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 -5/6 -3/4 -2/3 -23/36 -5/9 -1/2 -11/24 -3/8 -1/3 -11/36 -3/10 -7/24 -1/4 -2/9 -5/24 -2/11 -1/6 -1/8 0/1 1/7 1/6 1/5 2/9 1/4 3/11 2/7 3/10 1/3 3/8 2/5 5/12 7/16 1/2 5/9 7/12 2/3 3/4 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -6/7 1/1 1/0 -5/6 0/1 -4/5 0/1 1/1 -11/14 1/5 1/3 -18/23 1/3 4/11 -7/9 1/2 -3/4 1/1 -11/15 1/0 -8/11 1/1 1/0 -13/18 1/0 -5/7 1/1 1/0 -7/10 -3/1 -1/1 -2/3 0/1 -9/14 3/11 1/3 -16/25 6/19 1/3 -23/36 1/3 -7/11 1/3 3/8 -5/8 1/2 -8/13 1/2 1/1 -3/5 2/3 1/1 -7/12 1/1 -4/7 1/1 3/2 -13/23 1/1 2/1 -9/16 2/1 -14/25 8/3 3/1 -5/9 1/0 -6/11 1/1 1/0 -1/2 -1/1 1/1 -6/13 1/1 1/0 -11/24 1/0 -5/11 -1/1 1/0 -4/9 0/1 -11/25 0/1 1/1 -7/16 0/1 -3/7 1/2 1/1 -5/12 1/1 -2/5 1/1 2/1 -7/18 2/1 -5/13 5/2 3/1 -3/8 1/0 -4/11 1/1 1/0 -1/3 1/0 -4/13 -3/1 1/0 -11/36 -3/1 -7/23 -3/1 -2/1 -3/10 -3/1 -1/1 -5/17 1/1 1/0 -7/24 1/0 -2/7 -3/1 1/0 -5/18 -2/1 -3/11 -1/1 1/0 -1/4 -1/1 -3/13 -1/1 -1/2 -2/9 0/1 -5/23 0/1 1/1 -3/14 -1/1 1/1 -4/19 -1/3 0/1 -5/24 0/1 -1/5 0/1 1/1 -2/11 1/1 1/0 -1/6 1/0 -1/7 -3/1 1/0 -1/8 -2/1 0/1 -1/1 0/1 1/7 -3/5 -1/2 1/6 -1/2 1/5 -1/3 0/1 3/14 -1/1 -1/3 5/23 -1/3 0/1 2/9 0/1 1/4 -1/1 4/15 -2/3 3/11 -1/1 -1/2 5/18 -2/3 2/7 -3/5 -1/2 3/10 -1/1 -3/5 1/3 -1/2 5/14 -3/7 -1/3 9/25 -2/5 -1/3 13/36 -1/3 4/11 -1/2 -1/3 3/8 -1/2 5/13 -3/7 -5/12 2/5 -2/5 -1/3 5/12 -1/3 3/7 -1/3 -1/4 10/23 -2/9 -1/5 7/16 0/1 11/25 -1/3 0/1 4/9 0/1 5/11 -1/1 -1/2 1/2 -1/1 -1/3 7/13 -1/1 -1/2 13/24 -1/2 6/11 -1/2 -1/3 5/9 -1/2 14/25 -3/7 -8/19 9/16 -2/5 4/7 -3/8 -1/3 7/12 -1/3 3/5 -1/3 -2/7 11/18 -1/4 8/13 -1/3 -1/4 5/8 -1/4 7/11 -3/14 -1/5 2/3 0/1 9/13 -3/2 -1/1 25/36 -1/1 16/23 -1/1 -6/7 7/10 -1/1 -3/5 12/17 -3/5 -1/2 17/24 -1/2 5/7 -1/2 -1/3 13/18 -1/2 8/11 -1/2 -1/3 3/4 -1/3 10/13 -1/3 -1/4 7/9 -1/4 18/23 -4/19 -1/5 11/14 -1/5 -1/7 15/19 -1/7 0/1 19/24 0/1 4/5 -1/3 0/1 9/11 -1/5 -1/6 5/6 0/1 6/7 -1/2 -1/3 7/8 0/1 1/1 -1/3 0/1 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(95,82,-168,-145) (-1/1,-6/7) -> (-4/7,-13/23) Hyperbolic Matrix(47,40,168,143) (-6/7,-5/6) -> (5/18,2/7) Hyperbolic Matrix(47,38,-120,-97) (-5/6,-4/5) -> (-2/5,-7/18) Hyperbolic Matrix(71,56,-336,-265) (-4/5,-11/14) -> (-3/14,-4/19) Hyperbolic Matrix(385,302,552,433) (-11/14,-18/23) -> (16/23,7/10) Hyperbolic Matrix(241,188,432,337) (-18/23,-7/9) -> (5/9,14/25) Hyperbolic Matrix(71,54,-96,-73) (-7/9,-3/4) -> (-3/4,-11/15) Parabolic Matrix(145,106,264,193) (-11/15,-8/11) -> (6/11,5/9) Hyperbolic Matrix(265,192,432,313) (-8/11,-13/18) -> (11/18,8/13) Hyperbolic Matrix(25,18,168,121) (-13/18,-5/7) -> (1/7,1/6) Hyperbolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(527,338,672,431) (-9/14,-16/25) -> (18/23,11/14) Hyperbolic