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: 16 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -4/1 -8/3 -16/7 -2/1 -4/3 0/1 1/1 4/3 2/1 16/7 8/3 4/1 32/7 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 1/28 -7/1 7/192 -6/1 3/80 -5/1 5/128 -4/1 1/24 -11/3 11/256 -18/5 9/208 -7/2 7/160 -3/1 3/64 -8/3 1/20 -5/2 5/96 -7/3 7/128 -16/7 1/18 -9/4 9/160 -2/1 1/16 -9/5 9/128 -16/9 1/14 -7/4 7/96 -5/3 5/64 -8/5 1/12 -3/2 3/32 -7/5 7/64 -4/3 1/8 -9/7 9/64 -32/25 1/7 -23/18 23/160 -14/11 7/48 -5/4 5/32 -6/5 3/16 -7/6 7/32 -8/7 1/4 -1/1 1/0 0/1 0/1 1/1 1/64 8/7 1/60 7/6 7/416 6/5 3/176 5/4 5/288 4/3 1/56 11/8 11/608 18/13 9/496 7/5 7/384 3/2 3/160 8/5 1/52 5/3 5/256 7/4 7/352 16/9 1/50 9/5 9/448 2/1 1/48 9/4 9/416 16/7 1/46 7/3 7/320 5/2 5/224 8/3 1/44 3/1 3/128 7/2 7/288 4/1 1/40 9/2 9/352 32/7 1/39 23/5 23/896 14/3 7/272 5/1 5/192 6/1 3/112 7/1 7/256 8/1 1/36 1/0 1/32 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,64,6,55) (-8/1,1/0) -> (8/7,7/6) Hyperbolic Matrix(9,64,8,57) (-8/1,-7/1) -> (1/1,8/7) Hyperbolic Matrix(19,128,4,27) (-7/1,-6/1) -> (14/3,5/1) Hyperbolic Matrix(11,64,6,35) (-6/1,-5/1) -> (9/5,2/1) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(53,192,8,29) (-11/3,-18/5) -> (6/1,7/1) Hyperbolic Matrix(143,512,-112,-401) (-18/5,-7/2) -> (-23/18,-14/11) Hyperbolic Matrix(19,64,8,27) (-7/2,-3/1) -> (7/3,5/2) Hyperbolic Matrix(23,64,14,39) (-3/1,-8/3) -> (8/5,5/3) Hyperbolic Matrix(25,64,16,41) (-8/3,-5/2) -> (3/2,8/5) Hyperbolic Matrix(27,64,8,19) (-5/2,-7/3) -> (3/1,7/2) Hyperbolic Matrix(111,256,62,143) (-7/3,-16/7) -> (16/9,9/5) Hyperbolic Matrix(113,256,64,145) (-16/7,-9/4) -> (7/4,16/9) Hyperbolic Matrix(29,64,24,53) (-9/4,-2/1) -> (6/5,5/4) Hyperbolic Matrix(35,64,6,11) (-2/1,-9/5) -> (5/1,6/1) Hyperbolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(37,64,26,45) (-7/4,-5/3) -> (7/5,3/2) Hyperbolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(45,64,26,37) (-3/2,-7/5) -> (5/3,7/4) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(799,1024,174,223) (-9/7,-32/25) -> (32/7,23/5) Hyperbolic Matrix(801,1024,176,225) (-32/25,-23/18) -> (9/2,32/7) Hyperbolic Matrix(101,128,86,109) (-14/11,-5/4) -> (7/6,6/5) Hyperbolic Matrix(53,64,24,29) (-5/4,-6/5) -> (2/1,9/4) Hyperbolic Matrix(163,192,118,139) (-6/5,-7/6) -> (11/8,18/13) Hyperbolic Matrix(55,64,6,7) (-7/6,-8/7) -> (8/1,1/0) Hyperbolic Matrix(57,64,8,9) (-8/7,-1/1) -> (7/1,8/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,64,6,55) -> Matrix(57,-2,3392,-119) Matrix(9,64,8,57) -> Matrix(55,-2,3328,-121) Matrix(19,128,4,27) -> Matrix(109,-4,4224,-155) Matrix(11,64,6,35) -> Matrix(53,-2,2624,-99) Matrix(15,64,-4,-17) -> Matrix(49,-2,1152,-47) Matrix(53,192,8,29) -> Matrix(139,-6,5120,-221) Matrix(143,512,-112,-401) -> Matrix(369,-16,2560,-111) Matrix(19,64,8,27) -> Matrix(45,-2,2048,-91) Matrix(23,64,14,39) -> Matrix(41,-2,2112,-103) Matrix(25,64,16,41) -> Matrix(39,-2,2048,-105) Matrix(27,64,8,19) -> Matrix(37,-2,1536,-83) Matrix(111,256,62,143) -> Matrix(145,-8,7232,-399) Matrix(113,256,64,145) -> Matrix(143,-8,7168,-401) Matrix(29,64,24,53) -> Matrix(35,-2,2048,-117) Matrix(35,64,6,11) -> Matrix(29,-2,1088,-75) Matrix(143,256,62,111) -> Matrix(113,-8,5184,-367) Matrix(145,256,64,113