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 -10/3 -2/1 -8/5 -4/3 -8/7 0/1 1/1 4/3 3/2 2/1 8/3 14/5 3/1 16/5 10/3 4/1 32/7 14/3 16/3 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 0/1 -5/1 1/0 -4/1 -1/2 -11/3 -3/8 -18/5 -3/8 -1/3 -7/2 -1/4 -10/3 -1/2 -1/3 -3/1 -1/4 -2/1 -1/4 0/1 -5/3 -1/4 -8/5 -1/4 -11/7 -1/4 -14/9 -1/4 -1/5 -3/2 -1/4 -16/11 -1/5 -13/9 -3/16 -10/7 -1/5 -1/6 -17/12 -3/16 -24/17 -1/6 -7/5 -1/4 -4/3 -1/6 -9/7 -3/20 -32/25 -1/7 -23/18 -5/36 -14/11 -1/7 -1/8 -5/4 -1/8 -16/13 0/1 -11/9 -1/4 -6/5 -1/6 0/1 -7/6 -1/4 -8/7 -1/6 -1/1 -1/8 0/1 0/1 1/1 1/8 6/5 0/1 1/6 5/4 1/8 4/3 1/6 11/8 3/16 18/13 3/16 1/5 7/5 1/4 10/7 1/6 1/5 3/2 1/4 2/1 0/1 1/4 5/2 1/4 8/3 1/4 11/4 1/4 14/5 1/4 1/3 3/1 1/4 16/5 1/3 13/4 3/8 10/3 1/3 1/2 17/5 3/8 24/7 1/2 7/2 1/4 4/1 1/2 9/2 3/4 32/7 1/1 23/5 5/4 14/3 1/1 1/0 5/1 1/0 16/3 0/1 11/2 1/4 6/1 0/1 1/2 7/1 1/4 8/1 1/2 1/0 1/0 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(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(57,208,20,73) (-11/3,-18/5) -> (14/5,3/1) Hyperbolic Matrix(143,512,-112,-401) (-18/5,-7/2) -> (-23/18,-14/11) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(79,128,-50,-81) (-5/3,-8/5) -> (-8/5,-11/7) Parabolic Matrix(143,224,30,47) (-11/7,-14/9) -> (14/3,5/1) Hyperbolic Matrix(135,208,98,151) (-14/9,-3/2) -> (11/8,18/13) 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(799,1024,174,223) (-9/7,-32/25) -> (32/7,23/5) Hyperbolic Matrix(801,1024,176,225) (-32/25,-23/18) -> (9/2,32/7) Hyperbolic Matrix(177,224,64,81) (-14/11,-5/4) -> (11/4,14/5) Hyperbolic 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(41,-48,6,-7) (1/1,6/5) -> (6/1,7/1) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,48,-6,-41) -> Matrix(1,0,-4,1) Matrix(17,96,-14,-79) -> Matrix(1,0,-4,1) Matrix(15,64,-4,-17) -> Matrix(3,2,-8,-5) Matrix(57,208,20,73) -> Matrix(5,2,12,5) Matrix(143,512,-112,-401) -> Matrix(11,4,-80,-29) Matrix(71,240,-50,-169) -> Matrix(5,2,-28,-11) Matrix(49,160,-34,-111) -> Matrix(5,2,-28,-11) Matrix(7,16,-4,-9) -> Matrix(1,0,0,1) Matrix(79,128,-50,-81) -> Matrix(7,2,-32,-9) Matrix(143,224,30,47) -> Matrix(9,2,4,1) Matrix(135,208,98,151) -> Matrix(11,2,60,11) Matrix(175,256,54,79) -> Matrix(19,4,52,11) Matrix(177,256,56,81) -> Matrix(21,4,68,13) Matrix(113,160,12,17) -> Matrix(11,2,16,3) Matrix(159,224,22,31) -> Matrix(1,0,8,1) Matrix(47,64,-36,-49) -> Matrix(11,2,-72,-13) Matrix(799,1024,174,223) -> Matrix(55,8,48,7) Matrix(801,1024,176,225) -> Matrix(57,8,64,9) Matrix(177,224,64,81) -> Matrix(15,2,52,7) Matrix(207,256,38,47) -> Matrix(1,0,12,1) Matrix(209,256,40,49) -> Matrix(1,0,4,1) Matrix(193,224,56,65) -> Matrix(1,0,8,1) Matrix(143,160,42,47) -> Matrix(13,2,32,5) Matrix(1,0,2,1) -> Matrix(1,0,16,1) Matrix(41,-48,6,-7) -> Matrix(1,0,-4,1) Matrix(79,-96,14,-17) -> Matrix(1,0,-4,1) Matrix(49,-64,36,-47) -> Matrix(13,-2,72,-11) Matrix(369,-512,80,-111) -> Matrix(21,-4,16,-3) Matrix(169,-240,50,-71) -> Matrix(11,-2,28,-5) Matrix(111,-160,34,-49) -> Matrix(11,-2,28,-5) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(49,-128,18,-47) -> Matrix(9,-2,32,-7) Matrix(17,-64,4,-15) -> Matrix(5,-2,8,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 