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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/14 -3/14 -1/6 -1/7 0/1 1/5 1/4 2/7 1/3 2/5 3/7 1/2 4/7 2/3 5/7 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -4/5 -3/4 -2/3 -3/4 -3/5 -1/2 -5/7 -1/2 -2/3 -1/2 -2/5 -5/8 -1/2 -1/3 -3/5 -3/8 -4/7 -1/3 -1/2 -1/3 -1/4 -3/7 -1/4 -2/5 -1/4 -2/9 -5/13 -5/24 -8/21 -1/5 -3/8 -1/5 -1/6 -4/11 -1/6 0/1 -5/14 0/1 -1/3 -1/4 -2/7 -1/5 -1/4 -1/5 -1/6 -2/9 -1/8 0/1 -3/14 0/1 -1/5 -1/4 -1/6 -1/5 -1/6 -1/7 -1/6 0/1 -1/6 0/1 1/5 -1/8 1/4 -1/6 -1/7 2/7 -1/7 1/3 -1/8 3/8 -1/6 -1/7 2/5 -2/15 -1/8 3/7 -1/8 1/2 -1/8 -1/9 4/7 -1/9 3/5 -3/28 8/13 -2/19 -1/10 13/21 -1/10 5/8 -1/9 -1/10 7/11 -3/28 9/14 -2/19 2/3 -2/19 -1/10 5/7 -1/10 3/4 -1/10 -3/31 7/9 -5/52 11/14 -2/21 4/5 -2/21 -3/32 5/6 -5/54 -1/11 6/7 -1/11 1/1 -1/12 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(27,22,-70,-57) (-1/1,-4/5) -> (-2/5,-5/13) Hyperbolic Matrix(13,10,-56,-43) (-4/5,-3/4) -> (-1/4,-2/9) Hyperbolic Matrix(41,30,56,41) (-3/4,-5/7) -> (5/7,3/4) Hyperbolic Matrix(29,20,42,29) (-5/7,-2/3) -> (2/3,5/7) Hyperbolic Matrix(41,26,-112,-71) (-2/3,-5/8) -> (-3/8,-4/11) Hyperbolic Matrix(13,8,-70,-43) (-5/8,-3/5) -> (-1/5,-1/6) Hyperbolic Matrix(41,24,70,41) (-3/5,-4/7) -> (4/7,3/5) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(13,6,28,13) (-1/2,-3/7) -> (3/7,1/2) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(99,38,112,43) (-5/13,-8/21) -> (6/7,1/1) Hyperbolic Matrix(153,58,182,69) (-8/21,-3/8) -> (5/6,6/7) Hyperbolic Matrix(127,46,196,71) (-4/11,-5/14) -> (9/14,2/3) Hyperbolic Matrix(125,44,196,69) (-5/14,-1/3) -> (7/11,9/14) Hyperbolic Matrix(13,4,42,13) (-1/3,-2/7) -> (2/7,1/3) Hyperbolic Matrix(15,4,56,15) (-2/7,-1/4) -> (1/4,2/7) Hyperbolic Matrix(155,34,196,43) (-2/9,-3/14) -> (11/14,4/5) Hyperbolic Matrix(153,32,196,41) (-3/14,-1/5) -> (7/9,11/14) Hyperbolic Matrix(113,18,182,29) (-1/6,-1/7) -> (13/21,5/8) Hyperbolic Matrix(69,8,112,13) (-1/7,0/1) -> (8/13,13/21) Hyperbolic Matrix(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(43,-10,56,-13) (1/5,1/4) -> (3/4,7/9) Hyperbolic Matrix(71,-26,112,-41) (1/3,3/8) -> (5/8,7/11) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-12,1) Matrix(27,22,-70,-57) -> Matrix(5,4,-24,-19) Matrix(13,10,-56,-43) -> Matrix(3,2,-20,-13) Matrix(41,30,56,41) -> Matrix(11,6,-112,-61) Matrix(29,20,42,29) -> Matrix(9,4,-88,-39) Matrix(41,26,-112,-71) -> Matrix(5,2,-28,-11) Matrix(13,8,-70,-43) -> Matrix(5,2,-28,-11) Matrix(41,24,70,41) -> Matrix(17,6,-156,-55) Matrix(15,8,28,15) -> Matrix(7,2,-60,-17) Matrix(13,6,28,13) -> Matrix(7,2,-60,-17) Matrix(29,12,70,29) -> Matrix(17,4,-132,-31) Matrix(99,38,112,43) -> Matrix(29,6,-324,-67) Matrix(153,58,182,69) -> Matrix(31,6,-336,-65) Matrix(127,46,196,71) -> Matrix(13,2,-124,-19) Matrix(125,44,196,69) -> Matrix(5,2,-48,-19) Matrix(13,4,42,13) -> Matrix(9,2,-68,-15) Matrix(15,4,56,15) -> Matrix(11,2,-72,-13) Matrix(155,34,196,43) -> Matrix(19,2,-200,-21) Matrix(153,32,196,41) -> Matrix(3,2,-32,-21) Matrix(113,18,182,29) -> Matrix(1,0,-4,1) Matrix(69,8,112,13) -> Matrix(13,2,-124,-19) Matrix(43,-8,70,-13) -> Matrix(13,2,-124,-19) Matrix(43,-10,56,-13) -> Matrix(11,2,-116,-21) Matrix(71,-26,112,-41) -> Matrix(13,2,-124,-19) Matrix(57,-22,70,-27) -> Matrix(29,4,-312,-43) 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: 24 Permutation triple for Y: ((1,4,16,24,17,5,2)(3,10,8,7,19,23,11)(6,20,18,22,9,14,13); (1,2,8,20,21,9,3)(4,14,23,19,6,5,15)(7,12,11,16,22,18,17); (2,6,13,4,3,12,7)(5,18,8,10,9,16,15)(11,14,21,20,19,17,24)) ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/5 