) -> Matrix(111,-8,5120,-369) Matrix(37,64,26,45) -> Matrix(27,-2,1472,-109) Matrix(39,64,14,23) -> Matrix(25,-2,1088,-87) Matrix(41,64,16,25) -> Matrix(23,-2,1024,-89) Matrix(45,64,26,37) -> Matrix(19,-2,960,-101) Matrix(47,64,-36,-49) -> Matrix(17,-2,128,-15) Matrix(799,1024,174,223) -> Matrix(225,-32,8768,-1247) Matrix(801,1024,176,225) -> Matrix(223,-32,8704,-1249) Matrix(101,128,86,109) -> Matrix(27,-4,1600,-237) Matrix(53,64,24,29) -> Matrix(11,-2,512,-93) Matrix(163,192,118,139) -> Matrix(29,-6,1600,-331) Matrix(55,64,6,7) -> Matrix(9,-2,320,-71) Matrix(57,64,8,9) -> Matrix(7,-2,256,-73) Matrix(1,0,2,1) -> Matrix(1,0,64,1) Matrix(49,-64,36,-47) -> Matrix(113,-2,6272,-111) Matrix(369,-512,80,-111) -> Matrix(881,-16,34304,-623) Matrix(17,-64,4,-15) -> Matrix(81,-2,3200,-79) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 16 Genus: 9 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 32 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,19,23,30,28,20,7)(3,12,17,27,24,22,13,4)(5,18)(8,21)(9,10)(15,16); (1,4,16,28,31,24,9,19,25,12,8,7,14,13,18,30,32,27,15,6,11,3,10,20,26,22,21,23,29,17,5,2); (1,2,8,22,31,28,18,17,25,19,15,4,14,7,10,24,32,30,21,12,11,6,5,13,26,20,16,27,29,23,9,3)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 4/3 8/5 2/1 16/7 8/3 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/64 4/3 1/56 7/5 7/384 3/2 3/160 8/5 1/52 5/3 5/256 7/4 7/352 2/1 1/48 9/4 9/416 16/7 1/46 7/3 7/320 5/2 5/224 8/3 1/44 3/1 3/128 4/1 1/40 5/1 5/192 1/0 1/32 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(45,-64,19,-27) (7/5,3/2) -> (7/3,5/2) Hyperbolic Matrix(41,-64,25,-39) (3/2,8/5) -> (8/5,5/3) Parabolic Matrix(19,-32,3,-5) (5/3,7/4) -> (5/1,1/0) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(113,-256,49,-111) (9/4,16/7) -> (16/7,7/3) Parabolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,32,1) Matrix(25,-32,18,-23) -> Matrix(57,-1,3136,-55) Matrix(45,-64,19,-27) -> Matrix(109,-2,4960,-91) Matrix(41,-64,25,-39) -> Matrix(105,-2,5408,-103) Matrix(19,-32,3,-5) -> Matrix(51,-1,1888,-37) Matrix(17,-32,8,-15) -> Matrix(49,-1,2304,-47) Matrix(113,-256,49,-111) -> Matrix(369,-8,16928,-367) Matrix(25,-64,9,-23) -> Matrix(89,-2,3872,-87) Matrix(9,-32,2,-7) -> Matrix(41,-1,1600,-39) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM 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: 8 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 32 1 2/1 1/48 2 16 16/7 1/46 16 2 7/3 7/320 1 32 5/2 5/224 1 32 8/3 1/44 8 4 3/1 3/128 1 32 4/1 1/40 4 8 5/1 5/192 1 32 1/0 1/32 1 32 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(15,-32,7,-15) (2/1,16/7) -> (2/1,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(13,-32,2,-5) (7/3,5/2) -> (5/1,1/0) Glide Reflection Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,64,-1) (0/1,1/0) -> (0/1,1/32) Matrix(1,0,1,-1) -> Matrix(1,0,96,-1) (0/1,2/1) -> (0/1,1/48) Matrix(15,-32,7,-15) -> Matrix(47,-1,2208,-47) (2/1,16/7) -> (1/48,1/46) Matrix(97,-224,42,-97) -> Matrix(321,-7,14720,-321) (16/7,7/3) -> (1/46,7/320) Matrix(13,-32,2,-5) -> Matrix(45,-1,1664,-37) Matrix(25,-64,9,-23) -> Matrix(89,-2,3872,-87) 1/44 Matrix(9,-32,2,-7) -> Matrix(41,-1,1600,-39) 1/40 ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.