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: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/4 1/2 2/3 1/1 3/2 5/3 19/11 2/1 7/3 3/1 4/1 5/1 17/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/5 -3/7 1/6 -2/5 1/5 2/9 -1/3 0/1 1/4 -1/4 2/7 1/3 -2/9 1/3 4/11 -1/5 1/2 0/1 0/1 1/3 1/5 0/1 1/2 1/4 0/1 3/11 0/1 1/6 2/7 0/1 1/5 1/3 1/4 1/2 1/3 2/5 5/9 2/5 1/2 4/7 2/5 3/7 3/5 1/2 2/3 1/2 5/7 1/2 8/11 3/5 2/3 11/15 1/2 2/3 3/4 2/3 1/1 1/1 0/1 1/2 4/3 1/3 2/5 3/2 1/2 8/5 3/5 2/3 13/8 2/3 1/1 18/11 0/1 1/1 5/3 1/2 12/7 4/7 3/5 19/11 3/5 26/15 3/5 8/13 7/4 3/5 2/3 2/1 2/3 1/1 7/3 1/1 12/5 1/1 10/9 5/2 1/1 4/3 3/1 1/0 7/2 -1/1 0/1 4/1 0/1 9/2 0/1 1/3 14/3 0/1 1/1 19/4 0/1 1/3 5/1 0/1 1/2 11/2 0/1 1/1 17/3 0/1 23/4 0/1 1/5 6/1 0/1 1/3 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(55,24,16,7) (-1/2,-3/7) -> (3/1,7/2) Hyperbolic Matrix(19,8,-88,-37) (-3/7,-2/5) -> (-2/9,-1/5) Hyperbolic Matrix(17,6,14,5) (-2/5,-1/3) -> (1/1,4/3) Hyperbolic Matrix(13,4,16,5) (-1/3,-1/4) -> (3/4,1/1) Hyperbolic Matrix(93,22,38,9) (-1/4,-2/9) -> (12/5,5/2) Hyperbolic Matrix(11,2,38,7) (-1/5,0/1) -> (2/7,1/3) Hyperbolic Matrix(25,-4,44,-7) (0/1,1/5) -> (5/9,4/7) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(165,-46,226,-63) (3/11,2/7) -> (8/11,11/15) Hyperbolic Matrix(21,-8,8,-3) (1/3,1/2) -> (5/2,3/1) Hyperbolic Matrix(109,-60,20,-11) (1/2,5/9) -> (5/1,11/2) Hyperbolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(181,-130,110,-79) (5/7,8/11) -> (18/11,5/3) Hyperbolic Matrix(305,-224,64,-47) (11/15,3/4) -> (19/4,5/1) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(157,-254,34,-55) (8/5,13/8) -> (9/2,14/3) Hyperbolic Matrix(285,-464,164,-267) (13/8,18/11) -> (26/15,7/4) Hyperbolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(25,-44,4,-7) (7/4,2/1) -> (6/1,1/0) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(93,-436,16,-75) (14/3,19/4) -> (23/4,6/1) Hyperbolic Matrix(103,-578,18,-101) (11/2,17/3) -> (17/3,23/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(55,24,16,7) -> Matrix(1,0,-6,1) Matrix(19,8,-88,-37) -> Matrix(11,-2,28,-5) Matrix(17,6,14,5) -> Matrix(1,0,-2,1) Matrix(13,4,16,5) -> Matrix(1,0,-2,1) Matrix(93,22,38,9) -> Matrix(19,-6,16,-5) Matrix(11,2,38,7) -> Matrix(1,0,2,1) Matrix(25,-4,44,-7) -> Matrix(3,-2,8,-5) Matrix(17,-4,64,-15) -> Matrix(1,0,4,1) Matrix(165,-46,226,-63) -> Matrix(13,-2,20,-3) Matrix(21,-8,8,-3) -> Matrix(7,-2,4,-1) Matrix(109,-60,20,-11) -> Matrix(5,-2,8,-3) Matrix(95,-56,56,-33) -> Matrix(13,-6,24,-11) Matrix(25,-16,36,-23) -> Matrix(9,-4,16,-7) Matrix(181,-130,110,-79) -> Matrix(3,-2,8,-5) Matrix(305,-224,64,-47) -> Matrix(3,-2,8,-5) Matrix(25,-36,16,-23) -> Matrix(9,-4,16,-7) Matrix(157,-254,34,-55) -> Matrix(3,-2,8,-5) Matrix(285,-464,164,-267) -> Matrix(11,-8,18,-13) Matrix(419,-722,242,-417) -> Matrix(61,-36,100,-59) Matrix(25,-44,4,-7) -> Matrix(3,-2,8,-5) Matrix(43,-98,18,-41) -> Matrix(13,-12,12,-11) Matrix(17,-64,4,-15) -> Matrix(1,0,4,1) Matrix(93,-436,16,-75) -> Matrix(1,0,2,1) Matrix(103,-578,18,-101) -> Matrix(1,0,4,1) 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 Degree of the the map X: 8 Degree of the the map Y: 24 Permutation triple for Y: ((1,6,17,15,21,22,23,16,7,2)(3,11,18,8,13,20,24,19,12,4)(5,14)(9,10); (1,4,8,7,5)(3,10,18,16,6)(9,19,17,23,20)(12,15,14,22,13); (1,2,8,10,20,22,21,12,9,3)(4,13)(5,15,19,24,23,14,7,18,11,6)(16,17)) ----------------------------------------------------------------------- 