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 -5/3 -3/2 -1/1 -2/3 -1/2 -1/3 0/1 1/4 1/3 3/8 1/2 2/3 3/4 1/1 4/3 3/2 5/3 2/1 8/3 3/1 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 -1/2 1/0 -3/1 -2/1 0/1 -8/3 0/1 -5/2 1/0 -7/3 0/1 -2/1 1/0 -7/4 -3/2 1/0 -5/3 -1/1 -8/5 0/1 -3/2 1/0 -7/5 -2/1 -4/3 -1/1 -1/1 -1/1 1/1 -4/5 -1/1 -3/4 -1/2 1/0 -8/11 0/1 -5/7 -2/1 0/1 -2/3 -1/2 1/0 -5/8 1/0 -8/13 -2/1 -3/5 -1/1 -7/12 -1/2 1/0 -4/7 -1/1 -1/2 -1/2 1/0 -3/7 -2/3 0/1 -2/5 -1/2 1/0 -3/8 -1/2 -4/11 -1/3 -1/3 0/1 -2/7 1/0 -1/4 -1/2 1/0 0/1 0/1 1/4 1/2 1/0 2/7 1/0 1/3 0/1 3/8 1/2 2/5 1/2 1/0 3/7 0/1 2/3 1/2 1/2 1/0 4/7 1/1 3/5 1/1 5/8 1/0 2/3 1/2 1/0 5/7 0/1 2/1 3/4 1/2 1/0 1/1 -1/1 1/1 5/4 1/2 1/0 4/3 1/1 11/8 3/2 7/5 2/1 3/2 1/0 8/5 0/1 13/8 1/2 5/3 1/1 12/7 1/1 7/4 3/2 1/0 2/1 1/0 7/3 0/1 5/2 1/0 8/3 0/1 11/4 1/2 1/0 3/1 0/1 2/1 7/2 1/2 1/0 4/1 1/1 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(9,32,16,57) (-4/1,-7/2) -> (1/2,4/7) Hyperbolic Matrix(7,24,16,55) (-7/2,-3/1) -> (3/7,1/2) Hyperbolic Matrix(23,64,-32,-89) (-3/1,-8/3) -> (-8/11,-5/7) Hyperbolic Matrix(25,64,16,41) (-8/3,-5/2) -> (3/2,8/5) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(7,16,24,55) (-7/3,-2/1) -> (2/7,1/3) Hyperbolic Matrix(9,16,32,57) (-2/1,-7/4) -> (1/4,2/7) Hyperbolic Matrix(33,56,-56,-95) (-7/4,-5/3) -> (-3/5,-7/12) Hyperbolic Matrix(39,64,-64,-105) (-5/3,-8/5) -> (-8/13,-3/5) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(23,32,-64,-89) (-7/5,-4/3) -> (-4/11,-1/3) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3) Hyperbolic Matrix(153,112,56,41) (-3/4,-8/11) -> (8/3,11/4) Hyperbolic Matrix(23,16,56,39) (-5/7,-2/3) -> (2/5,3/7) Hyperbolic Matrix(25,16,64,41) (-2/3,-5/8) -> (3/8,2/5) Hyperbolic Matrix(233,144,144,89) (-5/8,-8/13) -> (8/5,13/8) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(57,32,16,9) (-4/7,-1/2) -> (7/2,4/1) Hyperbolic Matrix(55,24,16,7) (-1/2,-3/7) -> (3/1,7/2) Hyperbolic Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(41,16,64,25) (-2/5,-3/8) -> (5/8,2/3) Hyperbolic Matrix(153,56,112,41) (-3/8,-4/11) -> (4/3,11/8) Hyperbolic Matrix(55,16,24,7) (-1/3,-2/7) -> (2/1,7/3) Hyperbolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(89,-32,64,-23) (1/3,3/8) -> (11/8,7/5) Hyperbolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic Matrix(89,-64,32,-23) (5/7,3/4) -> (11/4,3/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,2,0,1) Matrix(9,32,16,57) -> Matrix(1,0,2,1) Matrix(7,24,16,55) -> Matrix(1,0,2,1) Matrix(23,64,-32,-89) -> Matrix(1,0,0,1) Matrix(25,64,16,41) -> Matrix(1,0,0,1) Matrix(23,56,16,39) -> Matrix(1,2,0,1) Matrix(7,16,24,55) -> Matrix(1,0,0,1) Matrix(9,16,32,57) -> Matrix(1,2,0,1) Matrix(33,56,-56,-95) -> Matrix(1,2,-2,-3) Matrix(39,64,-64,-105) -> Matrix(3,2,-2,-1) Matrix(41,64,16,25) -> Matrix(1,0,0,1) Matrix(39,56,16,23) -> Matrix(1,2,0,1) Matrix(23,32,-64,-89) -> Matrix(1,2,-4,-7) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(41,32,32,25) -> Matrix(1,0,2,1) Matrix(153,112,56,41) -> Matrix(1,0,2,1) Matrix(23,16,56,39) -> Matrix(1,0,2,1) Matrix(25,16,64,41) -> Matrix(1,0,2,1) Matrix(233,144,144,89) -> Matrix(1,2,2,5) Matrix(193,112,112,65) -> Matrix(1,2,0,1) Matrix(57,32,16,9) -> Matrix(1,0,2,1) Matrix(55,24,16,7) -> Matrix(1,0,2,1) Matrix(39,16,56,23) -> Matrix(1,0,2,1) Matrix(41,16,64,25) -> Matrix(1,0,2,1) Matrix(153,56,112,41) -> Matrix(11,4,8,3) Matrix(55,16,24,7) -> Matrix(1,0,0,1) Matrix(57,16,32,9) -> Matrix(1,2,0,1) Matrix(1,0,8,1) -> Matrix(1,0,2,1) Matrix(89,-32,64,-23) -> Matrix(7,-2,4,-1) Matrix(95,-56,56,-33) -> Matrix(3,-2,2,-1) Matrix(105,-64,64,-39) -> Matrix(1,-2,2,-3) Matrix(89,-64,32,-23) -> Matrix(1,0,0,1) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) 