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 -3/4 -1/2 -7/16 -5/16 -1/4 -3/16 -1/6 -1/7 -1/8 0/1 1/6 1/5 1/4 2/7 1/3 3/8 2/5 1/2 5/8 2/3 3/4 7/8 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 -5/6 -3/4 -4/5 -1/1 -2/3 -3/4 -1/2 -5/7 -1/3 0/1 -7/10 1/0 -2/3 -1/1 0/1 -5/8 -1/1 -3/5 -1/1 -2/3 -4/7 -1/1 -2/3 -1/2 -1/2 -4/9 -1/3 0/1 -7/16 -1/2 -1/4 -3/7 -1/3 0/1 -5/12 -1/2 -2/5 -1/3 0/1 -3/8 0/1 -1/3 -1/1 0/1 -5/16 -1/2 1/0 -4/13 -1/1 0/1 -3/10 1/0 -5/17 -4/3 -1/1 -7/24 -1/1 -2/7 -1/1 -2/3 -1/4 -1/2 -2/9 -1/3 0/1 -1/5 -1/3 0/1 -3/16 -1/2 -1/4 -2/11 -1/3 0/1 -1/6 -1/4 -1/7 -1/7 0/1 -1/8 0/1 0/1 -1/1 0/1 1/6 1/2 1/5 0/1 1/1 1/4 1/0 2/7 -2/1 -1/1 3/10 -1/2 1/3 -1/1 0/1 3/8 0/1 2/5 0/1 1/1 3/7 0/1 1/1 1/2 1/0 5/9 -2/1 -1/1 9/16 -3/2 1/0 4/7 -2/1 -1/1 7/12 1/0 3/5 -2/1 -1/1 5/8 -1/1 2/3 -1/1 0/1 11/16 -1/2 1/0 9/13 -1/1 0/1 7/10 -1/2 12/17 -1/5 0/1 17/24 0/1 5/7 0/1 1/1 3/4 1/0 7/9 -2/1 -1/1 4/5 -2/1 -1/1 13/16 -3/2 1/0 9/11 -2/1 -1/1 5/6 -3/2 6/7 -6/5 -1/1 7/8 -1/1 1/1 -1/1 0/1 1/0 -1/2 1/0 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(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(17,14,-96,-79) (-5/6,-4/5) -> (-2/11,-1/6) Hyperbolic Matrix(33,26,-80,-63) (-4/5,-3/4) -> (-5/12,-2/5) Hyperbolic Matrix(47,34,-112,-81) (-3/4,-5/7) -> (-3/7,-5/12) Hyperbolic Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(49,34,-160,-111) (-7/10,-2/3) -> (-4/13,-3/10) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(17,10,-80,-47) (-3/5,-4/7) -> (-2/9,-1/5) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(143,62,256,111) (-7/16,-3/7) -> (5/9,9/16) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(177,56,256,81) (-1/3,-5/16) -> (11/16,9/13) Hyperbolic Matrix(175,54,256,79) (-5/16,-4/13) -> (2/3,11/16) Hyperbolic Matrix(143,42,160,47) (-5/17,-7/24) -> (7/8,1/1) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(15,4,-64,-17) (-2/7,-1/4) -> (-1/4,-2/9) 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(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(113,12,160,17) (-1/8,0/1) -> (12/17,17/24) Hyperbolic Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(79,-14,96,-17) (1/6,1/5) -> (9/11,5/6) Hyperbolic Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(65,-18,112,-31) (1/4,2/7) -> (4/7,7/12) Hyperbolic Matrix(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(111,-34,160,-49) (3/10,1/3) -> (9/13,7/10) Hyperbolic Matrix(63,-26,80,-33) (2/5,3/7) -> (7/9,4/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(33,28,-112,-95) -> Matrix(5,4,-4,-3) Matrix(17,14,-96,-79) -> Matrix(3,2,-8,-5) Matrix(33,26,-80,-63) -> Matrix(3,2,-8,-5) Matrix(47,34,-112,-81) -> Matrix(1,0,0,1) Matrix(17,12,-112,-79) -> Matrix(1,0,-4,1) Matrix(49,34,-160,-111) -> Matrix(1,0,0,1) Matrix(31,20,48,31) -> Matrix(1,0,0,1) Matrix(49,30,80,49) -> Matrix(5,4,-4,-3) Matrix(17,10,-80,-47) -> Matrix(3,2,-8,-5) Matrix(15,8,-32,-17) -> Matrix(3,2,-8,-5) Matrix(145,64,256,113) -> Matrix(7,2,-4,-1) Matrix(143,62,256,111) -> Matrix(7,2,-4,-1) Matrix(31,12,80,31) -> Matrix(1,0,4,1) Matrix(17,6,48,17) -> Matrix(1,0,0,1) Matrix(177,56,256,81) -> Matrix(1,0,0,1) Matrix(175,54,256,79) -> Matrix(1,0,0,1) Matrix(143,42,160,47) -> Matrix(3,4,-4,-5) Matrix(193,56