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: 20 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -3/1 -8/3 -2/1 -3/2 -6/5 0/1 1/1 3/2 12/7 2/1 12/5 8/3 3/1 24/7 7/2 4/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 1/1 -7/2 1/0 -10/3 -1/1 -1/3 -3/1 0/1 -11/4 1/2 -8/3 1/1 -13/5 0/1 -5/2 1/0 -12/5 1/0 -7/3 -2/1 -9/4 1/0 -2/1 -1/1 1/1 -9/5 0/1 -7/4 -1/2 -12/7 0/1 -5/3 0/1 -13/8 1/0 -8/5 1/1 -11/7 2/1 -3/2 1/0 -10/7 -3/1 -1/1 -17/12 -3/2 -24/17 -1/1 -7/5 0/1 -18/13 -1/1 -11/8 1/0 -4/3 -1/1 1/1 -5/4 1/0 -11/9 -2/1 -6/5 -1/1 -7/6 -1/2 -1/1 0/1 0/1 -1/1 1/1 1/1 0/1 4/3 -1/1 1/1 7/5 0/1 10/7 1/1 3/1 3/2 1/0 11/7 -2/1 8/5 -1/1 13/8 1/0 5/3 0/1 12/7 0/1 7/4 1/2 9/5 0/1 2/1 -1/1 1/1 9/4 1/0 7/3 2/1 12/5 1/0 5/2 1/0 13/5 0/1 8/3 -1/1 11/4 -1/2 3/1 0/1 10/3 1/3 1/1 17/5 2/3 24/7 1/1 7/2 1/0 18/5 1/1 11/3 0/1 4/1 -1/1 1/1 5/1 0/1 11/2 1/2 6/1 1/1 7/1 2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(29,96,16,53) (-10/3,-3/1) -> (9/5,2/1) Hyperbolic Matrix(43,120,24,67) (-3/1,-11/4) -> (7/4,9/5) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(47,120,-38,-97) (-13/5,-5/2) -> (-5/4,-11/9) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(53,120,34,77) (-7/3,-9/4) -> (3/2,11/7) Hyperbolic Matrix(43,96,30,67) (-9/4,-2/1) -> (10/7,3/2) Hyperbolic Matrix(53,96,16,29) (-2/1,-9/5) -> (3/1,10/3) Hyperbolic Matrix(67,120,24,43) (-9/5,-7/4) -> (11/4,3/1) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(29,48,-26,-43) (-5/3,-13/8) -> (-7/6,-1/1) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(77,120,34,53) (-11/7,-3/2) -> (9/4,7/3) Hyperbolic Matrix(67,96,30,43) (-3/2,-10/7) -> (2/1,9/4) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(121,168,18,25) (-7/5,-18/13) -> (6/1,7/1) Hyperbolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,0,0,1) Matrix(19,72,-14,-53) -> Matrix(1,0,0,1) Matrix(71,240,-50,-169) -> Matrix(3,2,-2,-1) Matrix(29,96,16,53) -> Matrix(1,0,2,1) Matrix(43,120,24,67) -> Matrix(1,0,0,1) Matrix(71,192,44,119) -> Matrix(1,0,-2,1) Matrix(73,192,46,121) -> Matrix(1,-2,0,1) Matrix(47,120,-38,-97) -> Matrix(1,-2,0,1) Matrix(49,120,20,49) -> Matrix(1,-2,0,1) Matrix(71,168,30,71) -> Matrix(1,4,0,1) Matrix(53,120,34,77) -> Matrix(1,0,0,1) Matrix(43,96,30,67) -> Matrix(1,2,0,1) Matrix(53,96,16,29) -> Matrix(1,0,2,1) Matrix(67,120,24,43) -> Matrix(1,0,0,1) Matrix(97,168,56,97) -> Matrix(1,0,4,1) Matrix(71,120,42,71) -> Matrix(1,0,-2,1) Matrix(29,48,-26,-43) -> Matrix(1,0,-2,1) Matrix(119,192,44,71) -> Matrix(1,0,-2,1) Matrix(121,192,46,73) -> Matrix(1,-2,0,1) Matrix(77,120,34,53) -> Matrix(1,0,0,1) Matrix(67,96,30,43) -> Matrix(1,2,0,1) Matrix(407,576,118,167) -> Matrix(3,4,2,3) Matrix(409,576,120,169) -> Matrix(3,2,4,3) Matrix(121,168,18,25) -> Matrix(1,2,0,1) Matrix(191,264,34,47) -> Matrix(1,0,2,1) Matrix(217,264,60,73) -> Matrix(1,2,0,1) Matrix(143,168,40,47) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(19,-24,4,-5) -> Matrix(1,0,0,1) Matrix(53,-72,14,-19) -> Matrix(1,0,0,1) Matrix(169,-240,50,-71) -> Matrix(1,-2,2,-3) Matrix(73,-120,14,-23) -> Matrix(1,0,2,1) Matrix(19,-48,2,-5) -> Matrix(1,2,0,1) 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): 7 Degree of the the map X: 7 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,22,31,23,7)(3,12,18,27,13,4)(5,8,17)(9,15,10)(11,20,19)(14,26,25)(16,30)(28,29); (1,4,9,29,14,13,21,6,20,16,5,2)(3,10,22,11)(7,26,27,8)(12,24,23,25,30,15,31,32,18,17,28,19); (1,2,8,16,25,7,24,12,11,28,9,3)(4,14,23,15)(5,18,19,6)(10,30,20,22,21,13,26,29,17,27,32,31)) ----------------------------------------------------------------------- 