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 -3/4 -2/3 -1/2 -1/3 -1/4 -2/9 -2/11 0/1 1/6 1/5 2/9 1/4 2/7 1/3 2/5 5/12 1/2 13/24 7/12 2/3 3/4 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -5/6 1/1 -4/5 1/1 1/0 -3/4 1/1 -2/3 0/1 1/1 1/0 -5/8 1/1 -8/13 1/1 2/1 -3/5 2/1 1/0 -7/12 1/0 -4/7 -1/1 1/0 -1/2 1/0 -5/11 -1/1 0/1 -4/9 -1/1 0/1 1/0 -3/7 0/1 1/0 -5/12 1/0 -2/5 -1/1 1/0 -3/8 -1/1 -4/11 -1/1 0/1 -1/3 0/1 -4/13 0/1 1/1 -11/36 1/1 -7/23 1/1 2/1 -3/10 1/0 -2/7 -1/1 1/0 -5/18 -1/1 -3/11 -1/1 0/1 -1/4 0/1 -3/13 0/1 1/1 -2/9 0/1 1/1 1/0 -1/5 0/1 1/0 -2/11 -1/1 0/1 -1/6 0/1 0/1 0/1 1/0 1/6 0/1 1/5 0/1 1/0 2/9 -1/1 0/1 1/0 1/4 0/1 4/15 0/1 1/2 1/1 3/11 0/1 1/1 2/7 1/1 1/0 1/3 0/1 4/11 0/1 1/1 3/8 1/1 2/5 1/1 1/0 5/12 1/0 3/7 0/1 1/0 4/9 0/1 1/1 1/0 5/11 0/1 1/1 1/2 1/0 7/13 -2/1 -1/1 13/24 -1/1 6/11 -1/1 0/1 5/9 0/1 4/7 1/1 1/0 7/12 1/0 3/5 -2/1 1/0 2/3 -1/1 0/1 1/0 5/7 -2/1 1/0 13/18 -2/1 8/11 -2/1 -1/1 3/4 -1/1 4/5 -1/1 1/0 5/6 -1/1 6/7 -1/1 -1/2 1/1 -1/1 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(23,20,-84,-73) (-1/1,-5/6) -> (-5/18,-3/11) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(23,18,60,47) (-4/5,-3/4) -> (3/8,2/5) Hyperbolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(71,44,192,119) (-5/8,-8/13) -> (4/11,3/8) Hyperbolic Matrix(23,14,-120,-73) (-8/13,-3/5) -> (-1/5,-2/11) Hyperbolic Matrix(71,42,120,71) (-3/5,-7/12) -> (7/12,3/5) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(25,14,-84,-47) (-4/7,-1/2) -> (-3/10,-2/7) Hyperbolic Matrix(47,22,-156,-73) (-1/2,-5/11) -> (-7/23,-3/10) Hyperbolic Matrix(71,32,264,119) (-5/11,-4/9) -> (4/15,3/11) Hyperbolic Matrix(23,10,108,47) (-4/9,-3/7) -> (1/5,2/9) Hyperbolic Matrix(71,30,168,71) (-3/7,-5/12) -> (5/12,3/7) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(47,18,60,23) (-2/5,-3/8) -> (3/4,4/5) Hyperbolic Matrix(97,36,132,49) (-3/8,-4/11) -> (8/11,3/4) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(385,118,708,217) (-4/13,-11/36) -> (13/24,6/11) Hyperbolic Matrix(551,168,1020,311) (-11/36,-7/23) -> (7/13,13/24) Hyperbolic Matrix(143,40,168,47) (-2/7,-5/18) -> (5/6,6/7) Hyperbolic Matrix(23,6,-96,-25) (-3/11,-1/4) -> (-1/4,-3/13) Parabolic Matrix(97,22,216,49) (-3/13,-2/9) -> (4/9,5/11) Hyperbolic Matrix(47,10,108,23) (-2/9,-1/5) -> (3/7,4/9) Hyperbolic Matrix(191,34,264,47) (-2/11,-1/6) -> (13/18,8/11) Hyperbolic Matrix(1,0,12,1) (-1/6,0/1) -> (0/1,1/6) Parabolic