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: 12 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 4/11 1/2 1/1 3/2 2/1 7/3 11/4 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/30 -3/7 4/105 -8/19 11/285 -5/12 7/180 -2/5 1/25 -1/3 2/45 -1/4 1/20 -2/9 7/135 -3/14 11/210 -1/5 4/75 0/1 1/15 1/3 4/45 4/11 1/11 7/19 26/285 3/8 11/120 2/5 7/75 3/7 2/21 4/9 13/135 1/2 1/10 3/5 8/75 5/8 13/120 2/3 1/9 1/1 2/15 4/3 7/45 11/8 19/120 7/5 4/25 3/2 1/6 11/7 6/35 19/12 31/180 8/5 13/75 5/3 8/45 2/1 1/5 9/4 13/60 16/7 23/105 7/3 2/9 12/5 17/75 5/2 7/30 8/3 11/45 11/4 1/4 14/5 19/75 3/1 4/15 4/1 1/3 9/2 11/30 14/3 17/45 19/4 23/60 5/1 2/5 11/2 13/30 6/1 7/15 1/0 1/0 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(41,18,66,29) (-1/2,-3/7) -> (3/5,5/8) Hyperbolic Matrix(155,66,54,23) (-3/7,-8/19) -> (14/5,3/1) Hyperbolic Matrix(339,142,74,31) (-8/19,-5/12) -> (9/2,14/3) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(15,4,26,7) (-1/3,-1/4) -> (1/2,3/5) Hyperbolic Matrix(25,6,54,13) (-1/4,-2/9) -> (4/9,1/2) Hyperbolic Matrix(283,62,178,39) (-2/9,-3/14) -> (19/12,8/5) Hyperbolic Matrix(113,24,306,65) (-3/14,-1/5) -> (7/19,3/8) Hyperbolic Matrix(45,8,28,5) (-1/5,0/1) -> (8/5,5/3) Hyperbolic Matrix(13,-4,10,-3) (0/1,1/3) -> (1/1,4/3) Hyperbolic Matrix(89,-32,242,-87) (1/3,4/11) -> (4/11,7/19) Parabolic Matrix(87,-34,64,-25) (3/8,2/5) -> (4/3,11/8) Hyperbolic Matrix(119,-50,50,-21) (2/5,3/7) -> (7/3,12/5) Hyperbolic Matrix(175,-76,76,-33) (3/7,4/9) -> (16/7,7/3) Hyperbolic Matrix(59,-38,14,-9) (5/8,2/3) -> (4/1,9/2) Hyperbolic Matrix(13,-10,4,-3) (2/3,1/1) -> (3/1,4/1) Hyperbolic Matrix(95,-132,18,-25) (11/8,7/5) -> (5/1,11/2) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(193,-304,40,-63) (11/7,19/12) -> (19/4,5/1) Hyperbolic Matrix(189,-430,40,-91) (9/4,16/7) -> (14/3,19/4) Hyperbolic Matrix(67,-162,12,-29) (12/5,5/2) -> (11/2,6/1) Hyperbolic Matrix(15,-38,2,-5) (5/2,8/3) -> (6/1,1/0) Hyperbolic Matrix(89,-242,32,-87) (8/3,11/4) -> (11/4,14/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,30,1) Matrix(41,18,66,29) -> Matrix(107,-4,990,-37) Matrix(155,66,54,23) -> Matrix(209,-8,810,-31) Matrix(339,142,74,31) -> Matrix(413,-16,1110,-43) Matrix(69,28,32,13) -> Matrix(101,-4,480,-19) Matrix(31,12,18,7) -> Matrix(49,-2,270,-11) Matrix(15,4,26,7) -> Matrix(41,-2,390,-19) Matrix(25,6,54,13) -> Matrix(79,-4,810,-41) Matrix(283,62,178,39) -> Matrix(461,-24,2670,-139) Matrix(113,24,306,65) -> Matrix(419,-22,4590,-241) Matrix(45,8,28,5) -> Matrix(73,-4,420,-23) Matrix(13,-4,10,-3) -> Matrix(23,-2,150,-13) Matrix(89,-32,242,-87) -> Matrix(331,-30,3630,-329) Matrix(87,-34,64,-25) -> Matrix(151,-14,960,-89) Matrix(119,-50,50,-21) -> Matrix(169,-16,750,-71) Matrix(175,-76,76,-33) -> Matrix(251,-24,1140,-109) Matrix(59,-38,14,-9) -> Matrix(73,-8,210,-23) Matrix(13,-10,4,-3) -> Matrix(17,-2,60,-7) Matrix(95,-132,18,-25) -> Matrix(113,-18,270,-43) Matrix(37,-54,24,-35) -> Matrix(61,-10,360,-59) Matrix(193,-304,40,-63) -> Matrix(233,-40,600,-103) Matrix(189,-430,40,-91) -> Matrix(229,-50,600,-131) Matrix(67,-162,12,-29) -> Matrix(79,-18,180,-41) Matrix(15,-38,2,-5) -> Matrix(17,-4,30,-7) Matrix(89,-242,32,-87) -> Matrix(121,-30,480,-119) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 12 Genus: 7 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 24 Degree of the the map Y: 24 Permutation triple for Y: ((1,6,13,16,15,19,10,21,20,22,17,23,12,7,2)(3,11,18,8,4)(5,9,14); (1,4,6,16,24,17,12,8,7,14,18,21,10,3,5)(9,19,13,23,20)(11,15,22); (1,2,8,14,20,18,22,23,24,16,19,11,10,9,3)(4,12,13)(5,7,17,15,6)) ----------------------------------------------------------------------- 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, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+Z/2Z. ----------------------------------------------------------------------- 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): 24 Minimal number of generators: 5 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: 4 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 2/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 1/15 1/2 1/10 2/3 1/9 1/1 2/15 3/2 1/6 2/1 1/5 3/1 4/15 4/1 1/3 1/0 1/0 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(8,-3,3,-1) (0/1,1/2) -> (2/1,3/1) Hyperbolic Matrix(12,-7,7,-4) (1/2,2/3) -> (3/2,2/1) Hyperbolic Matrix(13,-10,4,-3) (2/3,1/1) -> (3/1,4/1) Hyperbolic Matrix(6,-7,1,-1) (1/1,3/2) -> (4/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,15,1) Matrix(8,-3,3,-1) -> Matrix(11,-1,45,-4) Matrix(12,-7,7,-4) -> Matrix(19,-2,105,-11) Matrix(13,-10,4,-3) -> Matrix(17,-2,60,-7) Matrix(6,-7,1,-1) -> Matrix(7,-1,15,-2) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 24 Minimal number of generators: 5 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: 4 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 15 1 1/1 2/15 1 15 3/2 1/6 5 3 2/1 1/5 3 5 4/1 1/3 5 3 1/0 1/0 1 15 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(6,-7,1,-1) (1/1,3/2) -> (4/1,1/0) Hyperbolic Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(3,-8,1,-3) (2/1,4/1) -> (2/1,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,15,-1) (-1/1,1/1) -> (0/1,2/15) Matrix(6,-7,1,-1) -> Matrix(7,-1,15,-2) Matrix(7,-12,4,-7) -> Matrix(11,-2,60,-11) (3/2,2/1) -> (1/6,1/5) Matrix(3,-8,1,-3) -> Matrix(4,-1,15,-4) (2/1,4/1) -> (1/5,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.