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): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -1/1 -1/2 0/1 1/3 3/8 3/7 1/2 3/5 2/3 3/4 5/6 6/7 1/1 7/6 6/5 4/3 3/2 5/3 2/1 7/3 8/3 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -8/3 0/1 1/2 -13/5 1/2 -5/2 1/2 -12/5 1/1 -7/3 0/1 1/1 -2/1 1/0 -7/4 -1/2 -19/11 -1/2 -12/7 0/1 -5/3 -1/1 0/1 -3/2 0/1 -4/3 0/1 1/1 -5/4 1/2 -6/5 1/1 -7/6 1/1 2/1 -1/1 1/0 -6/7 -2/1 -5/6 -2/1 -1/1 -4/5 -3/2 -7/9 -4/3 -1/1 -3/4 -1/1 -2/3 -1/1 0/1 -3/5 -1/1 -7/12 -1/1 -4/5 -4/7 -3/4 -1/2 -1/2 -4/9 -1/3 0/1 -3/7 0/1 -5/12 -1/1 0/1 -2/5 -1/2 -7/18 -1/3 0/1 -5/13 -1/4 -8/21 -1/4 0/1 -3/8 0/1 -1/3 -1/2 0/1 0/1 0/1 1/3 0/1 1/2 3/8 0/1 5/13 1/4 2/5 1/2 5/12 0/1 1/1 3/7 0/1 1/2 1/2 4/7 3/4 11/19 5/6 7/12 4/5 1/1 3/5 1/1 2/3 0/1 1/1 3/4 1/1 4/5 3/2 5/6 1/1 2/1 6/7 2/1 1/1 1/0 7/6 -2/1 -1/1 6/5 -1/1 5/4 -1/2 9/7 0/1 4/3 -1/1 0/1 3/2 0/1 5/3 0/1 1/1 12/7 0/1 7/4 1/2 2/1 1/0 9/4 -1/1 7/3 -1/1 0/1 12/5 -1/1 5/2 -1/2 18/7 0/1 13/5 -1/2 21/8 0/1 8/3 -1/2 0/1 3/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(55,144,-144,-377) (-8/3,-13/5) -> (-5/13,-8/21) Hyperbolic Matrix(73,186,-42,-107) (-13/5,-5/2) -> (-7/4,-19/11) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(19,42,-24,-53) (-7/3,-2/1) -> (-4/5,-7/9) Hyperbolic Matrix(17,30,30,53) (-2/1,-7/4) -> (1/2,4/7) Hyperbolic Matrix(73,126,84,145) (-19/11,-12/7) -> (6/7,1/1) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(19,24,-42,-53) (-4/3,-5/4) -> (-1/2,-4/9) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(71,84,60,71) (-6/5,-7/6) -> (7/6,6/5) Hyperbolic Matrix(73,84,126,145) (-7/6,-1/1) -> (11/19,7/12) Hyperbolic Matrix(109,96,42,37) (-1/1,-6/7) -> (18/7,13/5) Hyperbolic Matrix(71,60,84,71) (-6/7,-5/6) -> (5/6,6/7) Hyperbolic Matrix(37,30,90,73) (-5/6,-4/5) -> (2/5,5/12) Hyperbolic Matrix(109,84,48,37) (-7/9,-3/4) -> (9/4,7/3) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(71,42,120,71) (-3/5,-7/12) -> (7/12,3/5) Hyperbolic Matrix(73,42,-186,-107) (-7/12,-4/7) -> (-2/5,-7/18) Hyperbolic Matrix(53,30,30,17) (-4/7,-1/2) -> (7/4,2/1) Hyperbolic Matrix(109,48,84,37) (-4/9,-3/7) -> (9/7,4/3) Hyperbolic Matrix(71,30,168,71) (-3/7,-5/12) -> (5/12,3/7) Hyperbolic Matrix(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) Hyperbolic Matrix(109,42,96,37) (-7/18,-5/13) -> (1/1,7/6) Hyperbolic Matrix(505,192,192,73) (-8/21,-3/8) -> (21/8,8/3) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(107,-42,186,-73) (5/13,2/5) -> (4/7,11/19) Hyperbolic Matrix(53,-24,42,-19) (3/7,1/2) -> (5/4,9/7) Hyperbolic Matrix(53,-42,24,-19) (3/4,4/5) -> (2/1,9/4) Hyperbolic Matrix(107,-186,42,-73) (12/7,7/4) -> (5/2,18/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(17,48,6,17) -> Matrix(1,0,-4,1) Matrix(55,144,-144,-377) -> Matrix(1,0,-6,1) Matrix(73,186,-42,-107) -> Matrix(1,0,-4,1) Matrix(37,90,30,73) -> Matrix(3,-2,-4,3) Matrix(71,168,30,71) -> Matrix(1,0,-2,1) Matrix(19,42,-24,-53) -> Matrix(3,-4,-2,3) Matrix(17,30,30,53) -> Matrix(3,2,4,3) Matrix(73,126,84,145) -> Matrix(5,2,2,1) Matrix(71,120,42,71) -> Matrix(1,0,2,1) Matrix(19,30,12,19) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,-2,1) Matrix(19,24,-42,-53) -> Matrix(1,0,-4,1) Matrix(73,90,30,37) -> Matrix(3,-2,-4,3) Matrix(71,84,60,71) -> Matrix(3,-4,-2,3) Matrix(73,84,126,145) -> Matrix(5,-6,6,-7) Matrix(109,96,42,37) -> Matrix(1,2,-2,-3) Matrix(71,60,84,71) -> Matrix(3,4,2,3) Matrix(37,30,90,73) -> Matrix(1,2,0,1) Matrix(109,84,48,37) -> Matrix(3,4,-4,-5) Matrix(17,12