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 -1/1 -3/7 -1/3 -1/5 -1/7 0/1 1/3 1/2 3/5 1/1 9/7 3/2 5/3 2/1 11/5 17/7 5/2 3/1 4/1 13/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/2 1/0 -1/2 -1/1 0/1 -5/11 -1/2 1/0 -4/9 -1/1 0/1 -3/7 -1/1 -5/12 -1/2 0/1 -7/17 -1/2 1/0 -2/5 -1/1 0/1 -3/8 -1/1 -2/3 -4/11 -3/5 -1/2 -1/3 -1/2 1/0 -4/13 -3/5 -1/2 -3/10 -1/2 -2/5 -5/17 -1/3 -2/7 -1/3 0/1 -1/4 -1/1 0/1 -4/17 -1/1 0/1 -3/13 -1/2 1/0 -5/22 -1/1 0/1 -2/9 -1/1 -1/2 -1/5 0/1 -2/11 1/1 1/0 -1/6 -1/1 0/1 -2/13 -1/1 0/1 -1/7 -1/2 1/0 -1/8 -1/2 0/1 0/1 -1/1 0/1 1/5 -1/2 1/0 1/4 -1/1 0/1 1/3 0/1 2/5 -1/1 1/0 3/7 -1/2 1/0 1/2 -1/1 0/1 3/5 0/1 5/8 0/1 1/3 2/3 0/1 1/1 3/4 2/1 1/0 1/1 -1/2 1/0 5/4 2/1 1/0 9/7 1/0 13/10 -8/1 1/0 4/3 -3/1 1/0 7/5 -2/1 3/2 -2/1 -1/1 5/3 -1/1 7/4 -1/1 -2/3 2/1 -1/1 0/1 11/5 -1/2 1/0 20/9 -1/1 0/1 9/4 -1/1 0/1 7/3 -1/2 1/0 12/5 -1/1 0/1 17/7 0/1 22/9 0/1 1/1 5/2 0/1 1/0 3/1 -1/1 7/2 -2/3 -1/2 4/1 -1/1 0/1 13/3 -1/2 1/0 22/5 -1/1 0/1 9/2 -1/1 0/1 5/1 -1/2 1/0 6/1 -1/1 1/0 7/1 -1/1 8/1 -1/1 -1/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(61,28,-268,-123) (-1/2,-5/11) -> (-3/13,-5/22) Hyperbolic Matrix(31,14,-206,-93) (-5/11,-4/9) -> (-2/13,-1/7) Hyperbolic Matrix(59,26,-202,-89) (-4/9,-3/7) -> (-5/17,-2/7) Hyperbolic Matrix(81,34,-274,-115) (-3/7,-5/12) -> (-3/10,-5/17) Hyperbolic Matrix(29,12,-220,-91) (-5/12,-7/17) -> (-1/7,-1/8) Hyperbolic Matrix(83,34,-354,-145) (-7/17,-2/5) -> (-4/17,-3/13) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(27,10,-154,-57) (-3/8,-4/11) -> (-2/11,-1/6) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(209,64,160,49) (-4/13,-3/10) -> (13/10,4/3) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(233,56,104,25) (-1/4,-4/17) -> (20/9,9/4) Hyperbolic Matrix(185,42,22,5) (-5/22,-2/9) -> (8/1,1/0) Hyperbolic Matrix(19,4,-100,-21) (-2/9,-1/5) -> (-1/5,-2/11) Parabolic Matrix(249,40,56,9) (-1/6,-2/13) -> (22/5,9/2) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(65,-14,14,-3) (1/5,1/4) -> (9/2,5/1) Hyperbolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(77,-34,34,-15) (3/7,1/2) -> (9/4,7/3) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(239,-578,98,-237) (12/5,17/7) -> (17/7,22/9) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic Matrix(15,-98,2,-13) (6/1,7/1) -> (7/1,8/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(61,28,-268,-123) -> Matrix(1,0,0,1) Matrix(31,14,-206,-93) -> Matrix(1,0,0,1) Matrix(59,26,-202,-89) -> Matrix(1,0,-2,1) Matrix(81,34,-274,-115) -> Matrix(3,2,-8,-5) Matrix(29,12,-220,-91) -> Matrix(1,0,0,1) Matrix(83,34,-354,-145) -> Matrix(1,0,0,1) Matrix(51,20,28,11) -> Matrix(1,0,0,1) Matrix(27,10,-154,-57) -> Matrix(3,2,-2,-1) Matrix(23