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: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 3/2 8/5 2/1 12/5 8/3 3/1 24/7 4/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 1/1 1/1 0/1 4/3 -1/1 1/1 7/5 0/1 10/7 1/1 3/1 3/2 1/0 11/7 -2/1 8/5 -1/1 13/8 1/0 5/3 0/1 12/7 0/1 7/4 1/2 9/5 0/1 2/1 -1/1 1/1 9/4 1/0 7/3 2/1 12/5 1/0 5/2 1/0 13/5 0/1 8/3 -1/1 11/4 -1/2 3/1 0/1 10/3 1/3 1/1 17/5 2/3 24/7 1/1 7/2 1/0 18/5 1/1 11/3 0/1 4/1 -1/1 1/1 5/1 0/1 11/2 1/2 6/1 1/1 7/1 2/1 1/0 1/0 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(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(67,-96,37,-53) (10/7,3/2) -> (9/5,2/1) Hyperbolic Matrix(77,-120,43,-67) (3/2,11/7) -> (7/4,9/5) Hyperbolic Matrix(121,-192,75,-119) (11/7,8/5) -> (8/5,13/8) Parabolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(71,-120,29,-49) (5/3,12/7) -> (12/5,5/2) Hyperbolic Matrix(97,-168,41,-71) (12/7,7/4) -> (7/3,12/5) Hyperbolic Matrix(43,-96,13,-29) (2/1,9/4) -> (3/1,10/3) Hyperbolic Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(73,-192,27,-71) (13/5,8/3) -> (8/3,11/4) Parabolic Matrix(169,-576,49,-167) (17/5,24/7) -> (24/7,7/2) Parabolic Matrix(47,-168,7,-25) (7/2,18/5) -> (6/1,7/1) Hyperbolic Matrix(73,-264,13,-47) (18/5,11/3) -> (11/2,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(0,-1,1,0) Matrix(19,-24,4,-5) -> Matrix(1,0,0,1) Matrix(53,-72,14,-19) -> Matrix(1,0,0,1) Matrix(169,-240,50,-71) -> Matrix(1,-2,2,-3) Matrix(67,-96,37,-53) -> Matrix(0,1,-1,2) Matrix(77,-120,43,-67) -> Matrix(0,-1,1,0) Matrix(121,-192,75,-119) -> Matrix(0,-1,1,2) Matrix(73,-120,14,-23) -> Matrix(1,0,2,1) Matrix(71,-120,29,-49) -> Matrix(2,1,-1,0) Matrix(97,-168,41,-71) -> Matrix(4,-1,1,0) Matrix(43,-96,13,-29) -> Matrix(0,1,-1,2) Matrix(53,-120,19,-43) -> Matrix(0,-1,1,0) Matrix(19,-48,2,-5) -> Matrix(1,2,0,1) Matrix(73,-192,27,-71) -> Matrix(0,-1,1,2) Matrix(169,-576,49,-167) -> Matrix(4,-3,3,-2) Matrix(47,-168,7,-25) -> Matrix(2,-1,1,0) Matrix(73,-264,13,-47) -> Matrix(0,1,-1,2) 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): 7 ----------------------------------------------------------------------- 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 (-1/1,1/1).(0/1,1/0) 0 1 2/1 (-1/1,1/1) 0 6 9/4 1/0 1 4 7/3 2/1 1 12 12/5 1/0 3 1 5/2 1/0 1 12 13/5 0/1 1 12 8/3 -1/1 1 3 11/4 -1/2 1 12 3/1 0/1 1 4 10/3 (1/3,1/1) 0 6 24/7 1/1 3 1 7/2 1/0 1 12 18/5 1/1 1 2 4/1 (-1/1,1/1) 0 3 6/1 1/1 1 2 7/1 2/1 1 12 1/0 1/0 1 12 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(43,-96,13,-29) (2/1,9/4) -> (3/1,10/3) Hyperbolic Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(73,-192,27,-71) (13/5,8/3) -> (8/3,11/4) Parabolic Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(47,-168,7,-25) (7/2,18/5) -> (6/1,7/1) Hyperbolic Matrix(19,-72,5,-19) (18/5,4/1) -> (18/5,4/1) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) 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(1,0,1,-1) -> Matrix(0,1,1,0) (0/1,2/1) -> (-1/1,1/1) Matrix(43,-96,13,-29) -> Matrix(0,1,-1,2) 1/1 Matrix(53,-120,19,-43) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(71,-168,30,-71) -> Matrix(-1,4,0,1) (7/3,12/5) -> (2/1,1/0) Matrix(49,-120,20,-49) -> Matrix(1,2,0,-1) (12/5,5/2) -> (-1/1,1/0) Matrix(19,-48,2,-5) -> Matrix(1,2,0,1) 1/0 Matrix(73,-192,27,-71) -> Matrix(0,-1,1,2) -1/1 Matrix(71,-240,21,-71) -> Matrix(2,-1,3,-2) (10/3,24/7) -> (1/3,1/1) Matrix(97,-336,28,-97) -> Matrix(-1,2,0,1) (24/7,7/2) -> (1/1,1/0) Matrix(47,-168,7,-25) -> Matrix(2,-1,1,0) 1/1 Matrix(19,-72,5,-19) -> Matrix(0,1,1,0) (18/5,4/1) -> (-1/1,1/1) Matrix(5,-24,1,-5) -> Matrix(0,1,1,0) (4/1,6/1) -> (-1/1,1/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.