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 -6/1 -4/1 -2/1 -4/3 -8/7 0/1 1/1 4/3 3/2 8/5 2/1 16/7 5/2 8/3 3/1 16/5 7/2 4/1 5/1 16/3 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 4/3 2/1 -5/1 3/2 5/3 2/1 -4/1 2/1 -7/2 2/1 7/3 5/2 -10/3 2/1 8/3 -3/1 2/1 3/1 1/0 -8/3 1/0 -5/2 1/1 2/1 1/0 -7/3 1/1 2/1 1/0 -2/1 2/1 -9/5 2/1 5/2 3/1 -16/9 3/1 -7/4 2/1 3/1 1/0 -12/7 2/1 -5/3 2/1 5/2 3/1 -8/5 3/1 -3/2 2/1 3/1 1/0 -16/11 3/1 -13/9 3/1 7/2 4/1 -10/7 2/1 4/1 -17/12 2/1 5/2 3/1 -24/17 3/1 -7/5 3/1 4/1 1/0 -4/3 2/1 4/1 -9/7 3/1 4/1 1/0 -5/4 3/1 4/1 1/0 -16/13 1/0 -11/9 1/1 2/1 1/0 -6/5 2/1 4/1 -7/6 4/1 5/1 1/0 -8/7 1/0 -1/1 2/1 3/1 1/0 0/1 1/0 1/1 0/1 1/1 1/0 6/5 0/1 5/4 -1/1 0/1 1/0 4/3 0/1 7/5 0/1 1/2 1/1 10/7 0/1 3/2 0/1 1/1 1/0 8/5 1/1 5/3 1/1 2/1 1/0 7/4 0/1 1/1 1/0 2/1 0/1 2/1 9/4 0/1 1/1 1/0 16/7 1/1 7/3 1/1 2/1 1/0 12/5 2/1 5/2 1/1 2/1 1/0 8/3 1/0 3/1 0/1 1/1 1/0 16/5 1/1 13/4 1/1 3/2 2/1 10/3 2/1 17/5 2/1 3/1 1/0 24/7 1/0 7/2 0/1 1/1 1/0 4/1 0/1 2/1 9/2 0/1 1/1 1/0 5/1 1/1 2/1 1/0 16/3 1/0 11/2 -1/1 0/1 1/0 6/1 0/1 7/1 0/1 1/2 1/1 8/1 1/1 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,112,-12,-79) (-6/1,1/0) -> (-10/7,-17/12) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(17,80,-10,-47) (-5/1,-4/1) -> (-12/7,-5/3) Hyperbolic Matrix(31,112,-18,-65) (-4/1,-7/2) -> (-7/4,-12/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(33,80,-26,-63) (-5/2,-7/3) -> (-9/7,-5/4) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic 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(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(175,256,54,79) (-3/2,-16/11) -> (16/5,13/4) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(143,160,42,47) (-8/7,-1/1) -> (17/5,24/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(5,-8,2,-3) Matrix(17,96,-14,-79) -> Matrix(5,-8,2,-3) Matrix(17,80,-10,-47) -> Matrix(9,-16,4,-7) Matrix(31,112,-18,-65) -> Matrix(3,-8,2,-5) Matrix(33,112,-28,-95) -> Matrix(7,-18,2,-5) Matrix(49,160,-34,-111) -> Matrix(7,-18,2,-5) Matrix(17,48,6,17) -> Matrix(1,-2,0,1) Matrix(31,80,12,31) -> Matrix(1,0,0,1) Matrix(33,80,-26,-63) -> Matrix(1,2,0,1) Matrix(15,32,-8,-17) -> Matrix(5,-8,2,-3) Matrix(143,256,62,111) -> Matrix(3,-8,2,-5) Matrix(145,256,64,113) -> Matrix(1,-2,0,1) Matrix(49,80,30,49) -> Matrix(3,-8,2,-5) Matrix(31,48,20,31) -> Matrix(1,-2,0,1) Matrix(175,256,54,79) -> Matrix(3,-8,2,-5) Matrix(177,256,56,81) -> Matrix(1,-4,2,-7) Matrix(113,160,12,17) -> Matrix(3