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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/14 -3/14 -1/6 -1/7 0/1 1/5 1/4 2/7 1/3 2/5 3/7 1/2 4/7 2/3 5/7 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/2 -4/5 1/3 1/2 -3/4 1/1 -5/7 0/1 -2/3 0/1 1/2 -5/8 1/1 -3/5 1/2 1/1 -4/7 0/1 2/3 -1/2 1/1 -3/7 0/1 2/1 -2/5 1/1 1/0 -5/13 0/1 1/0 -8/21 0/1 -3/8 1/1 -4/11 2/1 1/0 -5/14 1/0 -1/3 0/1 1/0 -2/7 0/1 -1/4 1/1 -2/9 3/1 1/0 -3/14 1/0 -1/5 -1/1 1/0 -1/6 -1/1 -1/7 0/1 0/1 0/1 1/0 1/5 1/1 1/0 1/4 -1/1 2/7 0/1 1/3 0/1 1/0 3/8 -1/1 2/5 -1/1 1/0 3/7 -2/1 0/1 1/2 -1/1 4/7 -2/3 0/1 3/5 -1/1 -1/2 8/13 -1/2 0/1 13/21 0/1 5/8 -1/1 7/11 -2/3 -1/2 9/14 -1/2 2/3 -1/2 0/1 5/7 0/1 3/4 -1/1 7/9 -3/5 -1/2 11/14 -1/2 4/5 -1/2 -1/3 5/6 -1/3 6/7 0/1 1/1 -1/2 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(27,22,-70,-57) (-1/1,-4/5) -> (-2/5,-5/13) Hyperbolic Matrix(13,10,-56,-43) (-4/5,-3/4) -> (-1/4,-2/9) Hyperbolic Matrix(41,30,56,41) (-3/4,-5/7) -> (5/7,3/4) Hyperbolic Matrix(29,20,42,29) (-5/7,-2/3) -> (2/3,5/7) Hyperbolic Matrix(41,26,-112,-71) (-2/3,-5/8) -> (-3/8,-4/11) Hyperbolic Matrix(13,8,-70,-43) (-5/8,-3/5) -> (-1/5,-1/6) Hyperbolic Matrix(41,24,70,41) (-3/5,-4/7) -> (4/7,3/5) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(13,6,28,13) (-1/2,-3/7) -> (3/7,1/2) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(99,38,112,43) (-5/13,-8/21) -> (6/7,1/1) Hyperbolic Matrix(153,58,182,69) (-8/21,-3/8) -> (5/6,6/7) Hyperbolic Matrix(127,46,196,71) (-4/11,-5/14) -> (9/14,2/3) Hyperbolic Matrix(125,44,196,69) (-5/14,-1/3) -> (7/11,9/14) Hyperbolic Matrix(13,4,42,13) (-1/3,-2/7) -> (2/7,1/3) Hyperbolic Matrix(15,4,56,15) (-2/7,-1/4) -> (1/4,2/7) Hyperbolic Matrix(155,34,196,43) (-2/9,-3/14) -> (11/14,4/5) Hyperbolic Matrix(153,32,196,41) (-3/14,-1/5) -> (7/9,11/14) Hyperbolic Matrix(113,18,182,29) (-1/6,-1/7) -> (13/21,5/8) Hyperbolic Matrix(69,8,112,13) (-1/7,0/1) -> (8/13,13/21) Hyperbolic Matrix(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(43,-10,56,-13) (1/5,1/4) -> (3/4,7/9) Hyperbolic Matrix(71,-26,112,-41) (1/3,3/8) -> (5/8,7/11) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(27,22,-70,-57) -> Matrix(1,0,-2,1) Matrix(13,10,-56,-43) -> Matrix(3,-2,2,-1) Matrix(41,30,56,41) -> Matrix(1,0,-2,1) Matrix(29,20,42,29) -> Matrix(1,0,-4,1) Matrix(41,26,-112,-71) -> Matrix(3,-2,2,-1) Matrix(13,8,-70,-43) -> Matrix(1,0,-2,1) Matrix(41,24,70,41) -> Matrix(3,-2,-4,3) Matrix(15,8,28,15) -> Matrix(3,-2,-4,3) Matrix(13,6,28,13) -> Matrix(1,-2,0,1) Matrix(29,12,70,29) -> Matrix(1,-2,0,1) Matrix(99,38,112,43) -> Matrix(1,0,-2,1) Matrix(153,58,182,69) -> Matrix(1,0,-4,1) Matrix(127,46,196,71) -> Matrix(1,-2,-2,5) Matrix(125,44,196,69) -> Matrix(1,2,-2,-3) Matrix(13,4,42,13) -> Matrix(1,0,0,1) Matrix(15,4,56,15) -> Matrix(1,0,-2,1) Matrix(155,34,196,43) -> Matrix(1,-4,-2,9) Matrix(153,32,196,41) -> Matrix(1,4,-2,-7) Matrix(113,18,182,29) -> Matrix(1,0,0,1) Matrix(69,8,112,13) -> Matrix(1,0,-2,1) Matrix(43,-8,70,-13) -> Matrix(1,0,-2,1) Matrix(43,-10,56,-13) -> Matrix(1,2,-2,-3) Matrix(71,-26,112,-41) -> Matrix(1,2,-2,-3) Matrix(57,-22,70,-27) -> Matrix(1,0,-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): 6 Degree of the the map X: 6 Degree of the the map Y: 24 Permutation triple for Y: ((1,4,16,24,17,5,2)(3,10,8,7,19,23,11)(6,20,18,22,9,14,13); (1,2,8,20,21,9,3)(4,14,23,19,6,5,15)(7,12,11,16,22,18,17); (2,6,13,4,3,12,7)(5,18,8,10,9,16,15)(11,14,21,20,19,17,24)) ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/5 