6 Degree of the the map X: 12 Degree of the the map Y: 32 Permutation triple for Y: ((2,6)(3,4)(5,14)(7,18)(8,9)(11,12)(13,19)(15,17)(20,21)(22,26)(23,24)(25,29); (1,4,12,20,30,15,14,29,32,26,11,23,27,13,5,2)(3,9,24,31,17,7,6,16,22,8,21,28,19,18,25,10); (1,2,7,19,27,23,9,22,32,29,18,17,30,20,8,3)(4,10,25,14,13,28,21,12,26,16,6,5,15,31,24,11)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1 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): 48 Minimal number of generators: 9 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: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 4/3 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 0/1 0/1 1/1 1/8 4/3 1/6 7/5 1/4 10/7 1/6 1/5 3/2 1/4 2/1 0/1 1/4 5/2 1/4 8/3 1/4 3/1 1/4 4/1 1/2 5/1 1/0 16/3 0/1 11/2 1/4 6/1 0/1 1/2 7/1 1/4 8/1 1/2 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(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(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(49,-256,9,-47) (5/1,16/3) -> (16/3,11/2) Parabolic Matrix(9,-64,1,-7) (7/1,8/1) -> (8/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,8,1) Matrix(25,-32,18,-23) -> Matrix(7,-1,36,-5) Matrix(79,-112,12,-17) -> Matrix(5,-1,16,-3) Matrix(89,-128,16,-23) -> Matrix(5,-1,16,-3) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) Matrix(25,-64,9,-23) -> Matrix(5,-1,16,-3) Matrix(9,-32,2,-7) -> Matrix(3,-1,4,-1) Matrix(49,-256,9,-47) -> Matrix(1,0,4,1) Matrix(9,-64,1,-7) -> Matrix(3,-1,4,-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 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): 3 ----------------------------------------------------------------------- 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 8 1 2/1 (0/1,1/4) 0 8 8/3 1/4 2 2 3/1 1/4 1 16 4/1 1/2 2 4 5/1 1/0 1 16 16/3 0/1 4 1 6/1 (0/1,1/2) 0 8 8/1 1/2 2 2 1/0 1/0 1 16 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(7,-16,3,-7) (2/1,8/3) -> (2/1,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) Reflection Matrix(-1,16,0,1) (8/1,1/0) -> (8/1,1/0) 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(1,0,1,-1) -> Matrix(1,0,8,-1) (0/1,2/1) -> (0/1,1/4) Matrix(7,-16,3,-7) -> Matrix(1,0,8,-1) (2/1,8/3) -> (0/1,1/4) Matrix(17,-48,6,-17) -> Matrix(3,-1,8,-3) (8/3,3/1) -> (1/4,1/2) Matrix(9,-32,2,-7) -> Matrix(3,-1,4,-1) 1/2 Matrix(31,-160,6,-31) -> Matrix(1,0,0,-1) (5/1,16/3) -> (0/1,1/0) Matrix(17,-96,3,-17) -> Matrix(1,0,4,-1) (16/3,6/1) -> (0/1,1/2) Matrix(7,-48,1,-7) -> Matrix(1,0,4,-1) (6/1,8/1) -> (0/1,1/2) Matrix(-1,16,0,1) -> Matrix(-1,1,0,1) (8/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.