Matrix(95,-18,132,-25) (1/6,1/5) -> (5/7,13/18) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(73,-20,84,-23) (3/11,2/7) -> (6/7,1/1) Hyperbolic Matrix(47,-14,84,-25) (2/7,1/3) -> (5/9,4/7) Hyperbolic Matrix(73,-26,132,-47) (1/3,4/11) -> (6/11,5/9) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(23,20,-84,-73) -> Matrix(1,0,-2,1) Matrix(49,40,60,49) -> Matrix(1,-2,0,1) Matrix(23,18,60,47) -> Matrix(1,0,0,1) Matrix(23,16,-36,-25) -> Matrix(1,0,0,1) Matrix(71,44,192,119) -> Matrix(1,-2,2,-3) Matrix(23,14,-120,-73) -> Matrix(1,-2,0,1) Matrix(71,42,120,71) -> Matrix(1,-4,0,1) Matrix(97,56,168,97) -> Matrix(1,2,0,1) Matrix(25,14,-84,-47) -> Matrix(1,0,0,1) Matrix(47,22,-156,-73) -> Matrix(1,2,0,1) Matrix(71,32,264,119) -> Matrix(1,0,2,1) Matrix(23,10,108,47) -> Matrix(1,0,0,1) Matrix(71,30,168,71) -> Matrix(1,0,0,1) Matrix(49,20,120,49) -> Matrix(1,2,0,1) Matrix(47,18,60,23) -> Matrix(1,0,0,1) Matrix(97,36,132,49) -> Matrix(3,2,-2,-1) Matrix(23,8,-72,-25) -> Matrix(1,0,2,1) Matrix(385,118,708,217) -> Matrix(1,0,-2,1) Matrix(551,168,1020,311) -> Matrix(3,-4,-2,3) Matrix(143,40,168,47) -> Matrix(1,2,-2,-3) Matrix(23,6,-96,-25) -> Matrix(1,0,2,1) Matrix(97,22,216,49) -> Matrix(1,0,0,1) Matrix(47,10,108,23) -> Matrix(1,0,0,1) Matrix(191,34,264,47) -> Matrix(3,2,-2,-1) Matrix(1,0,12,1) -> Matrix(1,0,0,1) Matrix(95,-18,132,-25) -> Matrix(1,-2,0,1) Matrix(25,-6,96,-23) -> Matrix(1,0,2,1) Matrix(73,-20,84,-23) -> Matrix(1,0,-2,1) Matrix(47,-14,84,-25) -> Matrix(1,0,0,1) Matrix(73,-26,132,-47) -> Matrix(1,0,-2,1) Matrix(25,-12,48,-23) -> Matrix(1,-2,0,1) Matrix(25,-16,36,-23) -> 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: ((1,4,15,16,5,2)(3,10)(6,9,25,13,12,21)(7,8)(11,23,19,18,26,14)(17,29)(20,31,32,28,24,27)(22,30); (1,2,8,25,18,29,32,31,30,26,9,3)(4,13,28,14)(5,19,20,6)(7,23,11,10,27,15,22,21,12,17,16,24); (2,6,22,31,19,7)(3,11,28,29,12,4)(5,17,18)(8,24,13)(9,20,10)(14,30,15)(16,27)(25,26)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 (0/1,1/0) 0 12 1/6 0/1 1 2 1/5 (0/1,1/0) 0 12 2/9 0 4 1/4 0/1 1 3 4/15 0 4 3/11 (0/1,1/1) 0 12 2/7 (1/1,1/0) 0 12 1/3 0/1 1 4 4/11 (0/1,1/1) 0 12 3/8 1/1 1 3 2/5 (1/1,1/0) 0 12 5/12 1/0 1 1 3/7 (0/1,1/0) 0 12 4/9 0 4 5/11 (0/1,1/1) 0 12 1/2 1/0 1 6 7/13 (-2/1,-1/1) 0 12 13/24 -1/1 2 1 6/11 (-1/1,0/1) 0 12 5/9 0/1 1 4 4/7 (1/1,1/0) 0 12 7/12 1/0 3 1 3/5 (-2/1,1/0) 0 12 2/3 0 4 5/7 (-2/1,1/0) 0 12 13/18 -2/1 1 2 8/11 (-2/1,-1/1) 0 12 3/4 -1/1 1 3 4/5 (-1/1,1/0) 0 12 5/6 -1/1 2 2 6/7 (-1/1,-1/2) 0 12 1/1 (-1/1,0/1) 0 12 1/0 0/1 1 1 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(1,0,12,-1) (0/1,1/6) -> (0/1,1/6) Reflection Matrix(95,-18,132,-25) (1/6,1/5) -> (5/7,13/18) Hyperbolic Matrix(47,-10,108,-23) (1/5,2/9) -> (3/7,4/9) Glide Reflection Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(119,-32,264,-71) (4/15,3/11) -> (4/9,5/11) Glide Reflection Matrix(73,-20,84,-23) (3/11,2/7) -> (6/7,1/1) Hyperbolic Matrix(47,-14,84,-25) (2/7,1/3) -> (5/9,4/7) Hyperbolic Matrix(73,-26,132,-47) (1/3,4/11) -> (6/11,5/9) Hyperbolic Matrix(97,-36,132,-49) (4/11,3/8) -> (8/11,3/4) Glide Reflection Matrix(47,-18,60,-23) (3/8,2/5) -> (3/4,4/5) Glide Reflection Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(337,-182,624,-337) (7/13,13/24) -> (7/13,13/24) Reflection Matrix(287,-156,528,-287) (13/24,6/11) -> (13/24,6/11) Reflection Matrix(97,-56,168,-97) (4/7,7/12) -> (4/7,7/12) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(287,-208,396,-287) (13/18,8/11) -> (13/18,8/11) Reflection Matrix(49,-40,60,-49) (4/5,5/6) -> (4/5,5/6) Reflection Matrix(71,-60,84,-71) (5/6,6/7) -> (5/6,6/7) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(1,0,12,-1) -> Matrix(1,0,0,-1) (0/1,1/6) -> (0/1,1/0) Matrix(95,-18,132,-25) -> Matrix(1,-2,0,1) 1/0 Matrix(47,-10,108,-23) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(25,-6,96,-23) -> Matrix(1,0,2,1) 0/1 Matrix(119,-32,264,-71) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(73,-20,84,-23) -> Matrix(1,0,-2,1) 0/1 Matrix(47,-14,84,-25) -> Matrix(1,0,0,1) Matrix(73,-26,132,-47) -> Matrix(1,0,-2,1) 0/1 Matrix(97,-36,132,-49) -> Matrix(3,-2,-2,1) Matrix(47,-18,60,-23) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(49,-20,120,-49) -> Matrix(-1,2,0,1) (2/5,5/12) -> (1/1,1/0) Matrix(71,-30,168,-71) -> Matrix(1,0,0,-1) (5/12,3/7) -> (0/1,1/0) Matrix(25,-12,48,-23) -> Matrix(1,-2,0,1) 1/0 Matrix(337,-182,624,-337) -> Matrix(3,4,-2,-3) (7/13,13/24) -> (-2/1,-1/1) Matrix(287,-156,528,-287) -> Matrix(-1,0,2,1) (13/24,6/11) -> (-1/1,0/1) Matrix(97,-56,168,-97) -> Matrix(-1,2,0,1) (4/7,7/12) -> (1/1,1/0) Matrix(71,-42,120,-71) -> Matrix(1,4,0,-1) (7/12,3/5) -> (-2/1,1/0) Matrix(25,-16,36,-23) -> Matrix(1,0,0,1) Matrix(287,-208,396,-287) -> Matrix(3,4,-2,-3) (13/18,8/11) -> (-2/1,-1/1) Matrix(49,-40,60,-49) -> Matrix(1,2,0,-1) (4/5,5/6) -> (-1/1,1/0) Matrix(71,-60,84,-71) -> Matrix(3,2,-4,-3) (5/6,6/7) -> (-1/1,-1/2) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.