1/4 2/7 1/3 3/7 1/2 9/14 11/14 5/6 6/7 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/0 1/5 1/1 1/0 1/4 -1/1 2/7 0/1 1/3 0/1 1/0 3/8 -1/1 2/5 -1/1 1/0 3/7 -2/1 0/1 1/2 -1/1 4/7 -2/3 0/1 3/5 -1/1 -1/2 8/13 -1/2 0/1 13/21 0/1 5/8 -1/1 7/11 -2/3 -1/2 9/14 -1/2 2/3 -1/2 0/1 5/7 0/1 3/4 -1/1 7/9 -3/5 -1/2 11/14 -1/2 4/5 -1/2 -1/3 5/6 -1/3 6/7 0/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(43,-10,56,-13) (1/5,1/4) -> (3/4,7/9) Hyperbolic Matrix(41,-11,56,-15) (1/4,2/7) -> (5/7,3/4) Hyperbolic Matrix(29,-9,42,-13) (2/7,1/3) -> (2/3,5/7) Hyperbolic Matrix(71,-26,112,-41) (1/3,3/8) -> (5/8,7/11) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic Matrix(41,-17,70,-29) (2/5,3/7) -> (4/7,3/5) Hyperbolic Matrix(15,-7,28,-13) (3/7,1/2) -> (1/2,4/7) Parabolic Matrix(99,-61,112,-69) (8/13,13/21) -> (6/7,1/1) Hyperbolic Matrix(153,-95,182,-113) (13/21,5/8) -> (5/6,6/7) Hyperbolic Matrix(127,-81,196,-125) (7/11,9/14) -> (9/14,2/3) Parabolic Matrix(155,-121,196,-153) (7/9,11/14) -> (11/14,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-2,1) Matrix(43,-8,70,-13) -> Matrix(1,0,-2,1) Matrix(43,-10,56,-13) -> Matrix(1,2,-2,-3) Matrix(41,-11,56,-15) -> Matrix(1,0,0,1) Matrix(29,-9,42,-13) -> Matrix(1,0,-2,1) Matrix(71,-26,112,-41) -> Matrix(1,2,-2,-3) Matrix(57,-22,70,-27) -> Matrix(1,0,-2,1) Matrix(41,-17,70,-29) -> Matrix(1,2,-2,-3) Matrix(15,-7,28,-13) -> Matrix(1,2,-2,-3) Matrix(99,-61,112,-69) -> Matrix(1,0,0,1) Matrix(153,-95,182,-113) -> Matrix(1,0,-2,1) Matrix(127,-81,196,-125) -> Matrix(3,2,-8,-5) Matrix(155,-121,196,-153) -> Matrix(7,4,-16,-9) 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): 3 ----------------------------------------------------------------------- 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 (0/1,1/0) 0 14 1/5 (1/1,1/0) 0 14 3/14 1/0 4 1 1/4 -1/1 2 7 2/7 0/1 2 2 1/3 (0/1,1/0) 0 14 5/14 1/0 2 1 3/8 -1/1 2 7 8/21 0/1 2 2 5/13 (0/1,1/0) 0 14 2/5 (-1/1,1/0) 0 14 3/7 (-1/1,1/0) 0 2 1/2 -1/1 2 7 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(27,-5,70,-13) (0/1,1/5) -> (5/13,2/5) Glide Reflection Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(13,-3,56,-13) (3/14,1/4) -> (3/14,1/4) Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(13,-4,42,-13) (2/7,1/3) -> (2/7,1/3) Reflection Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(41,-15,112,-41) (5/14,3/8) -> (5/14,3/8) Reflection Matrix(127,-48,336,-127) (3/8,8/21) -> (3/8,8/21) Reflection Matrix(209,-80,546,-209) (8/21,5/13) -> (8/21,5/13) Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(13,-6,28,-13) (3/7,1/2) -> (3/7,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(27,-5,70,-13) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(29,-6,140,-29) -> Matrix(-1,2,0,1) (1/5,3/14) -> (1/1,1/0) Matrix(13,-3,56,-13) -> Matrix(1,2,0,-1) (3/14,1/4) -> (-1/1,1/0) Matrix(15,-4,56,-15) -> Matrix(-1,0,2,1) (1/4,2/7) -> (-1/1,0/1) Matrix(13,-4,42,-13) -> Matrix(1,0,0,-1) (2/7,1/3) -> (0/1,1/0) Matrix(29,-10,84,-29) -> Matrix(1,0,0,-1) (1/3,5/14) -> (0/1,1/0) Matrix(41,-15,112,-41) -> Matrix(1,2,0,-1) (5/14,3/8) -> (-1/1,1/0) Matrix(127,-48,336,-127) -> Matrix(-1,0,2,1) (3/8,8/21) -> (-1/1,0/1) Matrix(209,-80,546,-209) -> Matrix(1,0,0,-1) (8/21,5/13) -> (0/1,1/0) Matrix(29,-12,70,-29) -> Matrix(1,2,0,-1) (2/5,3/7) -> (-1/1,1/0) Matrix(13,-6,28,-13) -> Matrix(1,2,0,-1) (3/7,1/2) -> (-1/1,1/0) Matrix(-1,1,0,1) -> Matrix(-1,0,2,1) (1/2,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.