Matrix(1201,768,1728,1105) (-16/25,-23/36) -> (25/36,16/23) Hyperbolic Matrix(599,382,864,551) (-23/36,-7/11) -> (9/13,25/36) Hyperbolic Matrix(73,46,192,121) (-7/11,-5/8) -> (3/8,5/13) Hyperbolic Matrix(71,44,192,119) (-5/8,-8/13) -> (4/11,3/8) Hyperbolic Matrix(23,14,-120,-73) (-8/13,-3/5) -> (-1/5,-2/11) Hyperbolic Matrix(71,42,120,71) (-3/5,-7/12) -> (7/12,3/5) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(337,190,768,433) (-13/23,-9/16) -> (7/16,11/25) Hyperbolic Matrix(335,188,768,431) (-9/16,-14/25) -> (10/23,7/16) Hyperbolic Matrix(337,188,432,241) (-14/25,-5/9) -> (7/9,18/23) 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(311,142,576,263) (-11/24,-5/11) -> (7/13,13/24) Hyperbolic Matrix(71,32,264,119) (-5/11,-4/9) -> (4/15,3/11) Hyperbolic Matrix(95,42,432,191) (-4/9,-11/25) -> (5/23,2/9) Hyperbolic Matrix(191,84,216,95) (-11/25,-7/16) -> (7/8,1/1) Hyperbolic Matrix(23,10,-168,-73) (-7/16,-3/7) -> (-1/7,-1/8) Hyperbolic Matrix(71,30,168,71) (-3/7,-5/12) -> (5/12,3/7) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(119,46,432,167) (-7/18,-5/13) -> (3/11,5/18) Hyperbolic Matrix(121,46,192,73) (-5/13,-3/8) -> (5/8,7/11) Hyperbolic Matrix(119,44,192,71) (-3/8,-4/11) -> (8/13,5/8) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(313,96,864,265) (-4/13,-11/36) -> (13/36,4/11) Hyperbolic Matrix(623,190,1728,527) (-11/36,-7/23) -> (9/25,13/36) Hyperbolic Matrix(119,36,552,167) (-7/23,-3/10) -> (3/14,5/23) Hyperbolic Matrix(409,120,576,169) (-5/17,-7/24) -> (17/24,5/7) Hyperbolic Matrix(407,118,576,167) (-7/24,-2/7) -> (12/17,17/24) Hyperbolic Matrix(143,40,168,47) (-2/7,-5/18) -> (5/6,6/7) Hyperbolic Matrix(217,60,264,73) (-5/18,-3/11) -> (9/11,5/6) Hyperbolic Matrix(23,6,-96,-25) (-3/11,-1/4) -> (-1/4,-3/13) Parabolic Matrix(97,22,216,49) (-3/13,-2/9) -> (4/9,5/11) Hyperbolic Matrix(191,42,432,95) (-2/9,-5/23) -> (11/25,4/9) Hyperbolic Matrix(241,52,672,145) (-5/23,-3/14) -> (5/14,9/25) Hyperbolic Matrix(457,96,576,121) (-4/19,-5/24) -> (19/24,4/5) Hyperbolic Matrix(455,94,576,119) (-5/24,-1/5) -> (15/19,19/24) Hyperbolic Matrix(191,34,264,47) (-2/11,-1/6) -> (13/18,8/11) Hyperbolic Matrix(121,18,168,25) (-1/6,-1/7) -> (5/7,13/18) Hyperbolic Matrix(121,14,216,25) (-1/8,0/1) -> (14/25,9/16) Hyperbolic Matrix(73,-10,168,-23) (0/1,1/7) -> (3/7,10/23) Hyperbolic Matrix(73,-14,120,-23) (1/6,1/5) -> (3/5,11/18) Hyperbolic