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: ((1,7,8,2)(3,13,28,14)(4,18,19,5)(6,23,9,24)(10,27,26,11)(12,22,21,29)(15,25,30,16)(17,32,20,31); (1,5,22,6)(2,11,12,3)(4,16,23,17)(7,27,21,28)(8,18,29,9)(10,30,13,31)(14,32,26,15)(19,25,24,20); (1,3,15,4)(2,9,16,10)(5,20,14,21)(6,25,26,7)(8,28,30,19)(11,32,23,22)(12,18,17,13)(24,29,27,31)) ----------------------------------------------------------------------- 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: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/2 -1/1 -1/2 -3/8 -1/3 0/1 1/4 1/3 1/2 2/3 3/4 1/1 4/3 3/2 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 -1/1 -3/2 1/0 -7/5 -2/1 -4/3 -1/1 -1/1 -1/1 1/1 -4/5 -1/1 -3/4 -1/2 1/0 -2/3 -1/2 1/0 -1/2 -1/2 1/0 -2/5 -1/2 1/0 -3/8 -1/2 -4/11 -1/3 -1/3 0/1 -2/7 1/0 -1/4 -1/2 1/0 0/1 0/1 1/4 1/2 1/0 1/3 0/1 1/2 1/2 1/0 3/5 1/1 5/8 1/0 2/3 1/2 1/0 3/4 1/2 1/0 1/1 -1/1 1/1 5/4 1/2 1/0 4/3 1/1 3/2 1/0 8/5 0/1 13/8 1/2 5/3 1/1 7/4 3/2 1/0 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(7,12,-24,-41) (-2/1,-5/3) -> (-1/3,-2/7) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(23,32,-64,-89) (-7/5,-4/3) -> (-4/11,-1/3) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(7,4,-16,-9) (-2/3,-1/2) -> (-1/2,-2/5) Parabolic Matrix(41,16,64,25) (-2/5,-3/8) -> (5/8,2/3) Hyperbolic Matrix(207,76,128,47) (-3/8,-4/11) -> (8/5,13/8) Hyperbolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,1,0,1) Matrix(7,12,-24,-41) -> Matrix(1,1,0,1) Matrix(23,36,-16,-25) -> Matrix(1,-1,0,1) Matrix(23,32,-64,-89) -> Matrix(1,2,-4,-7) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(41,32,32,25) -> Matrix(1,0,2,1) Matrix(17,12,24,17) -> Matrix(1,1,0,1) Matrix(7,4,-16,-9) -> Matrix(1,1,-2,-1) Matrix(41,16,64,25) -> Matrix(1,0,2,1) Matrix(207,76,128,47) -> Matrix(3,1,8,3) Matrix(57,16,32,9) -> Matrix(1,2,0,1) Matrix(1,0,8,1) -> Matrix(1,0,2,1) Matrix(41,-12,24,-7) -> Matrix(1,1,0,1) Matrix(9,-4,16,-7) -> Matrix(1,-1,2,-1) Matrix(105,-64,64,-39) -> Matrix(1,-2,2,-3) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,-1,0,1) 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 0/1 1 4 1/4 (0/1,1/1).(1/2,1/0) 0 4 1/3 0/1 2 4 1/2 (0/1,1/1).(1/2,1/0) 0 4 3/5 1/1 2 4 5/8 1/0 3 4 2/3 (1/2,1/0) 0 4 3/4 (0/1,1/1).(1/2,1/0) 0 4 1/1 0 4 5/4 (0/1,1/1).(1/2,1/0) 0 4 4/3 1/1 3 4 3/2 1/0 1 4 8/5 0/1 3 4 13/8 1/2 3 4 5/3 1/1 2 4 7/4 (1/1,2/1).(3/2,1/0) 0 4 2/1 1/0 1 4 1/0 1/0 1 4 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(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic 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(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) 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(1,0,8,-1) -> Matrix(1,0,2,-1) (0/1,1/4) -> (0/1,1/1) Matrix(41,-12,24,-7) -> Matrix(1,1,0,1) 1/0 Matrix(9,-4,16,-7) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(105,-64,64,-39) -> Matrix(1,-2,2,-3) 1/1 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(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(31,-40,24,-31) -> Matrix(1,0,2,-1) (5/4,4/3) -> (0/1,1/1) Matrix(25,-36,16,-23) -> Matrix(1,-1,0,1) 1/0 Matrix(129,-208,80,-129) -> Matrix(1,0,4,-1) (8/5,13/8) -> (0/1,1/2) 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 has no extra symmetries.