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 -9/2 -4/1 -3/1 -2/1 -3/2 -4/3 0/1 1/1 6/5 4/3 3/2 12/7 2/1 12/5 5/2 3/1 36/11 7/2 4/1 9/2 5/1 11/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -5/1 1/1 -9/2 0/1 1/1 1/0 -4/1 1/0 -15/4 -2/1 -1/1 1/0 -11/3 -1/1 -7/2 -1/1 1/0 -3/1 -1/1 1/1 -11/4 -1/1 0/1 -8/3 0/1 -5/2 0/1 1/0 -12/5 1/0 -7/3 -1/1 -9/4 -1/1 0/1 1/0 -11/5 -1/1 -2/1 -1/1 1/1 -13/7 -1/1 -24/13 -1/1 -11/6 -1/1 0/1 -9/5 -1/1 1/1 -7/4 -1/1 1/0 -12/7 -1/1 -5/3 -1/1 -3/2 -1/1 0/1 1/0 -7/5 -1/1 -18/13 0/1 -11/8 -1/1 0/1 -4/3 0/1 -5/4 0/1 1/0 -6/5 1/0 -7/6 -1/1 1/0 -1/1 -1/1 0/1 0/1 1/1 1/1 6/5 1/0 5/4 0/1 1/0 4/3 0/1 3/2 0/1 1/1 1/0 8/5 0/1 13/8 0/1 1/1 5/3 1/1 12/7 1/1 7/4 1/1 1/0 2/1 -1/1 1/1 11/5 1/1 9/4 0/1 1/1 1/0 7/3 1/1 12/5 1/0 5/2 0/1 1/0 8/3 0/1 11/4 0/1 1/1 3/1 -1/1 1/1 13/4 0/1 1/1 36/11 1/1 23/7 1/1 10/3 -1/1 1/1 7/2 1/1 1/0 18/5 1/0 11/3 1/1 4/1 1/0 13/3 -1/1 9/2 -1/1 0/1 1/0 5/1 -1/1 11/2 -1/1 0/1 6/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(25,132,-18,-95) (-6/1,-5/1) -> (-7/5,-18/13) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(23,96,-6,-25) (-9/2,-4/1) -> (-4/1,-15/4) Parabolic Matrix(71,264,32,119) (-15/4,-11/3) -> (11/5,9/4) Hyperbolic Matrix(23,84,-20,-73) (-11/3,-7/2) -> (-7/6,-1/1) Hyperbolic Matrix(25,84,-14,-47) (-7/2,-3/1) -> (-9/5,-7/4) Hyperbolic Matrix(47,132,-26,-73) (-3/1,-11/4) -> (-11/6,-9/5) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) 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(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(97,216,22,49) (-9/4,-11/5) -> (13/3,9/2) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(551,1020,168,311) (-13/7,-24/13) -> (36/11,23/7) Hyperbolic Matrix(385,708,118,217) (-24/13,-11/6) -> (13/4,36/11) 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(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) 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(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(73,-156,22,-47) (2/1,11/5) -> (23/7,10/3) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(25,132,-18,-95) -> Matrix(1,0,-2,1) Matrix(23,108,10,47) -> Matrix(1,0,0,1) Matrix(23,96,-6,-25) -> Matrix(1,-2,0,1) Matrix(71,264,32,119) -> Matrix(1,2,0,1) Matrix(23,84,-20,-73) -> Matrix(1,0,0,1) Matrix(25,84,-14,-47) -> Matrix(1,0,0,1) Matrix(47,132,-26,-73) -> Matrix(1,0,0,1) Matrix(71,192,44,119) -> Matrix(1,0,2,1) Matrix(23,60,18,47) -> Matrix(1,0,0,1) Matrix(49,120,20,49) -> Matrix(1,0,0,1) Matrix(71,168,30,71) -> Matrix(1,2,0,1) Matrix(47,108,10,23) -> Matrix(1,0,0,1) Matrix(97,216,22,49) -> Matrix(1,0,0,1) Matrix(23,48,-12,-25) -> Matrix(1,0,0,1) Matrix(551,1020,168,311) -> Matrix(1,2,0,1) Matrix(385,708,118,217) -> Matrix(1,0,2,1) Matrix(97,168,56,97) -> Matrix(1,2,0,1) Matrix(71,120,42,71) -> Matrix(1,0,2,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(191,264,34,47) -> Matrix(1,0,0,1) Matrix(97,132,36,49) -> Matrix(1,0,2,1) Matrix(47,60,18,23) -> Matrix(1,0,0,1) Matrix(49,60,40,49) -> Matrix(1,0,0,1) Matrix(143,168,40,47) -> Matrix(1,2,0,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(73,-84,20,-23) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(73,-120,14,-23) -> Matrix(1,0,-2,1) Matrix(47,-84,14,-25) -> Matrix(1,0,0,1) Matrix(73,-156,22,-47) -> Matrix(1,0,0,1) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(25,-96,6,-23) -> 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): 4 Degree of the the map X: 4 