Matrix(265,-56,336,-71) (1/5,3/14) -> (11/14,15/19) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(145,-82,168,-95) (9/16,4/7) -> (6/7,7/8) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(95,82,-168,-145) -> Matrix(3,-2,2,-1) Matrix(47,40,168,143) -> Matrix(1,2,-2,-3) Matrix(47,38,-120,-97) -> Matrix(3,-2,2,-1) Matrix(71,56,-336,-265) -> Matrix(1,0,-4,1) Matrix(385,302,552,433) -> Matrix(7,-2,-10,3) Matrix(241,188,432,337) -> Matrix(9,-4,-20,9) Matrix(71,54,-96,-73) -> Matrix(3,-2,2,-1) Matrix(145,106,264,193) -> Matrix(1,-2,-2,5) Matrix(265,192,432,313) -> Matrix(1,0,-4,1) Matrix(25,18,168,121) -> Matrix(1,2,-2,-3) Matrix(71,50,-240,-169) -> Matrix(1,0,0,1) Matrix(47,32,-72,-49) -> Matrix(1,0,4,1) Matrix(527,338,672,431) -> Matrix(7,-2,-38,11) Matrix(1201,768,1728,1105) -> Matrix(37,-12,-40,13) Matrix(599,382,864,551) -> Matrix(17,-6,-14,5) Matrix(73,46,192,121) -> Matrix(9,-4,-20,9) Matrix(71,44,192,119) -> Matrix(1,0,-4,1) Matrix(23,14,-120,-73) -> Matrix(3,-2,2,-1) Matrix(71,42,120,71) -> Matrix(5,-4,-16,13) Matrix(97,56,168,97) -> Matrix(5,-6,-14,17) Matrix(337,190,768,433) -> Matrix(1,-2,-2,5) Matrix(335,188,768,431) -> Matrix(1,-2,-6,13) Matrix(337,188,432,241) -> Matrix(1,-4,-4,17) Matrix(167,92,216,119) -> Matrix(1,0,-4,1) Matrix(23,12,-48,-25) -> Matrix(1,0,0,1) Matrix(313,144,576,265) -> Matrix(1,-2,-2,5) Matrix(311,142,576,263) -> Matrix(1,2,-2,-3) Matrix(71,32,264,119) -> Matrix(1,2,-2,-3) Matrix(95,42,432,191) -> Matrix(1,0,-4,1) Matrix(191,84,216,95) -> Matrix(1,0,-4,1) Matrix(23,10,-168,-73) -> Matrix(5,-2,-2,1) Matrix(71,30,168,71) -> Matrix(3,-2,-10,7) Matrix(49,20,120,49) -> Matrix(3,-4,-8,11) Matrix(119,46,432,167) -> Matrix(3,-8,-4,11) Matrix(121,46,192,73) -> Matrix(1,-4,-4,17) Matrix(119,44,192,71) -> Matrix(1,0,-4,1) Matrix(23,8,-72,-25) -> Matrix(1,-4,0,1) Matrix(313,96,864,265) -> Matrix(1,2,-2,-3) Matrix(623,190,1728,527) -> Matrix(3,8,-8,-21) Matrix(119,36,552,167) -> Matrix(1,2,-2,-3) Matrix(409,120,576,169) -> Matrix(1,-2,-2,5) Matrix(407,118,576,167) -> Matrix(1,6,-2,-11) Matrix(143,40,168,47) -> Matrix(1,2,-2,-3) Matrix(217,60,264,73) -> Matrix(1,2,-6,-11) Matrix(23,6,-96,-25) -> Matrix(1,2,-2,-3) Matrix(97,22,216,49) -> Matrix(1,0,0,1) Matrix(191,42,432,95) -> Matrix(1,0,-4,1) Matrix(241,52,672,145) -> Matrix(1,-2,-2,5) Matrix(457,96,576,121) -> Matrix(1,0,0,1) Matrix(455,94,576,119) -> Matrix(1,0,-8,1) Matrix(191,34,264,47) -> Matrix(1,-2,-2,5) Matrix(121,18,168,25) -> Matrix(1,2,-2,-3) Matrix(121,14,216,25) -> Matrix(5,8,-12,-19) Matrix(73,-10,168,-23) -> Matrix(3,2,-14,-9) Matrix(73,-14,120,-23) -> Matrix(5,2,-18,-7) Matrix(265,-56,336,-71) -> Matrix(1,0,-4,1) Matrix(25,-6,96,-23) -> Matrix(1,2,-2,-3) Matrix(169,-50,240,-71) -> Matrix(1,0,0,1) Matrix(25,-8,72,-23) -> Matrix(7,4,-16,-9) Matrix(97,-38,120,-47) -> Matrix(5,2,-18,-7) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(145,-82,168,-95) -> Matrix(5,2,-18,-7) Matrix(49,-32,72,-47) -> Matrix(1,0,4,1) Matrix(73,-54,96,-71) -> Matrix(5,2,-18,-7) 