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 -3/1 -2/1 -1/1 -2/3 0/1 1/2 4/7 2/3 3/4 4/5 1/1 5/4 4/3 3/2 8/5 7/4 2/1 16/7 5/2 8/3 3/1 7/2 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -7/2 -2/1 1/0 -3/1 -1/1 1/0 -8/3 -1/1 -5/2 -1/1 0/1 -7/3 0/1 1/0 -2/1 1/0 -9/5 -2/1 1/0 -16/9 -2/1 -7/4 -3/2 -5/3 -1/1 1/0 -8/5 -1/1 -3/2 -1/1 0/1 -4/3 0/1 -5/4 1/0 -1/1 0/1 1/0 -5/6 -1/1 0/1 -4/5 0/1 -3/4 1/2 -8/11 1/1 -5/7 1/1 1/0 -7/10 0/1 1/0 -2/3 1/2 1/0 -9/14 0/1 1/0 -16/25 0/1 -7/11 0/1 1/2 -5/8 1/2 -8/13 1/1 -3/5 1/2 1/1 -7/12 3/4 -4/7 1/1 -1/2 1/1 1/0 0/1 1/0 1/2 -1/1 1/0 4/7 -1/1 3/5 -1/1 -1/2 5/8 -1/2 2/3 -1/2 1/0 5/7 -1/1 1/0 3/4 -1/2 4/5 0/1 1/1 0/1 1/0 6/5 1/2 1/0 5/4 1/0 4/3 0/1 3/2 0/1 1/1 8/5 1/1 13/8 3/2 5/3 1/1 1/0 12/7 1/1 7/4 3/2 2/1 1/0 9/4 -1/2 16/7 0/1 7/3 0/1 1/0 12/5 0/1 5/2 0/1 1/1 8/3 1/1 11/4 1/0 3/1 1/1 1/0 10/3 3/2 1/0 17/5 3/2 2/1 24/7 2/1 7/2 2/1 1/0 4/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(17,56,-24,-79) (-7/2,-3/1) -> (-5/7,-7/10) Hyperbolic Matrix(17,48,-28,-79) (-3/1,-8/3) -> (-8/13,-3/5) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(17,40,-20,-47) (-5/2,-7/3) -> (-1/1,-5/6) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,-224,-401) (-9/5,-16/9) -> (-16/25,-7/11) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(33,56,-56,-95) (-7/4,-5/3) -> (-3/5,-7/12) Hyperbolic Matrix(49,80,-68,-111) (-5/3,-8/5) -> (-8/11,-5/7) 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(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(33,40,-52,-63) (-5/4,-1/1) -> (-7/11,-5/8) Hyperbolic Matrix(97,80,40,33) (-5/6,-4/5) -> (12/5,5/2) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(175,128,108,79) (-3/4,-8/11) -> (8/5,13/8) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(511,328,148,95) (-9/14,-16/25) -> (24/7,7/2) Hyperbolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(63,-40,52,-33) (5/8,2/3) -> (6/5,5/4) Hyperbolic Matrix(79,-56,24,-17) (2/3,5/7) -> (3/1,10/3) Hyperbolic Matrix(111,-80,68,-49) (5/7,3/4) -> (13/8,5/3) Hyperbolic Matrix(47,-40,20,-17) (4/5,1/1) -> (7/3,12/5) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(191,-440,56,-129) (16/7,7/3) -> (17/5,24/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,2,0,1) Matrix(15,56,4,15) -> Matrix(1,4,0,1) Matrix(17,56,-24,-79) -> Matrix(1,2,0,1) Matrix(17,48,-28,-79) -> Matrix(1,0,2,1) Matrix(31,80,12,31) -> Matrix(1,0,2,1) Matrix(17,40,-20,-47) -> Matrix(1,0,0,1) Matrix(15,32,-8,-17) -> Matrix(1,-2,0,1) Matrix(143,256,-224,-401) -> Matrix(1,2,2,5) Matrix(145,256,64,113) -> Matrix(1,2,-4,-7) Matrix(33,56,-56,-95) -> Matrix(1,0,2,1) Matrix(49,80,-68,-111) -> Matrix(1,2,0,1) Matrix(31,48,20,31) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(31,40,24,31) -> Matrix(1,0,0,1) Matrix(33,40,-52,-63) -> Matrix(1,0,2,1) Matrix(97,80,40,33) -> Matrix(1,0,2,1) Matrix(31,24,40,31) -> Matrix(1,0,-4,1) Matrix(175,128,108,79) -> Matrix(5,-4,4,-3) Matrix(47,32,-72,-49) -> Matrix(1,0,0,1) Matrix(511,328,148,95) -> Matrix(1,2,0,1) Matrix(207,128,76,47) -> Matrix(3,-2,2,-1) Matrix(193,112,112,65) -> Matrix(7,-6,6,-5) Matrix(15,8,28,15) -> Matrix(1,-2,0,1) Matrix(1,0,4,1) -> Matrix(1,-2,0,1) Matrix(95,-56,56,-33) -> Matrix(1,0,2,1) Matrix(79,-48,28,-17) -> Matrix(1,0,2,1) Matrix(63,-40,52,-33) -> Matrix(1,0,2,1) Matrix(79,-56,24,-17) -> Matrix(1,2,0,1) Matrix(111,-80,68,-49) -> Matrix(1,2,0,1) Matrix(47,-40,20,-17) -> Matrix(1,0,0,1) Matrix(95,-112,28,-33) -> Matrix(3,-2,2,-1) Matrix(17,-32,8,-15) -> Matrix(1,-2,0,1) Matrix(191,-440,56,-129) -> Matrix(3,-2,2,-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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 Degree of the the map X: 8 