,-8,2,-5) Matrix(159,224,22,31) -> Matrix(1,-4,2,-7) Matrix(47,64,-36,-49) -> Matrix(1,0,0,1) Matrix(207,256,38,47) -> Matrix(1,-4,0,1) Matrix(209,256,40,49) -> Matrix(1,0,0,1) Matrix(193,224,56,65) -> Matrix(1,-4,0,1) Matrix(143,160,42,47) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) Matrix(95,-112,28,-33) -> Matrix(1,2,0,1) Matrix(79,-96,14,-17) -> Matrix(1,0,0,1) Matrix(63,-80,26,-33) -> Matrix(1,2,0,1) Matrix(81,-112,34,-47) -> Matrix(3,-2,2,-1) Matrix(79,-112,12,-17) -> Matrix(1,0,0,1) Matrix(111,-160,34,-49) -> Matrix(3,-2,2,-1) Matrix(47,-80,10,-17) -> Matrix(1,0,0,1) Matrix(17,-32,8,-15) -> Matrix(1,0,0,1) Matrix(17,-64,4,-15) -> Matrix(1,0,0,1) 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): 6 Degree of the the map X: 6 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,21,7)(3,12,13,4)(5,18)(8,20)(9,10)(15,16)(17,22,19,23)(24,27,26,25); (1,4,16,26,29,17,5,2)(3,10,27,32,22,8,7,11)(6,14,13,9,25,30,23,20)(12,15,24,31,19,18,21,28); (1,2,8,23,31,24,9,3)(4,14,6,5,19,32,27,15)(7,18,17,30,25,16,12,11)(10,13,28,21,20,22,29,26)) ----------------------------------------------------------------------- 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: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -10/3 -2/1 -4/3 0/1 1/1 4/3 3/2 2/1 8/3 3/1 4/1 16/3 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 2/1 -7/2 2/1 7/3 5/2 -10/3 2/1 8/3 -3/1 2/1 3/1 1/0 -2/1 2/1 -5/3 2/1 5/2 3/1 -8/5 3/1 -3/2 2/1 3/1 1/0 -4/3 2/1 4/1 -5/4 3/1 4/1 1/0 -16/13 1/0 -11/9 1/1 2/1 1/0 -6/5 2/1 4/1 -7/6 4/1 5/1 1/0 -8/7 1/0 -1/1 2/1 3/1 1/0 0/1 1/0 1/1 0/1 1/1 1/0 4/3 0/1 7/5 0/1 1/2 1/1 10/7 0/1 3/2 0/1 1/1 1/0 2/1 0/1 2/1 5/2 1/1 2/1 1/0 8/3 1/0 3/1 0/1 1/1 1/0 4/1 0/1 2/1 5/1 1/1 2/1 1/0 16/3 1/0 11/2 -1/1 0/1 1/0 6/1 0/1 7/1 0/1 1/2 1/1 8/1 1/1 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,32,-2,-9) (-4/1,1/0) -> (-4/1,-7/2) Parabolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(39,128,-32,-105) (-10/3,-3/1) -> (-11/9,-6/5) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic 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(23,32,-18,-25) (-3/2,-4/3) -> (-4/3,-5/4) Parabolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) 