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): 11 Degree of the the map X: 22 Degree of the the map Y: 64 Permutation triple for Y: ((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); (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)) ----------------------------------------------------------------------- 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 -3/4 -2/3 -3/8 -1/3 -7/24 -1/4 -1/6 0/1 1/6 1/5 1/4 3/10 1/3 13/36 3/8 2/5 5/12 1/2 7/12 2/3 25/36 3/4 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -5/6 0/1 -4/5 0/1 1/1 -3/4 1/1 -5/7 1/1 1/0 -7/10 -3/1 -1/1 -2/3 0/1 -9/14 3/11 1/3 -7/11 1/3 3/8 -5/8 1/2 -8/13 1/2 1/1 -3/5 2/3 1/1 -1/2 -1/1 1/1 -2/5 1/1 2/1 -5/13 5/2 3/1 -3/8 1/0 -4/11 1/1 1/0 -1/3 1/0 -4/13 -3/1 1/0 -3/10 -3/1 -1/1 -5/17 1/1 1/0 -7/24 1/0 -2/7 -3/1 1/0 -1/4 -1/1 -1/5 0/1 1/1 -1/6 1/0 0/1 -1/1 0/1 1/7 -3/5 -1/2 1/6 -1/2 1/5 -1/3 0/1 1/4 -1/1 2/7 -3/5 -1/2 3/10 -1/1 -3/5 1/3 -1/2 5/14 -3/7 -1/3 9/25 -2/5 -1/3 13/36 -1/3 4/11 -1/2 -1/3 3/8 -1/2 5/13 -3/7 -5/12 2/5 -2/5 -1/3 5/12 -1/3 3/7 -1/3 -1/4 1/2 -1/1 -1/3 4/7 -3/8 -1/3 7/12 -1/3 3/5 -1/3 -2/7 11/18 -1/4 8/13 -1/3 -1/4 5/8 -1/4 7/11 -3/14 -1/5 2/3 0/1 9/13 -3/2 -1/1 25/36 -1/1 16/23 -1/1 -6/7 7/10 -1/1 -3/5 12/17 -3/5 -1/2 17/24 -1/2 5/7 -1/2 -1/3 3/4 -1/3 4/5 -1/3 0/1 9/11 -1/5 -1/6 5/6 0/1 6/7 -1/2 -1/3 1/1 -1/3 0/1 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(59,51,96,83) (-1/1,-5/6) -> (11/18,8/13) Hyperbolic Matrix(11,9,72,59) (-5/6,-4/5) -> (1/7,1/6) Hyperbolic Matrix(35,27,-48,-37) (-4/5,-3/4) -> (-3/4,-5/7) Parabolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(301,193,432,277) (-9/14,-7/11) -> (16/23,7/10) Hyperbolic Matrix(73,46,192,121) (-7/11,-5/8) -> (3/8,5/13) Hyperbolic Matrix(71,44,192,119) (-5/8,-8/13) -> (4/11,3/8) Hyperbolic Matrix(83,51,96,59) (-8/13,-3/5) -> (6/7,1/1) Hyperbolic Matrix(13,7,24,13) (-3/5,-1/2) -> (1/2,4/7) Hyperbolic Matrix(11,5,24,11) (-1/2,-2/5) -> (3/7,1/2) Hyperbolic Matrix(13,5,96,37) (-2/5,-5/13) -> (0/1,1/7) Hyperbolic Matrix(121,46,192,73) (-5/13,-3/8) -> (5/8,7/11) Hyperbolic Matrix(119,44,192,71) (-3/8,-4/11) -> (8/13,5/8) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(155,47,432,131) (-4/13,-3/10) -> (5/14,9/25) Hyperbolic Matrix(409,120,576,169) (-5/17,-7/24) -> (17/24,5/7) Hyperbolic Matrix(407,118,576,167) (-7/24,-2/7) -> (12/17,17/24) Hyperbolic Matrix(11,3,-48,-13) (-2/7,-1/4) -> (-1/4,-1/5) Parabolic Matrix(61,11,72,13) (-1/5,-1/6) -> (5/6,6/7) Hyperbolic Matrix(59,9,72,11) (-1/6,0/1) -> (9/11,5/6) Hyperbolic Matrix(73,-14,120,-23) (1/6,1/5) -> (3/5,11/18) Hyperbolic