,224,65) -> Matrix(9,8,-8,-7) Matrix(15,4,-64,-17) -> Matrix(3,2,-8,-5) Matrix(209,40,256,49) -> Matrix(7,2,-4,-1) Matrix(207,38,256,47) -> Matrix(7,2,-4,-1) Matrix(159,22,224,31) -> Matrix(1,0,8,1) Matrix(113,12,160,17) -> Matrix(1,0,-4,1) Matrix(79,-12,112,-17) -> Matrix(1,0,-4,1) Matrix(79,-14,96,-17) -> Matrix(1,-2,0,1) Matrix(47,-10,80,-17) -> Matrix(1,-2,0,1) Matrix(65,-18,112,-31) -> Matrix(1,0,0,1) Matrix(95,-28,112,-33) -> Matrix(5,4,-4,-3) Matrix(111,-34,160,-49) -> Matrix(1,0,0,1) Matrix(63,-26,80,-33) -> Matrix(1,-2,0,1) Matrix(17,-8,32,-15) -> Matrix(1,-2,0,1) Matrix(49,-36,64,-47) -> Matrix(1,-2,0,1) 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: 32 Permutation triple for Y: ((1,4,16,26,29,17,5,2)(3,10,27,32,22,8,7,11)(6,14,13,9,25,30,23,20)(12,15,24,31,19,18,21,28); (1,2,8,23,31,24,9,3)(4,14,6,5,19,32,27,15)(7,18,17,30,25,16,12,11)(10,13,28,21,20,22,29,26); (2,6,21,7)(3,12,13,4)(5,18)(8,20)(9,10)(15,16)(17,22,19,23)(24,27,26,25)) ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/4 1/3 1/2 5/8 3/4 13/16 5/6 7/8 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 0/1 1/4 1/0 2/7 -2/1 -1/1 3/10 -1/2 1/3 -1/1 0/1 1/2 1/0 3/5 -2/1 -1/1 5/8 -1/1 2/3 -1/1 0/1 3/4 1/0 4/5 -2/1 -1/1 13/16 -3/2 1/0 9/11 -2/1 -1/1 5/6 -3/2 6/7 -6/5 -1/1 7/8 -1/1 1/1 -1/1 0/1 1/0 -1/2 1/0 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(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(41,-25,64,-39) (3/5,5/8) -> (5/8,2/3) Parabolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(209,-169,256,-207) (4/5,13/16) -> (13/16,9/11) 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(1,1,-2,-1) Matrix(9,-2,32,-7) -> Matrix(1,-1,0,1) Matrix(95,-28,112,-33) -> Matrix(5,4,-4,-3) Matrix(105,-32,128,-39) -> Matrix(1,-1,0,1) Matrix(9,-4,16,-7) -> Matrix(1,-1,0,1) Matrix(41,-25,64,-39) -> Matrix(1,2,-2,-3) Matrix(25,-18,32,-23) -> Matrix(1,-1,0,1) Matrix(209,-169,256,-207) -> Matrix(3,5,-2,-3) Matrix(57,-49,64,-55) -> Matrix(5,6,-6,-7) 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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 (-1/1,0/1) 0 8 1/4 1/0 1 2 2/7 (-2/1,-1/1) 0 8 7/24 -1/1 6 1 3/10 -1/2 1 4 5/16 (-1/1,0/1).(-1/2,1/0) 0 1 1/3 (-1/1,0/1) 0 8 3/8 0/1 2 1 1/2 1/0 1 4 1/0 (-1/1,0/1).(-1/2,1/0) 0 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(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(97,-28,336,-97) (2/7,7/24) -> (2/7,7/24) Reflection Matrix(71,-21,240,-71) (7/24,3/10) -> (7/24,3/10) Reflection Matrix(49,-15,160,-49) (3/10,5/16) -> (3/10,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(7,-3,16,-7) (3/8,1/2) -> (3/8,1/2) 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(9,-2,32,-7) -> Matrix(1,-1,0,1) 1/0 Matrix(97,-28,336,-97) -> Matrix(3,4,-2,-3) (2/7,7/24) -> (-2/1,-1/1) Matrix(71,-21,240,-71) -> Matrix(3,2,-4,-3) (7/24,3/10) -> (-1/1,-1/2) Matrix(49,-15,160,-49) -> Matrix(1,1,0,-1) (3/10,5/16) -> (-1/2,1/0) Matrix(31,-10,96,-31) -> Matrix(-1,0,2,1) (5/16,1/3) -> (-1/1,0/1) Matrix(17,-6,48,-17) -> Matrix(-1,0,2,1) (1/3,3/8) -> (-1/1,0/1) Matrix(7,-3,16,-7) -> Matrix(1,0,0,-1) (3/8,1/2) -> (0/1,1/0) Matrix(-1,1,0,1) -> Matrix(1,1,0,-1) (1/2,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.