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,19,31,20,7)(3,12,30,29,13,4)(5,11,17)(8,23,24)(9,27,10)(14,22,15)(16,28)(25,26); (1,4,15,28,27,31,32,30,23,16,5,2)(3,10,29,11)(6,17,25,24,12,21,20,9,26,14,13,18)(7,22,19,8); (1,2,8,25,9,3)(4,14)(5,6)(7,21,12,11,16,15)(10,28,23,19,18,13)(17,29,32,31,22,26)(20,27)(24,30)) ----------------------------------------------------------------------- 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 -4/1 -3/2 0/1 1/1 3/2 12/7 2/1 3/1 7/2 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -7/2 -1/1 1/0 -3/1 -1/1 1/1 -2/1 -1/1 1/1 -9/5 -1/1 1/1 -7/4 -1/1 1/0 -12/7 -1/1 -5/3 -1/1 -3/2 -1/1 0/1 1/0 -7/5 -1/1 -11/8 -1/1 0/1 -4/3 0/1 -5/4 0/1 1/0 -6/5 1/0 -7/6 -1/1 1/0 -1/1 -1/1 0/1 0/1 1/1 1/1 4/3 0/1 3/2 0/1 1/1 1/0 8/5 0/1 5/3 1/1 12/7 1/1 7/4 1/1 1/0 2/1 -1/1 1/1 3/1 -1/1 1/1 10/3 -1/1 1/1 7/2 1/1 1/0 11/3 1/1 4/1 1/0 5/1 -1/1 11/2 -1/1 0/1 6/1 0/1 1/0 0/1 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(25,84,-14,-47) (-7/2,-3/1) -> (-9/5,-7/4) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(53,96,16,29) (-2/1,-9/5) -> (3/1,10/3) 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(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(43,60,-38,-53) (-7/5,-11/8) -> (-7/6,-1/1) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(91,108,16,19) (-6/5,-7/6) -> (11/2,6/1) 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(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(43,-156,8,-29) (7/2,11/3) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(0,-1,1,0) Matrix(19,72,-14,-53) -> Matrix(0,-1,1,2) Matrix(25,84,-14,-47) -> Matrix(1,0,0,1) Matrix(5,12,2,5) -> Matrix(0,-1,1,0) Matrix(53,96,16,29) -> Matrix(0,-1,1,0) Matrix(97,168,56,97) -> Matrix(1,2,0,1) Matrix(71,120,42,71) -> Matrix(1,0,2,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(43,60,-38,-53) -> Matrix(2,1,-1,0) Matrix(29,36,4,5) -> Matrix(0,-1,1,0) Matrix(91,108,16,19) -> Matrix(0,-1,1,2) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(19,-24,4,-5) -> Matrix(0,-1,1,0) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(67,-108,18,-29) -> Matrix(2,-1,1,0) Matrix(47,-84,14,-25) -> Matrix(1,0,0,1) Matrix(43,-156,8,-29) -> Matrix(0,-1,1,0) 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 0/1 1 1 1/1 1/1 1 12 4/3 0/1 1 3 3/2 0 4 8/5 0/1 1 3 5/3 1/1 1 12 12/7 1/1 1 1 7/4 (1/1,1/0) 0 12 2/1 (-1/1,1/1) 0 6 3/1 (-1/1,1/1) 0 4 10/3 (-1/1,1/1) 0 6 7/2 (1/1,1/0) 0 12 11/3 1/1 1 12 4/1 1/0 1 3 5/1 -1/1 1 12 11/2 (-1/1,0/1) 0 12 6/1 0/1 1 2 1/0 (0/1,1/0) 0 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(71,-120,42,-71) (5/3,12/7) -> (5/3,12/7) Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(43,-156,8,-29) (7/2,11/3) -> (5/1,11/2) Hyperbolic Matrix(23,-132,4,-23) (11/2,6/1) -> (11/2,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(1,0,2,-1) -> Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Matrix(19,-24,4,-5) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(67,-108,18,-29) -> Matrix(2,-1,1,0) 1/1 Matrix(71,-120,42,-71) -> Matrix(1,0,2,-1) (5/3,12/7) -> (0/1,1/1) Matrix(97,-168,56,-97) -> Matrix(-1,2,0,1) (12/7,7/4) -> (1/1,1/0) Matrix(47,-84,14,-25) -> Matrix(1,0,0,1) Matrix(5,-12,2,-5) -> Matrix(0,1,1,0) (2/1,3/1) -> (-1/1,1/1) Matrix(19,-60,6,-19) -> Matrix(0,1,1,0) (3/1,10/3) -> (-1/1,1/1) Matrix(43,-156,8,-29) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(23,-132,4,-23) -> Matrix(-1,0,2,1) (11/2,6/1) -> (-1/1,0/1) Matrix(-1,12,0,1) -> Matrix(1,0,0,-1) (6/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.