Matrix(13,-3,48,-11) (1/5,1/4) -> (1/4,2/7) Parabolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(469,-169,1296,-467) (9/25,13/36) -> (13/36,4/11) Parabolic Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(61,-25,144,-59) (2/5,5/12) -> (5/12,3/7) Parabolic Matrix(85,-49,144,-83) (4/7,7/12) -> (7/12,3/5) Parabolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(901,-625,1296,-899) (9/13,25/36) -> (25/36,16/23) Parabolic Matrix(37,-27,48,-35) (5/7,3/4) -> (3/4,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(59,51,96,83) -> Matrix(2,-1,-7,4) Matrix(11,9,72,59) -> Matrix(4,-1,-7,2) Matrix(35,27,-48,-37) -> Matrix(2,-1,1,0) Matrix(71,50,-240,-169) -> Matrix(1,0,0,1) Matrix(47,32,-72,-49) -> Matrix(1,0,4,1) Matrix(301,193,432,277) -> Matrix(10,-3,-13,4) Matrix(73,46,192,121) -> Matrix(9,-4,-20,9) Matrix(71,44,192,119) -> Matrix(1,0,-4,1) Matrix(83,51,96,59) -> Matrix(2,-1,-7,4) Matrix(13,7,24,13) -> Matrix(0,-1,1,2) Matrix(11,5,24,11) -> Matrix(0,-1,1,2) Matrix(13,5,96,37) -> Matrix(2,-5,-3,8) Matrix(121,46,192,73) -> Matrix(1,-4,-4,17) Matrix(119,44,192,71) -> Matrix(1,0,-4,1) Matrix(23,8,-72,-25) -> Matrix(1,-4,0,1) Matrix(155,47,432,131) -> Matrix(2,5,-5,-12) Matrix(409,120,576,169) -> Matrix(1,-2,-2,5) Matrix(407,118,576,167) -> Matrix(1,6,-2,-11) Matrix(11,3,-48,-13) -> Matrix(0,-1,1,2) Matrix(61,11,72,13) -> Matrix(0,-1,1,2) Matrix(59,9,72,11) -> Matrix(0,-1,1,6) Matrix(73,-14,120,-23) -> Matrix(5,2,-18,-7) Matrix(13,-3,48,-11) -> Matrix(0,-1,1,2) Matrix(169,-50,240,-71) -> Matrix(1,0,0,1) Matrix(25,-8,72,-23) -> Matrix(7,4,-16,-9) Matrix(469,-169,1296,-467) -> Matrix(2,1,-9,-4) Matrix(97,-38,120,-47) -> Matrix(5,2,-18,-7) Matrix(61,-25,144,-59) -> Matrix(8,3,-27,-10) Matrix(85,-49,144,-83) -> Matrix(14,5,-45,-16) Matrix(49,-32,72,-47) -> Matrix(1,0,4,1) Matrix(901,-625,1296,-899) -> Matrix(8,9,-9,-10) Matrix(37,-27,48,-35) -> Matrix(2,1,-9,-4) 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): 11 ----------------------------------------------------------------------- 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 (-1/1,0/1) 0 12 1/6 -1/2 2 2 1/5 (-1/3,0/1) 0 12 1/4 -1/1 1 3 2/7 (-3/5,-1/2) 0 12 3/10 (-1/1,-3/5) 0 6 1/3 -1/2 2 4 5/14 (-3/7,-1/3) 0 6 13/36 -1/3 1 1 4/11 (-1/2,-1/3) 0 12 3/8 -1/2 2 3 5/13 (-3/7,-5/12) 0 12 2/5 (-2/5,-1/3) 0 12 5/12 -1/3 3 1 1/2 (-1/1,-1/3) 0 6 7/12 -1/3 5 1 3/5 (-1/3,-2/7) 0 12 8/13 (-1/3,-1/4) 0 12 5/8 -1/4 2 3 7/11 (-3/14,-1/5) 0 12 2/3 0/1 2 4 9/13 (-3/2,-1/1) 0 12 25/36 -1/1 9 1 7/10 (-1/1,-3/5) 0 6 12/17 (-3/5,-1/2) 0 12 17/24 -1/2 4 1 5/7 (-1/2,-1/3) 0 12 3/4 -1/3 1 3 4/5 (-1/3,0/1) 0 12 9/11 (-1/5,-1/6) 0 12 5/6 0/1 2 2 6/7 (-1/2,-1/3) 0 12 1/1 (-1/3,0/1) 0 12 1/0 