,24,17) -> Matrix(1,0,2,1) Matrix(19,12,30,19) -> Matrix(1,0,2,1) Matrix(71,42,120,71) -> Matrix(9,8,10,9) Matrix(73,42,-186,-107) -> Matrix(5,4,-14,-11) Matrix(53,30,30,17) -> Matrix(3,2,4,3) Matrix(109,48,84,37) -> Matrix(1,0,2,1) Matrix(71,30,168,71) -> Matrix(1,0,2,1) Matrix(73,30,90,37) -> Matrix(1,2,0,1) Matrix(109,42,96,37) -> Matrix(7,2,-4,-1) Matrix(505,192,192,73) -> Matrix(1,0,2,1) Matrix(17,6,48,17) -> Matrix(1,0,4,1) Matrix(1,0,6,1) -> Matrix(1,0,4,1) Matrix(377,-144,144,-55) -> Matrix(1,0,-6,1) Matrix(107,-42,186,-73) -> Matrix(11,-4,14,-5) Matrix(53,-24,42,-19) -> Matrix(1,0,-4,1) Matrix(53,-42,24,-19) -> Matrix(3,-4,-2,3) Matrix(107,-186,42,-73) -> Matrix(1,0,-4,1) 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): 12 Degree of the the map X: 12 Degree of the the map Y: 36 Permutation triple for Y: ((1,7,2)(3,12,19,36,21,20,27,26,13)(4,15,18,23,8,22,25,16,5)(6,10,9)(11,24,17)(14,31,32)(28,34,30)(29,33,35); (1,5,17,34,22,31,35,18,6)(2,10,27,33,32,36,28,11,3)(4,13,14)(7,21,8)(9,25,19)(12,16,29)(15,30,26)(20,24,23); (1,3,4)(2,8,24,28,15,14,33,16,9)(5,12,11)(6,19,29,31,13,30,17,20,7)(10,18,26)(21,32,22)(23,35,27)(25,34,36)) ----------------------------------------------------------------------- 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): 36 Minimal number of generators: 7 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: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 2/3 3/4 1/1 4/3 3/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 0/1 0/1 1/2 1/2 3/5 1/1 2/3 0/1 1/1 3/4 1/1 4/5 3/2 1/1 1/0 4/3 -1/1 0/1 3/2 0/1 5/3 0/1 1/1 7/4 1/2 2/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,0,1) (-1/1,1/0) -> (2/1,1/0) Parabolic Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(26,-15,33,-19) (1/2,3/5) -> (3/4,4/5) Hyperbolic Matrix(19,-12,27,-17) (3/5,2/3) -> (2/3,3/4) Parabolic Matrix(37,-30,21,-17) (4/5,1/1) -> (7/4,2/1) Hyperbolic Matrix(26,-33,15,-19) (1/1,4/3) -> (5/3,7/4) Hyperbolic Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,3,0,1) -> Matrix(1,0,0,1) Matrix(1,0,3,1) -> Matrix(1,0,2,1) Matrix(26,-15,33,-19) -> Matrix(5,-4,4,-3) Matrix(19,-12,27,-17) -> Matrix(1,0,0,1) Matrix(37,-30,21,-17) -> Matrix(1,-2,2,-3) Matrix(26,-33,15,-19) -> Matrix(1,0,2,1) Matrix(19,-27,12,-17) -> 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 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): 2 ----------------------------------------------------------------------- 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 2 3 2/3 (0/1,1/1) 0 9 3/4 1/1 4 3 1/1 1/0 2 9 4/3 (-1/1,0/1) 0 9 3/2 0/1 2 3 1/0 (0/1,1/0) 0 9 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(1,0,3,-1) (0/1,2/3) -> (0/1,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(19,-15,24,-19) (3/4,5/6) -> (3/4,5/6) Reflection Matrix(10,-9,9,-8) (4/5,1/1) -> (1/1,5/4) Parabolic Matrix(19,-24,15,-19) (6/5,4/3) -> (6/5,4/3) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(-1,3,0,1) (3/2,1/0) -> (3/2,1/0) 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(1,0,3,-1) -> Matrix(1,0,2,-1) (0/1,2/3) -> (0/1,1/1) Matrix(17,-12,24,-17) -> Matrix(1,0,2,-1) (2/3,3/4) -> (0/1,1/1) Matrix(19,-15,24,-19) -> Matrix(3,-4,2,-3) (3/4,5/6) -> (1/1,2/1) Matrix(10,-9,9,-8) -> Matrix(1,-2,0,1) 1/0 Matrix(19,-24,15,-19) -> Matrix(-1,0,2,1) (6/5,4/3) -> (-1/1,0/1) Matrix(17,-24,12,-17) -> Matrix(-1,0,2,1) (4/3,3/2) -> (-1/1,0/1) Matrix(-1,3,0,1) -> Matrix(1,0,0,-1) (3/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.