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 -1/1 -7/2 -4/5 -3/1 -2/3 -8/3 -1/2 -5/2 0/1 -7/3 -2/3 -2/1 -1/1 -1/2 0/1 -9/5 -2/3 -16/9 -1/2 -7/4 -1/2 0/1 -5/3 0/1 -8/5 -1/2 -3/2 0/1 -4/3 -1/1 -5/4 -1/1 -1/2 -1/1 0/1 -5/6 -2/1 -4/5 -1/1 -3/4 -1/1 -1/2 -8/11 -1/2 -5/7 0/1 -7/10 -2/3 -2/3 -1/1 -1/2 0/1 -9/14 -2/3 -16/25 -1/2 -7/11 0/1 -5/8 -1/1 -1/2 -8/13 -1/2 -3/5 0/1 -7/12 -1/2 0/1 -4/7 -1/1 -1/3 -1/2 0/1 0/1 -1/2 1/0 1/2 0/1 4/7 -1/1 1/1 3/5 0/1 5/8 -1/1 1/0 2/3 -1/1 0/1 1/0 5/7 0/1 3/4 -1/1 1/0 4/5 -1/1 1/1 0/1 6/5 -1/1 0/1 1/0 5/4 -1/1 1/0 4/3 -1/1 3/2 0/1 8/5 1/0 13/8 -1/1 1/0 5/3 0/1 12/7 -1/1 1/1 7/4 0/1 1/0 2/1 -1/1 0/1 1/0 9/4 0/1 1/0 16/7 1/0 7/3 -2/1 12/5 -1/1 5/2 0/1 8/3 1/0 11/4 -3/1 1/0 3/1 -2/1 10/3 -2/1 -3/2 -1/1 17/5 -2/1 24/7 -3/2 7/2 -4/3 4/1 -1/1 1/0 -1/1 0/1 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(1,0,0,1) Matrix(15,56,4,15) -> Matrix(9,8,-8,-7) Matrix(17,56,-24,-79) -> Matrix(3,2,-2,-1) Matrix(17,48,-28,-79) -> Matrix(3,2,-8,-5) Matrix(31,80,12,31) -> Matrix(1,0,2,1) Matrix(17,40,-20,-47) -> Matrix(3,2,-2,-1) Matrix(15,32,-8,-17) -> Matrix(1,0,0,1) Matrix(143,256,-224,-401) -> Matrix(3,2,-8,-5) Matrix(145,256,64,113) -> Matrix(1,0,2,1) Matrix(33,56,-56,-95) -> Matrix(1,0,0,1) Matrix(49,80,-68,-111) -> Matrix(1,0,0,1) Matrix(31,48,20,31) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,0,1) Matrix(31,40,24,31) -> Matrix(3,2,-2,-1) Matrix(33,40,-52,-63) -> Matrix(1,0,0,1) Matrix(97,80,40,33) -> Matrix(1,2,-2,-3) Matrix(31,24,40,31) -> Matrix(3,2,-2,-1) Matrix(175,128,108,79) -> Matrix(3,2,-2,-1) Matrix(47,32,-72,-49) -> Matrix(1,0,0,1) Matrix(511,328,148,95) -> Matrix(17,10,-12,-7) Matrix(207,128,76,47) -> Matrix(7,4,-2,-1) Matrix(193,112,112,65) -> Matrix(1,0,2,1) Matrix(15,8,28,15) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(95,-56,56,-33) -> Matrix(1,0,0,1) Matrix(79,-48,28,-17) -> Matrix(1,-2,0,1) Matrix(63,-40,52,-33) -> Matrix(1,0,0,1) Matrix(79,-56,24,-17) -> Matrix(3,2,-2,-1) Matrix(111,-80,68,-49) -> Matrix(1,0,0,1) Matrix(47,-40,20,-17) -> Matrix(3,2,-2,-1) Matrix(95,-112,28,-33) -> Matrix(3,2,-2,-1) Matrix(17,-32,8,-15) -> Matrix(1,0,0,1) Matrix(191,-440,56,-129) -> Matrix(3,8,-2,-5) 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 2/3 3/4 1/1 4/3 3/2 2/1 5/2 8/3 7/2 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 -7/2 -4/5 -3/1 -2/3 -8/3 -1/2 -5/2 0/1 -2/1 -1/1 -1/2 0/1 -3/2 0/1 -4/3 -1/1 -5/4 -1/1 -1/2 -1/1 0/1 -4/5 -1/1 -3/4 -1/1 -1/2 -5/7 0/1 -7/10 -2/3 -2/3 -1/1 -1/2 0/1 -1/2 0/1 0/1 -1/2 1/0 1/2 0/1 2/3 -1/1 0/1 1/0 5/7 0/1 3/4 -1/1 1/0 4/5 -1/1 1/1 0/1 5/4 -1/1 1/0 4/3 -1/1 3/2 0/1 2/1 -1/1 0/1 1/0 5/2 0/1 8/3 1/0 3/1 -2/1 10/3 -2/1 -3/2 -1/1 7/2 -4/3 4/1 -1/1 1/0 -1/1 0/1 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(7,20,8,23) (-3/1,-8/3) -> (4/5,1/1) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) 