2/7 1/3 3/7 1/2 4/7 13/21 2/3 5/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -2/3 -1/2 -2/5 -3/5 -3/8 -4/7 -1/3 -1/2 -1/3 -1/4 -1/3 -1/4 -2/7 -1/5 -1/4 -1/5 -1/6 0/1 -1/6 0/1 1/5 -1/8 1/4 -1/6 -1/7 2/7 -1/7 1/3 -1/8 2/5 -2/15 -1/8 3/7 -1/8 1/2 -1/8 -1/9 4/7 -1/9 3/5 -3/28 8/13 -2/19 -1/10 13/21 -1/10 5/8 -1/9 -1/10 2/3 -2/19 -1/10 5/7 -1/10 3/4 -1/10 -3/31 1/1 -1/12 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(6,5,7,6) (-1/1,-2/3) -> (3/4,1/1) Hyperbolic Matrix(8,5,35,22) (-2/3,-3/5) -> (1/5,1/4) Hyperbolic Matrix(41,24,70,41) (-3/5,-4/7) -> (4/7,3/5) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(8,3,21,8) (-1/2,-1/3) -> (1/3,2/5) Hyperbolic Matrix(13,4,42,13) (-1/3,-2/7) -> (2/7,1/3) Hyperbolic Matrix(15,4,56,15) (-2/7,-1/4) -> (1/4,2/7) Hyperbolic Matrix(22,5,35,8) (-1/4,0/1) -> (5/8,2/3) Hyperbolic Matrix(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(22,-9,49,-20) (2/5,3/7) -> (3/7,1/2) Parabolic Matrix(274,-169,441,-272) (8/13,13/21) -> (13/21,5/8) Parabolic Matrix(36,-25,49,-34) (2/3,5/7) -> (5/7,3/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-12,1) Matrix(6,5,7,6) -> Matrix(1,1,-12,-11) Matrix(8,5,35,22) -> Matrix(3,1,-16,-5) Matrix(41,24,70,41) -> Matrix(17,6,-156,-55) Matrix(15,8,28,15) -> Matrix(7,2,-60,-17) Matrix(8,3,21,8) -> Matrix(5,1,-36,-7) Matrix(13,4,42,13) -> Matrix(9,2,-68,-15) Matrix(15,4,56,15) -> Matrix(11,2,-72,-13) Matrix(22,5,35,8) -> Matrix(7,1,-64,-9) Matrix(43,-8,70,-13) -> Matrix(13,2,-124,-19) Matrix(22,-9,49,-20) -> Matrix(23,3,-192,-25) Matrix(274,-169,441,-272) -> Matrix(9,1,-100,-11) Matrix(36,-25,49,-34) -> Matrix(49,5,-500,-51) 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): 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 0/1 (-1/6,0/1) 0 7 1/4 (-1/6,-1/7) 0 7 2/7 -1/7 2 1 1/3 -1/8 1 7 3/7 -1/8 3 1 1/2 (-1/8,-1/9) 0 7 4/7 -1/9 4 1 3/5 -3/28 1 7 13/21 -1/10 1 1 5/8 (-1/9,-1/10) 0 7 2/3 (-2/19,-1/10) 0 7 5/7 -1/10 5 1 1/1 -1/12 1 7 1/0 0/1 6 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(22,-5,35,-8) (0/1,1/4) -> (5/8,2/3) Glide Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(13,-4,42,-13) (2/7,1/3) -> (2/7,1/3) Reflection Matrix(8,-3,21,-8) (1/3,3/7) -> (1/3,3/7) Reflection Matrix(13,-6,28,-13) (3/7,1/2) -> (3/7,1/2) Reflection Matrix(15,-8,28,-15) (1/2,4/7) -> (1/2,4/7) Reflection Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(64,-39,105,-64) (3/5,13/21) -> (3/5,13/21) Reflection Matrix(209,-130,336,-209) (13/21,5/8) -> (13/21,5/8) Reflection Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection Matrix(6,-5,7,-6) (5/7,1/1) -> (5/7,1/1) 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(-1,0,12,1) (0/1,1/0) -> (-1/6,0/1) Matrix(22,-5,35,-8) -> Matrix(5,1,-44,-9) Matrix(15,-4,56,-15) -> Matrix(13,2,-84,-13) (1/4,2/7) -> (-1/6,-1/7) Matrix(13,-4,42,-13) -> Matrix(15,2,-112,-15) (2/7,1/3) -> (-1/7,-1/8) Matrix(8,-3,21,-8) -> Matrix(7,1,-48,-7) (1/3,3/7) -> (-1/6,-1/8) Matrix(13,-6,28,-13) -> Matrix(17,2,-144,-17) (3/7,1/2) -> (-1/8,-1/9) Matrix(15,-8,28,-15) -> Matrix(17,2,-144,-17) (1/2,4/7) -> (-1/8,-1/9) Matrix(41,-24,70,-41) -> Matrix(55,6,-504,-55) (4/7,3/5) -> (-1/9,-3/28) Matrix(64,-39,105,-64) -> Matrix(29,3,-280,-29) (3/5,13/21) -> (-3/28,-1/10) Matrix(209,-130,336,-209) -> Matrix(19,2,-180,-19) (13/21,5/8) -> (-1/9,-1/10) Matrix(29,-20,42,-29) -> Matrix(39,4,-380,-39) (2/3,5/7) -> (-2/19,-1/10) Matrix(6,-5,7,-6) -> Matrix(11,1,-120,-11) (5/7,1/1) -> (-1/10,-1/12) Matrix(-1,2,0,1) -> Matrix(-1,0,24,1) (1/1,1/0) -> (-1/12,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.