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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/4 -1/2 -3/8 -1/4 -3/16 -1/6 -1/8 0/1 1/4 1/3 1/2 2/3 3/4 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 -3/4 -1/2 -5/7 -2/5 -1/3 -7/10 -1/2 -2/3 -2/5 -1/3 -1/2 -1/4 -2/5 -4/19 -1/5 -3/8 -1/5 -1/3 -1/5 -2/11 -1/4 -1/6 -1/5 -1/7 0/1 -3/16 0/1 -2/11 -1/5 0/1 -1/6 -1/6 -1/7 -2/13 -1/7 -1/8 -1/7 0/1 -1/7 0/1 1/4 -1/8 2/7 -2/17 -1/9 3/10 -1/8 1/3 -2/17 -1/9 1/2 -1/10 3/5 -4/43 -1/11 5/8 -1/11 2/3 -1/11 -2/23 3/4 -1/12 4/5 -1/13 0/1 13/16 0/1 9/11 -1/11 0/1 5/6 -1/12 6/7 -2/25 -1/13 7/8 -1/13 1/1 -1/13 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(23,18,-32,-25) (-1/1,-3/4) -> (-3/4,-5/7) Parabolic Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(23,16,-128,-89) (-7/10,-2/3) -> (-2/11,-1/6) 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(39,14,64,23) (-3/8,-1/3) -> (3/5,5/8) Hyperbolic Matrix(7,2,-32,-9) (-1/3,-1/4) -> (-1/4,-1/5) 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(57,8,64,9) (-1/7,-1/8) -> (7/8,1/1) Hyperbolic Matrix(55,6,64,7) (-1/8,0/1) -> (6/7,7/8) Hyperbolic Matrix(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(105,-32,128,-39) (3/10,1/3) -> (9/11,5/6) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-12,1) Matrix(23,18,-32,-25) -> Matrix(3,2,-8,-5) Matrix(17,12,-112,-79) -> Matrix(1,0,-4,1) Matrix(23,16,-128,-89) -> Matrix(5,2,-28,-11) Matrix(7,4,-16,-9) -> Matrix(7,2,-32,-9) Matrix(41,16,64,25) -> Matrix(29,6,-324,-67) Matrix(39,14,64,23) -> Matrix(31,6,-336,-65) Matrix(7,2,-32,-9) -> Matrix(11,2,-72,-13) Matrix(209,40,256,49) -> Matrix(1,0,-4,1) Matrix(207,38,256,47) -> Matrix(1,0,-8,1) Matrix(57,8,64,9) -> Matrix(13,2,-176,-27) Matrix(55,6,64,7) -> Matrix(15,2,-188,-25) Matrix(9,-2,32,-7) -> Matrix(15,2,-128,-17) Matrix(95,-28,112,-33) -> Matrix(1,0,-4,1) Matrix(105,-32,128,-39) -> Matrix(17,2,-196,-23) Matrix(9,-4,16,-7) -> Matrix(19,2,-200,-21) Matrix(25,-18,32,-23) -> Matrix(23,2,-288,-25) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 4 Degree of the the map X: 8 Degree of the the map Y: 16 Permutation triple for Y: ((1,4,10,13,15,11,5,2)(3,8,14,16,12,7,6,9); (1,2,7,12,15,13,8,3)(4,9,6,5,11,16,14,10); (2,6)(3,4)(11,12)(13,14)) ----------------------------------------------------------------------- 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): 24 Minimal number of generators: 5 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: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/2 3/4 5/6 7/8 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/7 0/1 1/2 -1/10 2/3 -1/11 -2/23 3/4 -1/12 4/5 -1/13 0/1 5/6 -1/12 6/7 -2/25 -1/13 7/8 -1/13 1/1 -1/13 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(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic Matrix(57,-49,64,-55) (6/7,7/8) -> (7/8,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-6,1) Matrix(5,-2,8,-3) -> Matrix(9,1,-100,-11) Matrix(25,-18,32,-23) -> Matrix(23,2,-288,-25) Matrix(61,-50,72,-59) -> Matrix(11,1,-144,-13) Matrix(57,-49,64,-55) -> Matrix(25,2,-338,-27) 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): 4 ----------------------------------------------------------------------- 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 (-1/7,0/1) 0 8 1/8 -1/7 2 1 1/6 -1/8 2 4 1/4 -1/8 2 2 1/2 -1/10 2 4 1/0 0/1 6 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(1,0,16,-1) (0/1,1/8) -> (0/1,1/8) Reflection Matrix(7,-1,48,-7) (1/8,1/6) -> (1/8,1/6) Reflection Matrix(5,-1,24,-5) (1/6,1/4) -> (1/6,1/4) Reflection Matrix(3,-1,8,-3) (1/4,1/2) -> (1/4,1/2) 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(-1,0,14,1) (0/1,1/0) -> (-1/7,0/1) Matrix(1,0,16,-1) -> Matrix(-1,0,14,1) (0/1,1/8) -> (-1/7,0/1) Matrix(7,-1,48,-7) -> Matrix(15,2,-112,-15) (1/8,1/6) -> (-1/7,-1/8) Matrix(5,-1,24,-5) -> Matrix(7,1,-48,-7) (1/6,1/4) -> (-1/6,-1/8) Matrix(3,-1,8,-3) -> Matrix(9,1,-80,-9) (1/4,1/2) -> (-1/8,-1/10) Matrix(-1,1,0,1) -> Matrix(-1,0,20,1) (1/2,1/0) -> (-1/10,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.