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(23,28,32,39) (-5/4,-1/1) -> (5/7,3/4) Hyperbolic Matrix(23,20,8,7) (-1/1,-4/5) -> (8/3,3/1) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(39,28,32,23) (-3/4,-5/7) -> (1/1,5/4) Hyperbolic Matrix(121,84,36,25) (-7/10,-2/3) -> (10/3,7/2) Hyperbolic Matrix(7,4,12,7) (-2/3,-1/2) -> (1/2,2/3) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(79,-56,24,-17) (2/3,5/7) -> (3/1,10/3) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(15,56,4,15) -> Matrix(9,8,-8,-7) Matrix(17,56,-24,-79) -> Matrix(3,2,-2,-1) Matrix(7,20,8,23) -> Matrix(3,2,-5,-3) Matrix(31,80,12,31) -> Matrix(1,0,2,1) Matrix(9,20,4,9) -> Matrix(1,0,1,1) Matrix(7,12,4,7) -> Matrix(1,0,1,1) Matrix(17,24,12,17) -> Matrix(1,0,0,1) Matrix(31,40,24,31) -> Matrix(3,2,-2,-1) Matrix(23,28,32,39) -> Matrix(1,0,1,1) Matrix(23,20,8,7) -> Matrix(1,2,-1,-1) Matrix(31,24,40,31) -> Matrix(3,2,-2,-1) Matrix(39,28,32,23) -> Matrix(1,0,1,1) Matrix(121,84,36,25) -> Matrix(5,2,-3,-1) Matrix(7,4,12,7) -> Matrix(1,0,1,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(79,-56,24,-17) -> Matrix(3,2,-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 (-1/1,0/1) 0 2 1/2 0/1 2 8 2/3 (-2/1,0/1) 0 4 5/7 0/1 1 8 3/4 (-1/1,1/0) 0 8 4/5 -1/1 3 2 1/1 0/1 1 8 5/4 (-1/1,1/0) 0 8 4/3 -1/1 1 2 3/2 0/1 2 8 2/1 (-2/1,0/1) 0 4 5/2 0/1 2 8 8/3 1/0 3 2 3/1 -2/1 1 8 10/3 (-2/1,-4/3) 0 4 7/2 -4/3 2 8 4/1 -1/1 4 2 1/0 (-1/1,0/1) 0 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(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(79,-56,24,-17) (2/3,5/7) -> (3/1,10/3) Hyperbolic Matrix(39,-28,32,-23) (5/7,3/4) -> (1/1,5/4) Glide Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(23,-20,8,-7) (4/5,1/1) -> (8/3,3/1) 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(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) 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(1,0,4,-1) -> Matrix(-1,0,2,1) (0/1,1/2) -> (-1/1,0/1) Matrix(7,-4,12,-7) -> Matrix(-1,0,1,1) (1/2,2/3) -> (-2/1,0/1) Matrix(79,-56,24,-17) -> Matrix(3,2,-2,-1) -1/1 Matrix(39,-28,32,-23) -> Matrix(-1,0,1,1) *** -> (-2/1,0/1) Matrix(31,-24,40,-31) -> Matrix(1,2,0,-1) (3/4,4/5) -> (-1/1,1/0) Matrix(23,-20,8,-7) -> Matrix(3,2,-1,-1) Matrix(31,-40,24,-31) -> Matrix(1,2,0,-1) (5/4,4/3) -> (-1/1,1/0) Matrix(17,-24,12,-17) -> Matrix(-1,0,2,1) (4/3,3/2) -> (-1/1,0/1) Matrix(7,-12,4,-7) -> Matrix(-1,0,1,1) (3/2,2/1) -> (-2/1,0/1) Matrix(9,-20,4,-9) -> Matrix(-1,0,1,1) (2/1,5/2) -> (-2/1,0/1) Matrix(31,-80,12,-31) -> Matrix(1,0,0,-1) (5/2,8/3) -> (0/1,1/0) Matrix(41,-140,12,-41) -> Matrix(5,8,-3,-5) (10/3,7/2) -> (-2/1,-4/3) Matrix(15,-56,4,-15) -> Matrix(7,8,-6,-7) (7/2,4/1) -> (-4/3,-1/1) Matrix(-1,8,0,1) -> Matrix(-1,0,2,1) (4/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.