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 -1/1 1/1 3/2 5/3 19/11 2/1 7/3 3/1 4/1 5/1 17/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/3 1/5 0/1 1/2 1/4 0/1 1/3 1/4 1/2 1/3 2/5 5/9 2/5 1/2 4/7 2/5 3/7 3/5 1/2 2/3 1/2 1/1 0/1 1/2 3/2 1/2 5/3 1/2 12/7 4/7 3/5 19/11 3/5 7/4 3/5 2/3 2/1 2/3 1/1 7/3 1/1 5/2 1/1 4/3 3/1 1/0 4/1 0/1 5/1 0/1 1/2 11/2 0/1 1/1 17/3 0/1 6/1 0/1 1/3 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(25,-4,44,-7) (0/1,1/5) -> (5/9,4/7) Hyperbolic Matrix(40,-9,9,-2) (1/5,1/4) -> (4/1,5/1) Hyperbolic Matrix(24,-7,7,-2) (1/4,1/3) -> (3/1,4/1) Hyperbolic Matrix(21,-8,8,-3) (1/3,1/2) -> (5/2,3/1) Hyperbolic Matrix(109,-60,20,-11) (1/2,5/9) -> (5/1,11/2) Hyperbolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(30,-19,19,-12) (3/5,2/3) -> (3/2,5/3) Hyperbolic Matrix(6,-5,5,-4) (2/3,1/1) -> (1/1,3/2) Parabolic Matrix(210,-361,121,-208) (12/7,19/11) -> (19/11,7/4) Parabolic Matrix(25,-44,4,-7) (7/4,2/1) -> (6/1,1/0) Hyperbolic Matrix(22,-49,9,-20) (2/1,7/3) -> (7/3,5/2) Parabolic Matrix(52,-289,9,-50) (11/2,17/3) -> (17/3,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(25,-4,44,-7) -> Matrix(3,-2,8,-5) Matrix(40,-9,9,-2) -> Matrix(1,0,0,1) Matrix(24,-7,7,-2) -> Matrix(1,0,-4,1) Matrix(21,-8,8,-3) -> Matrix(7,-2,4,-1) Matrix(109,-60,20,-11) -> Matrix(5,-2,8,-3) Matrix(95,-56,56,-33) -> Matrix(13,-6,24,-11) Matrix(30,-19,19,-12) -> Matrix(9,-4,16,-7) Matrix(6,-5,5,-4) -> Matrix(1,0,0,1) Matrix(210,-361,121,-208) -> Matrix(31,-18,50,-29) Matrix(25,-44,4,-7) -> Matrix(3,-2,8,-5) Matrix(22,-49,9,-20) -> Matrix(7,-6,6,-5) Matrix(52,-289,9,-50) -> Matrix(1,0,2,1) 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): 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 -1/1 0/1 2 1 1/1 (0/1,1/2) 0 5 3/2 1/2 2 2 5/3 1/2 2 5 19/11 3/5 2 1 7/4 (3/5,2/3) 0 10 2/1 (2/3,1/1) 0 10 7/3 1/1 6 1 3/1 1/0 2 5 4/1 0/1 2 2 5/1 (0/1,1/2) 0 5 17/3 0/1 2 1 6/1 (0/1,1/3) 0 10 1/0 (0/1,1/1) 0 10 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(56,-95,33,-56) (5/3,19/11) -> (5/3,19/11) Reflection Matrix(153,-266,88,-153) (19/11,7/4) -> (19/11,7/4) Reflection Matrix(25,-44,4,-7) (7/4,2/1) -> (6/1,1/0) Hyperbolic Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(8,-21,3,-8) (7/3,3/1) -> (7/3,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection Matrix(16,-85,3,-16) (5/1,17/3) -> (5/1,17/3) Reflection Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,2,-1) (-1/1,1/0) -> (0/1,1/1) Matrix(0,1,1,0) -> Matrix(1,0,4,-1) (-1/1,1/1) -> (0/1,1/2) Matrix(5,-6,4,-5) -> Matrix(1,0,4,-1) (1/1,3/2) -> (0/1,1/2) Matrix(19,-30,12,-19) -> Matrix(7,-4,12,-7) (3/2,5/3) -> (1/2,2/3) Matrix(56,-95,33,-56) -> Matrix(11,-6,20,-11) (5/3,19/11) -> (1/2,3/5) Matrix(153,-266,88,-153) -> Matrix(19,-12,30,-19) (19/11,7/4) -> (3/5,2/3) Matrix(25,-44,4,-7) -> Matrix(3,-2,8,-5) 1/2 Matrix(13,-28,6,-13) -> Matrix(5,-4,6,-5) (2/1,7/3) -> (2/3,1/1) Matrix(8,-21,3,-8) -> Matrix(-1,2,0,1) (7/3,3/1) -> (1/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,0,-1) (3/1,4/1) -> (0/1,1/0) Matrix(9,-40,2,-9) -> Matrix(1,0,4,-1) (4/1,5/1) -> (0/1,1/2) Matrix(16,-85,3,-16) -> Matrix(1,0,4,-1) (5/1,17/3) -> (0/1,1/2) Matrix(35,-204,6,-35) -> Matrix(1,0,6,-1) (17/3,6/1) -> (0/1,1/3) ----------------------------------------------------------------------- The pullback map has no extra symmetries.