,7/12) Glide Reflection Matrix(11,-2,60,-11) (1/6,1/5) -> (1/6,1/5) Reflection Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(121,-46,192,-73) (3/8,5/13) -> (5/8,7/11) Glide Reflection Matrix(181,-70,468,-181) (5/13,7/18) -> (5/13,7/18) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(43,-28,66,-43) (7/11,2/3) -> (7/11,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(11,10,-12,-11) -> Matrix(-1,0,2,1) (-1/1,-5/6) -> (-1/1,0/1) Matrix(47,38,120,97) -> Matrix(7,6,-22,-19) Matrix(61,48,108,85) -> Matrix(3,2,-2,-1) -1/1 Matrix(29,22,54,41) -> Matrix(7,5,-10,-7) (-1/1,-2/3).(-3/4,-1/2) Matrix(19,14,42,31) -> Matrix(1,1,-4,-3) -1/2 Matrix(29,20,-42,-29) -> Matrix(5,3,-8,-5) (-5/7,-2/3) -> (-3/4,-1/2) Matrix(19,12,-30,-19) -> Matrix(1,1,0,-1) (-2/3,-3/5) -> (-1/2,1/0) Matrix(7,4,30,17) -> Matrix(1,-1,-2,1) Matrix(5,2,-18,-7) -> Matrix(1,1,-4,-3) -1/2 Matrix(37,8,60,13) -> Matrix(1,4,-2,-7) Matrix(11,2,-60,-11) -> Matrix(3,4,-2,-3) (-1/5,-1/6) -> (-2/1,-1/1) Matrix(17,2,42,5) -> Matrix(1,1,-6,-5) Matrix(31,-4,54,-7) -> Matrix(1,-1,-2,1) Matrix(11,-2,60,-11) -> Matrix(7,6,-8,-7) (1/6,1/5) -> (-1/1,-3/4) Matrix(13,-4,36,-11) -> Matrix(7,4,-16,-9) -1/2 Matrix(121,-46,192,-73) -> Matrix(37,14,-66,-25) Matrix(181,-70,468,-181) -> Matrix(29,10,-84,-29) (5/13,7/18) -> (-5/14,-1/3) Matrix(71,-30,168,-71) -> Matrix(1,0,0,-1) (5/12,3/7) -> (0/1,1/0) Matrix(71,-42,120,-71) -> Matrix(19,12,-30,-19) (7/12,3/5) -> (-2/3,-3/5) Matrix(43,-28,66,-43) -> Matrix(21,11,-40,-21) (7/11,2/3) -> (-11/20,-1/2) Matrix(5,-4,6,-5) -> Matrix(1,1,0,-1) (2/3,1/1) -> (-1/2,1/0) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.