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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/4 -1/2 -7/16 -5/16 -1/4 -3/16 -1/6 -1/7 -1/8 0/1 1/6 1/5 1/4 2/7 1/3 3/8 2/5 1/2 5/8 2/3 3/4 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/8 -5/6 3/20 -4/5 2/13 3/19 -3/4 1/6 -5/7 1/6 -7/10 1/6 -2/3 2/11 1/5 -5/8 1/5 -3/5 3/14 -4/7 1/5 2/9 -1/2 1/4 -4/9 0/1 1/3 -7/16 0/1 -3/7 1/4 -5/12 1/4 -2/5 2/7 1/3 -3/8 1/3 -1/3 1/2 -5/16 0/1 -4/13 0/1 1/5 -3/10 1/4 -5/17 5/16 -7/24 1/3 -2/7 1/3 2/5 -1/4 1/2 -2/9 0/1 1/1 -1/5 1/0 -3/16 0/1 -2/11 0/1 1/5 -1/6 1/2 -1/7 1/2 -1/8 1/1 0/1 0/1 1/1 1/6 1/0 1/5 1/2 1/4 1/0 2/7 -2/1 -1/1 3/10 -1/2 1/3 1/0 3/8 -1/1 2/5 -1/1 -2/3 3/7 -1/2 1/2 -1/2 5/9 -1/2 9/16 -2/5 4/7 -2/5 -1/3 7/12 -1/2 3/5 -3/8 5/8 -1/3 2/3 -1/3 -2/7 11/16 -2/7 9/13 -5/18 7/10 -1/4 12/17 -2/7 -3/11 17/24 -3/11 5/7 -1/4 3/4 -1/4 7/9 -1/4 4/5 -3/13 -2/9 13/16 -2/9 9/11 -7/32 5/6 -3/14 6/7 -6/29 -1/5 7/8 -1/5 1/1 -1/6 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(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(17,14,-96,-79) (-5/6,-4/5) -> (-2/11,-1/6) Hyperbolic Matrix(33,26,-80,-63) (-4/5,-3/4) -> (-5/12,-2/5) Hyperbolic Matrix(47,34,-112,-81) (-3/4,-5/7) -> (-3/7,-5/12) Hyperbolic Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(49,34,-160,-111) (-7/10,-2/3) -> (-4/13,-3/10) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(17,10,-80,-47) (-3/5,-4/7) -> (-2/9,-1/5) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(143,62,256,111) (-7/16,-3/7) -> (5/9,9/16) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(177,56,256,81) (-1/3,-5/16) -> (11/16,9/13) Hyperbolic Matrix(175,54,256,79) (-5/16,-4/13) -> (2/3,11/16) Hyperbolic Matrix(143,42,160,47) (-5/17,-7/24) -> (7/8,1/1) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(15,4,-64,-17) (-2/7,-1/4) -> (-1/4,-2/9) Parabolic Matrix(209,40,256,49) (-1/5,-3/16) -> (13/16,9/11) Hyperbolic Matrix(207,38,256,47) (-3/16,-2/11) -> (4/5,13/16) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(113,12,160,17) (-1/8,0/1) -> (12/17,17/24) Hyperbolic Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(79,-14,96,-17) (1/6,1/5) -> (9/11,5/6) Hyperbolic Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(65,-18,112,-31) (1/4,2/7) -> (4/7,7/12) Hyperbolic Matrix(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(111,-34,160,-49) (3/10,1/3) -> (9/13,7/10) Hyperbolic