0/1 2 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,-9,72,-11) (0/1,1/6) -> (9/11,5/6) Glide Reflection Matrix(61,-11,72,-13) (1/6,1/5) -> (5/6,6/7) Glide Reflection Matrix(13,-3,48,-11) (1/5,1/4) -> (1/4,2/7) Parabolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(181,-65,504,-181) (5/14,13/36) -> (5/14,13/36) Reflection Matrix(287,-104,792,-287) (13/36,4/11) -> (13/36,4/11) Reflection Matrix(119,-44,192,-71) (4/11,3/8) -> (8/13,5/8) Glide Reflection Matrix(121,-46,192,-73) (3/8,5/13) -> (5/8,7/11) Glide Reflection Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(11,-5,24,-11) (5/12,1/2) -> (5/12,1/2) Reflection Matrix(13,-7,24,-13) (1/2,7/12) -> (1/2,7/12) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(83,-51,96,-59) (3/5,8/13) -> (6/7,1/1) Glide Reflection Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(649,-450,936,-649) (9/13,25/36) -> (9/13,25/36) Reflection Matrix(251,-175,360,-251) (25/36,7/10) -> (25/36,7/10) Reflection Matrix(577,-408,816,-577) (12/17,17/24) -> (12/17,17/24) Reflection Matrix(239,-170,336,-239) (17/24,5/7) -> (17/24,5/7) Reflection Matrix(37,-27,48,-35) (5/7,3/4) -> (3/4,4/5) Parabolic 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(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(59,-9,72,-11) -> Matrix(2,1,-11,-6) Matrix(61,-11,72,-13) -> Matrix(2,1,-3,-2) *** -> (-1/1,-1/3) Matrix(13,-3,48,-11) -> Matrix(0,-1,1,2) -1/1 Matrix(169,-50,240,-71) -> Matrix(1,0,0,1) Matrix(25,-8,72,-23) -> Matrix(7,4,-16,-9) -1/2 Matrix(181,-65,504,-181) -> Matrix(8,3,-21,-8) (5/14,13/36) -> (-3/7,-1/3) Matrix(287,-104,792,-287) -> Matrix(5,2,-12,-5) (13/36,4/11) -> (-1/2,-1/3) Matrix(119,-44,192,-71) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(121,-46,192,-73) -> Matrix(9,4,-38,-17) Matrix(97,-38,120,-47) -> Matrix(5,2,-18,-7) -1/3 Matrix(49,-20,120,-49) -> Matrix(11,4,-30,-11) (2/5,5/12) -> (-2/5,-1/3) Matrix(11,-5,24,-11) -> Matrix(2,1,-3,-2) (5/12,1/2) -> (-1/1,-1/3) Matrix(13,-7,24,-13) -> Matrix(2,1,-3,-2) (1/2,7/12) -> (-1/1,-1/3) Matrix(71,-42,120,-71) -> Matrix(13,4,-42,-13) (7/12,3/5) -> (-1/3,-2/7) Matrix(83,-51,96,-59) -> Matrix(4,1,-15,-4) *** -> (-1/3,-1/5) Matrix(49,-32,72,-47) -> Matrix(1,0,4,1) 0/1 Matrix(649,-450,936,-649) -> Matrix(5,6,-4,-5) (9/13,25/36) -> (-3/2,-1/1) Matrix(251,-175,360,-251) -> Matrix(4,3,-5,-4) (25/36,7/10) -> (-1/1,-3/5) Matrix(577,-408,816,-577) -> Matrix(11,6,-20,-11) (12/17,17/24) -> (-3/5,-1/2) Matrix(239,-170,336,-239) -> Matrix(5,2,-12,-5) (17/24,5/7) -> (-1/2,-1/3) Matrix(37,-27,48,-35) -> Matrix(2,1,-9,-4) -1/3 Matrix(-1,2,0,1) -> Matrix(-1,0,6,1) (1/1,1/0) -> (-1/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.