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(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,32,-2,-9) -> Matrix(7,-12,3,-5) Matrix(33,112,-28,-95) -> Matrix(7,-18,2,-5) Matrix(39,128,-32,-105) -> Matrix(1,-4,1,-3) Matrix(7,16,-4,-9) -> Matrix(3,-4,1,-1) Matrix(39,64,14,23) -> Matrix(1,-2,-1,3) Matrix(41,64,16,25) -> Matrix(1,-4,1,-3) Matrix(23,32,-18,-25) -> Matrix(3,-10,1,-3) Matrix(207,256,38,47) -> Matrix(1,-4,0,1) Matrix(209,256,40,49) -> Matrix(1,0,0,1) Matrix(55,64,6,7) -> Matrix(1,-6,1,-5) Matrix(57,64,8,9) -> Matrix(1,-2,1,-1) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) Matrix(25,-32,18,-23) -> Matrix(1,0,1,1) Matrix(79,-112,12,-17) -> Matrix(1,0,0,1) Matrix(89,-128,16,-23) -> Matrix(1,0,-1,1) Matrix(9,-16,4,-7) -> Matrix(1,-2,1,-1) Matrix(9,-32,2,-7) -> Matrix(1,-2,1,-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): 6 ----------------------------------------------------------------------- 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 -6/1 (1/1,3/2).(4/3,2/1) 0 4 -4/1 2/1 3 2 -2/1 2/1 1 4 -4/3 (2/1,4/1).(3/1,1/0) 0 2 -6/5 (2/1,4/1).(3/1,1/0) 0 4 -7/6 0 8 -8/7 1/0 4 2 -1/1 0 8 0/1 1/0 2 2 1/1 0 8 6/5 0/1 1 4 4/3 0/1 1 2 2/1 (0/1,2/1).(1/1,1/0) 0 4 4/1 (0/1,2/1).(1/1,1/0) 0 2 6/1 0/1 1 4 7/1 0 8 8/1 1/1 4 2 1/0 0 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,48,-6,-41) (-6/1,1/0) -> (-6/5,-7/6) Hyperbolic Matrix(5,24,-1,-5) (-6/1,-4/1) -> (-6/1,-4/1) Reflection Matrix(3,8,-1,-3) (-4/1,-2/1) -> (-4/1,-2/1) Reflection Matrix(5,8,-3,-5) (-2/1,-4/3) -> (-2/1,-4/3) Reflection Matrix(19,24,-15,-19) (-4/3,-6/5) -> (-4/3,-6/5) Reflection 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(41,-48,6,-7) (1/1,6/5) -> (6/1,7/1) Hyperbolic Matrix(19,-24,15,-19) (6/5,4/3) -> (6/5,4/3) Reflection Matrix(5,-8,3,-5) (4/3,2/1) -> (4/3,2/1) Reflection Matrix(3,-8,1,-3) (2/1,4/1) -> (2/1,4/1) 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(7,48,-6,-41) -> Matrix(5,-6,1,-1) Matrix(5,24,-1,-5) -> Matrix(5,-8,3,-5) (-6/1,-4/1) -> (4/3,2/1) Matrix(3,8,-1,-3) -> Matrix(-1,4,0,1) (-4/1,-2/1) -> (2/1,1/0) Matrix(5,8,-3,-5) -> Matrix(3,-8,1,-3) (-2/1,-4/3) -> (2/1,4/1) Matrix(19,24,-15,-19) -> Matrix(-1,6,0,1) (-4/3,-6/5) -> (3/1,1/0) Matrix(55,64,6,7) -> Matrix(1,-6,1,-5) Matrix(57,64,8,9) -> Matrix(1,-2,1,-1) (0/1,2/1).(1/1,1/0) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) 1/0 Matrix(41,-48,6,-7) -> Matrix(1,0,1,1) 0/1 Matrix(19,-24,15,-19) -> Matrix(1,0,0,-1) (6/5,4/3) -> (0/1,1/0) Matrix(5,-8,3,-5) -> Matrix(1,0,1,-1) (4/3,2/1) -> (0/1,2/1) Matrix(3,-8,1,-3) -> Matrix(-1,2,0,1) (2/1,4/1) -> (1/1,1/0) Matrix(5,-24,1,-5) -> Matrix(1,0,1,-1) (4/1,6/1) -> (0/1,2/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.