,8,-72,-25) -> Matrix(1,0,0,1) Matrix(209,64,160,49) -> Matrix(11,6,-2,-1) Matrix(43,12,68,19) -> Matrix(1,0,4,1) Matrix(233,56,104,25) -> Matrix(1,0,0,1) Matrix(185,42,22,5) -> Matrix(1,0,0,1) Matrix(19,4,-100,-21) -> Matrix(1,0,2,1) Matrix(249,40,56,9) -> Matrix(1,0,0,1) Matrix(181,22,74,9) -> Matrix(1,0,2,1) Matrix(67,-12,28,-5) -> Matrix(1,0,0,1) Matrix(65,-14,14,-3) -> Matrix(1,0,0,1) Matrix(37,-10,26,-7) -> Matrix(3,2,-2,-1) Matrix(47,-18,34,-13) -> Matrix(1,-2,0,1) Matrix(67,-28,12,-5) -> Matrix(1,0,0,1) Matrix(77,-34,34,-15) -> Matrix(1,0,0,1) Matrix(31,-18,50,-29) -> Matrix(1,0,4,1) Matrix(37,-26,10,-7) -> Matrix(1,0,-2,1) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(127,-162,98,-125) -> Matrix(1,-10,0,1) Matrix(31,-50,18,-29) -> Matrix(3,4,-4,-5) Matrix(111,-242,50,-109) -> Matrix(1,0,0,1) Matrix(239,-578,98,-237) -> Matrix(1,0,2,1) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) Matrix(79,-338,18,-77) -> Matrix(1,0,0,1) Matrix(15,-98,2,-13) -> Matrix(1,2,-2,-3) 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): 5 Degree of the the map X: 5 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,17,26,29,18,7,2)(3,11,27,32,22,8,12,4)(5,10,9,25,30,23,20,15)(13,16,24,31,19,21,28,14); (1,4,14,5)(3,10)(6,13)(7,20,21,8)(9,16,11,17)(18,19)(22,23)(29,32,31,30); (1,2,8,23,31,24,9,3)(4,12,21,18,30,25,17,13)(5,15,7,19,32,27,16,6)(10,14,28,20,22,29,26,11)) ----------------------------------------------------------------------- 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: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/3 1/1 9/7 5/3 2/1 11/5 3/1 4/1 13/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/2 1/0 0/1 -1/1 0/1 1/5 -1/2 1/0 1/4 -1/1 0/1 1/3 0/1 2/5 -1/1 1/0 3/7 -1/2 1/0 1/2 -1/1 0/1 3/5 0/1 2/3 0/1 1/1 3/4 2/1 1/0 1/1 -1/2 1/0 5/4 2/1 1/0 9/7 1/0 4/3 -3/1 1/0 7/5 -2/1 3/2 -2/1 -1/1 5/3 -1/1 2/1 -1/1 0/1 11/5 -1/2 1/0 9/4 -1/1 0/1 7/3 -1/2 1/0 12/5 -1/1 0/1 17/7 0/1 5/2 0/1 1/0 3/1 -1/1 7/2 -2/3 -1/2 4/1 -1/1 0/1 13/3 -1/2 1/0 9/2 -1/1 0/1 5/1 -1/2 1/0 6/1 -1/1 1/0 7/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(65,-14,14,-3) (1/5,1/4) -> (9/2,5/1) Hyperbolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(77,-34,34,-15) (3/7,1/2) -> (9/4,7/3) Hyperbolic Matrix(20,-11,11,-6) (1/2,3/5) -> (5/3,2/1) Hyperbolic Matrix(30,-19,19,-12) (3/5,2/3) -> (3/2,5/3) Hyperbolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(64,-81,49,-62) (5/4,9/7) -> (9/7,4/3) Parabolic Matrix(56,-121,25,-54) (2/1,11/5) -> (11/5,9/4) Parabolic Matrix(42,-101,5,-12) (12/5,17/7) -> (7/1,1/0) Hyperbolic