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,10,29,11)(4,15,16,5)(6,20,30,21)(7,25,26,8)(9,18)(12,13)(14,24)(17,27)(19,28)(22,23)(31,32); (1,5,18,30,31,15,19,6)(2,8,28,29,32,25,9,3)(4,13,20,17,16,22,21,14)(7,23,10,27,26,12,11,24); (1,3,12,4)(2,6,22,7)(5,17,10,9)(8,27,20,19)(11,28,15,14)(13,26,32,30)(16,31,29,23)(18,25,24,21)) ----------------------------------------------------------------------- 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 -1/1 0/1 1/2 4/7 2/3 3/4 4/5 1/1 5/4 4/3 3/2 7/4 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -7/4 -3/2 -5/3 -1/1 1/0 -8/5 -1/1 -3/2 -1/1 0/1 -4/3 0/1 -5/4 1/0 -1/1 0/1 1/0 -4/5 0/1 -3/4 1/2 -2/3 1/2 1/0 -5/8 1/2 -3/5 1/2 1/1 -7/12 3/4 -4/7 1/1 -1/2 1/1 1/0 0/1 1/0 1/2 -1/1 1/0 4/7 -1/1 3/5 -1/1 -1/2 5/8 -1/2 2/3 -1/2 1/0 3/4 -1/2 4/5 0/1 1/1 0/1 1/0 5/4 1/0 4/3 0/1 3/2 0/1 1/1 8/5 1/1 5/3 1/1 1/0 12/7 1/1 7/4 3/2 2/1 1/0 1/0 1/0 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(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(33,56,-56,-95) (-7/4,-5/3) -> (-3/5,-7/12) Hyperbolic Matrix(17,28,20,33) (-5/3,-8/5) -> (4/5,1/1) 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(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(17,20,28,33) (-5/4,-1/1) -> (3/5,5/8) Hyperbolic Matrix(33,28,20,17) (-1/1,-4/5) -> (8/5,5/3) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(33,20,28,17) (-5/8,-3/5) -> (1/1,5/4) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,1,0,1) Matrix(15,28,8,15) -> Matrix(1,3,0,1) Matrix(33,56,-56,-95) -> Matrix(1,0,2,1) Matrix(17,28,20,33) -> Matrix(1,1,0,1) Matrix(31,48,20,31) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(31,40,24,31) -> Matrix(1,0,0,1) Matrix(17,20,28,33) -> Matrix(1,1,-2,-1) Matrix(33,28,20,17) -> Matrix(1,1,0,1) Matrix(31,24,40,31) -> Matrix(1,0,-4,1) Matrix(17,12,24,17) -> Matrix(1,-1,0,1) Matrix(31,20,48,31) -> Matrix(1,-1,0,1) Matrix(33,20,28,17) -> Matrix(1,-1,2,-1) Matrix(193,112,112,65) -> Matrix(7,-6,6,-5) Matrix(15,8,28,15) -> Matrix(1,-2,0,1) Matrix(1,0,4,1) -> Matrix(1,-2,0,1) Matrix(95,-56,56,-33) -> Matrix(1,0,2,1) 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): 8 ----------------------------------------------------------------------- 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 1/0 1 2 1/2 (-1/1,1/0) 0 8 4/7 -1/1 4 2 3/5 0 8 5/8 -1/2 2 8 2/3 (-1/2,1/0) 0 4 3/4 -1/2 2 8 4/5 0/1 3 2 1/1 0 8 5/4 1/0 2 8 4/3 0/1 1 2 3/2 (0/1,1/1) 0 8 8/5 1/1 3 2 5/3 0 8 12/7 1/1 4 2 7/4 3/2 2 8 2/1 1/0 1 4 1/0 1/0 2 8 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(15,-8,28,-15) (1/2,4/7) -> (1/2,4/7) Reflection Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(33,-20,28,-17) (3/5,5/8) -> (1/1,5/4) Glide Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(33,-28,20,-17) (4/5,1/1) -> (8/5,5/3) Glide Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) 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(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,4,-1) -> Matrix(1,2,0,-1) (0/1,1/2) -> (-1/1,1/0) Matrix(15,-8,28,-15) -> Matrix(1,2,0,-1) (1/2,4/7) -> (-1/1,1/0) Matrix(95,-56,56,-33) -> Matrix(1,0,2,1) 0/1 Matrix(33,-20,28,-17) -> Matrix(1,1,2,1) Matrix(31,-20,48,-31) -> Matrix(1,1,0,-1) (5/8,2/3) -> (-1/2,1/0) Matrix(17,-12,24,-17) -> Matrix(1,1,0,-1) (2/3,3/4) -> (-1/2,1/0) Matrix(31,-24,40,-31) -> Matrix(-1,0,4,1) (3/4,4/5) -> (-1/2,0/1) Matrix(33,-28,20,-17) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(31,-40,24,-31) -> Matrix(1,0,0,-1) (5/4,4/3) -> (0/1,1/0) Matrix(17,-24,12,-17) -> Matrix(1,0,2,-1) (4/3,3/2) -> (0/1,1/1) Matrix(31,-48,20,-31) -> Matrix(1,0,2,-1) (3/2,8/5) -> (0/1,1/1) Matrix(97,-168,56,-97) -> Matrix(5,-6,4,-5) (12/7,7/4) -> (1/1,3/2) Matrix(15,-28,8,-15) -> Matrix(-1,3,0,1) (7/4,2/1) -> (3/2,1/0) Matrix(-1,4,0,1) -> Matrix(-1,1,0,1) (2/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.