Matrix(63,-26,80,-33) (2/5,3/7) -> (7/9,4/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-14,1) Matrix(33,28,-112,-95) -> Matrix(27,-4,88,-13) Matrix(17,14,-96,-79) -> Matrix(13,-2,46,-7) Matrix(33,26,-80,-63) -> Matrix(25,-4,94,-15) Matrix(47,34,-112,-81) -> Matrix(11,-2,50,-9) Matrix(17,12,-112,-79) -> Matrix(11,-2,28,-5) Matrix(49,34,-160,-111) -> Matrix(11,-2,50,-9) Matrix(31,20,48,31) -> Matrix(21,-4,-68,13) Matrix(49,30,80,49) -> Matrix(29,-6,-82,17) Matrix(17,10,-80,-47) -> Matrix(9,-2,14,-3) Matrix(15,8,-32,-17) -> Matrix(9,-2,32,-7) Matrix(145,64,256,113) -> Matrix(5,-2,-12,5) Matrix(143,62,256,111) -> Matrix(9,-2,-22,5) Matrix(31,12,80,31) -> Matrix(13,-4,-16,5) Matrix(17,6,48,17) -> Matrix(5,-2,-2,1) Matrix(177,56,256,81) -> Matrix(1,2,-4,-7) Matrix(175,54,256,79) -> Matrix(11,-2,-38,7) Matrix(143,42,160,47) -> Matrix(19,-6,-98,31) Matrix(193,56,224,65) -> Matrix(23,-8,-112,39) Matrix(15,4,-64,-17) -> Matrix(5,-2,8,-3) Matrix(209,40,256,49) -> Matrix(7,2,-32,-9) Matrix(207,38,256,47) -> Matrix(13,-2,-58,9) Matrix(159,22,224,31) -> Matrix(7,-4,-26,15) Matrix(113,12,160,17) -> Matrix(1,2,-4,-7) Matrix(79,-12,112,-17) -> Matrix(1,2,-4,-7) Matrix(79,-14,96,-17) -> Matrix(3,2,-14,-9) Matrix(47,-10,80,-17) -> Matrix(1,-2,-2,5) Matrix(65,-18,112,-31) -> Matrix(1,0,-2,1) Matrix(95,-28,112,-33) -> Matrix(5,4,-24,-19) Matrix(111,-34,160,-49) -> Matrix(5,2,-18,-7) Matrix(63,-26,80,-33) -> Matrix(7,4,-30,-17) Matrix(17,-8,32,-15) -> Matrix(3,2,-8,-5) Matrix(49,-36,64,-47) -> Matrix(23,6,-96,-25) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 32 Permutation triple for Y: ((1,4,16,26,29,17,5,2)(3,10,27,32,22,8,7,11)(6,14,13,9,25,30,23,20)(12,15,24,31,19,18,21,28); (1,2,8,23,31,24,9,3)(4,14,6,5,19,32,27,15)(7,18,17,30,25,16,12,11)(10,13,28,21,20,22,29,26); (2,6,21,7)(3,12,13,4)(5,18)(8,20)(9,10)(15,16)(17,22,19,23)(24,27,26,25)) ----------------------------------------------------------------------- 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): 96 Minimal number of generators: 17 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: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/2 -7/16 0/1 1/5 1/4 1/3 3/8 3/7 1/2 5/8 2/3 3/4 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/8 -3/4 1/6 -5/7 1/6 -2/3 2/11 1/5 -5/8 1/5 -3/5 3/14 -4/7 1/5 2/9 -1/2 1/4 -4/9 0/1 1/3 -7/16 0/1 -3/7 1/4 -2/5 2/7 1/3 -3/8 1/3 -1/3 1/2 -1/4 1/2 -1/5 1/0 0/1 0/1 1/1 1/5 1/2 1/4 1/0 1/3 1/0 3/8 -1/1 2/5 -1/1 -2/3 3/7 -1/2 1/2 -1/2 5/9 -1/2 9/16 -2/5 4/7 -2/5 -1/3 3/5 -3/8 5/8 -1/3 2/3 -1/3 -2/7 5/7 -1/4 3/4 -1/4 1/1 -1/6 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(7,6,8,7) (-1/1,-3/4) -> (3/4,1/1) Hyperbolic Matrix(41,30,56,41) (-3/4,-5/7) -> (5/7,3/4) Hyperbolic Matrix(23,16,56,39) (-5/7,-2/3) -> (2/5,3/7) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(7,4,40,23) (-3/5,-4/7) -> (0/1,1/5) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(143,62,256,111) (-7/16,-3/7) -> (5/9,9/16) Hyperbolic Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(7,2,24,7) (-1/3,-1/4) -> (1/4,1/3) Hyperbolic