Matrix(56,-137,9,-22) (17/7,5/2) -> (6/1,7/1) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(40,-169,9,-38) (4/1,13/3) -> (13/3,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,1,-2,-1) Matrix(67,-12,28,-5) -> Matrix(1,0,0,1) Matrix(65,-14,14,-3) -> Matrix(1,0,0,1) Matrix(37,-10,26,-7) -> Matrix(3,2,-2,-1) Matrix(47,-18,34,-13) -> Matrix(1,-2,0,1) Matrix(67,-28,12,-5) -> Matrix(1,0,0,1) Matrix(77,-34,34,-15) -> Matrix(1,0,0,1) Matrix(20,-11,11,-6) -> Matrix(1,1,-2,-1) Matrix(30,-19,19,-12) -> Matrix(3,-1,-2,1) Matrix(37,-26,10,-7) -> Matrix(1,0,-2,1) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(64,-81,49,-62) -> Matrix(1,-5,0,1) Matrix(56,-121,25,-54) -> Matrix(1,1,-2,-1) Matrix(42,-101,5,-12) -> Matrix(1,1,-2,-1) Matrix(56,-137,9,-22) -> Matrix(1,-1,0,1) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) Matrix(40,-169,9,-38) -> Matrix(1,1,-2,-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): 5 ----------------------------------------------------------------------- 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 -1/1 (-1/1,0/1).(-1/2,1/0) 0 1 1/1 (-1/2,1/0) 0 4 9/7 1/0 5 1 4/3 (-3/1,1/0) 0 8 3/2 (-2/1,-1/1) 0 8 5/3 -1/1 2 1 2/1 (-1/1,0/1) 0 8 11/5 (-1/1,0/1).(-1/2,1/0) 0 1 7/3 (-1/2,1/0) 0 4 5/2 (0/1,1/0) 0 8 3/1 -1/1 1 2 7/2 (-2/3,-1/2) 0 8 4/1 (-1/1,0/1) 0 8 13/3 (-1/1,0/1).(-1/2,1/0) 0 1 5/1 (-1/2,1/0) 0 4 6/1 (-1/1,1/0) 0 8 7/1 -1/1 1 1 1/0 (-1/1,0/1) 0 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(8,-9,7,-8) (1/1,9/7) -> (1/1,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(26,-37,7,-10) (4/3,3/2) -> (7/2,4/1) Glide Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(21,-44,10,-21) (2/1,11/5) -> (2/1,11/5) Reflection Matrix(34,-77,15,-34) (11/5,7/3) -> (11/5,7/3) Reflection Matrix(28,-67,5,-12) (7/3,5/2) -> (5/1,6/1) Glide Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(14,-65,3,-14) (13/3,5/1) -> (13/3,5/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(0,1,1,0) -> Matrix(1,1,0,-1) (-1/1,1/1) -> (-1/2,1/0) Matrix(8,-9,7,-8) -> Matrix(1,1,0,-1) (1/1,9/7) -> (-1/2,1/0) Matrix(55,-72,42,-55) -> Matrix(1,6,0,-1) (9/7,4/3) -> (-3/1,1/0) Matrix(26,-37,7,-10) -> Matrix(1,1,-2,-3) Matrix(19,-30,12,-19) -> Matrix(3,4,-2,-3) (3/2,5/3) -> (-2/1,-1/1) Matrix(11,-20,6,-11) -> Matrix(-1,0,2,1) (5/3,2/1) -> (-1/1,0/1) Matrix(21,-44,10,-21) -> Matrix(-1,0,2,1) (2/1,11/5) -> (-1/1,0/1) Matrix(34,-77,15,-34) -> Matrix(1,1,0,-1) (11/5,7/3) -> (-1/2,1/0) Matrix(28,-67,5,-12) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) -1/1 Matrix(25,-104,6,-25) -> Matrix(-1,0,2,1) (4/1,13/3) -> (-1/1,0/1) Matrix(14,-65,3,-14) -> Matrix(1,1,0,-1) (13/3,5/1) -> (-1/2,1/0) Matrix(13,-84,2,-13) -> Matrix(1,2,0,-1) (6/1,7/1) -> (-1/1,1/0) Matrix(-1,14,0,1) -> Matrix(-1,0,2,1) (7/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.