Matrix(9,2,40,9) (-1/4,-1/5) -> (1/5,1/4) Hyperbolic Matrix(23,4,40,7) (-1/5,0/1) -> (4/7,3/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-14,1) Matrix(7,6,8,7) -> Matrix(7,-1,-34,5) Matrix(41,30,56,41) -> Matrix(29,-5,-110,19) Matrix(23,16,56,39) -> Matrix(17,-3,-28,5) Matrix(31,20,48,31) -> Matrix(21,-4,-68,13) Matrix(49,30,80,49) -> Matrix(29,-6,-82,17) Matrix(7,4,40,23) -> Matrix(5,-1,-4,1) Matrix(15,8,-32,-17) -> Matrix(9,-2,32,-7) Matrix(145,64,256,113) -> Matrix(5,-2,-12,5) Matrix(143,62,256,111) -> Matrix(9,-2,-22,5) Matrix(39,16,56,23) -> Matrix(11,-3,-40,11) Matrix(31,12,80,31) -> Matrix(13,-4,-16,5) Matrix(17,6,48,17) -> Matrix(5,-2,-2,1) Matrix(7,2,24,7) -> Matrix(3,-1,-2,1) Matrix(9,2,40,9) -> Matrix(1,-1,2,-1) Matrix(23,4,40,7) -> Matrix(3,-1,-8,3) Matrix(17,-8,32,-15) -> Matrix(3,2,-8,-5) 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): 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 (0/1,1/1) 0 8 1/5 1/2 1 8 1/4 1/0 1 2 1/3 1/0 1 8 3/8 -1/1 3 1 2/5 (-1/1,-2/3) 0 8 3/7 -1/2 1 8 1/2 -1/2 1 4 5/9 -1/2 1 8 9/16 -2/5 1 1 4/7 (-2/5,-1/3) 0 8 3/5 -3/8 1 8 5/8 -1/3 5 1 2/3 (-1/3,-2/7) 0 8 5/7 -1/4 1 8 3/4 -1/4 3 2 1/1 -1/6 1 8 1/0 0/1 7 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(23,-4,40,-7) (0/1,1/5) -> (4/7,3/5) Glide Reflection Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(39,-16,56,-23) (2/5,3/7) -> (2/3,5/7) Glide Reflection Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(41,-30,56,-41) (5/7,3/4) -> (5/7,3/4) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) 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(1,0,0,-1) -> Matrix(1,0,2,-1) (0/1,1/0) -> (0/1,1/1) Matrix(23,-4,40,-7) -> Matrix(1,1,-2,-3) Matrix(9,-2,40,-9) -> Matrix(-1,1,0,1) (1/5,1/4) -> (1/2,1/0) Matrix(7,-2,24,-7) -> Matrix(1,1,0,-1) (1/4,1/3) -> (-1/2,1/0) Matrix(17,-6,48,-17) -> Matrix(1,2,0,-1) (1/3,3/8) -> (-1/1,1/0) Matrix(31,-12,80,-31) -> Matrix(5,4,-6,-5) (3/8,2/5) -> (-1/1,-2/3) Matrix(39,-16,56,-23) -> Matrix(5,3,-18,-11) Matrix(17,-8,32,-15) -> Matrix(3,2,-8,-5) -1/2 Matrix(161,-90,288,-161) -> Matrix(9,4,-20,-9) (5/9,9/16) -> (-1/2,-2/5) Matrix(127,-72,224,-127) -> Matrix(11,4,-30,-11) (9/16,4/7) -> (-2/5,-1/3) Matrix(49,-30,80,-49) -> Matrix(17,6,-48,-17) (3/5,5/8) -> (-3/8,-1/3) Matrix(31,-20,48,-31) -> Matrix(13,4,-42,-13) (5/8,2/3) -> (-1/3,-2/7) Matrix(41,-30,56,-41) -> Matrix(19,5,-72,-19) (5/7,3/4) -> (-5/18,-1/4) Matrix(7,-6,8,-7) -> Matrix(5,1,-24,-5) (3/4,1/1) -> (-1/4,-1/6) Matrix(-1,2,0,1) -> Matrix(-1,0,12,1) (1/1,1/0) -> (-1/6,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.