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 0/1 1/1 15/11 3/2 2/1 5/2 3/1 10/3 15/4 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -4/15 -15/4 -1/4 -11/3 -11/45 -7/2 -7/30 -10/3 -2/9 -3/1 -1/5 -5/2 -1/6 -2/1 -2/15 -7/4 -7/60 -19/11 -19/165 -12/7 -4/35 -5/3 -1/9 -3/2 -1/10 -10/7 -2/21 -17/12 -17/180 -7/5 -7/75 -11/8 -11/120 -15/11 -1/11 -4/3 -4/45 -5/4 -1/12 -6/5 -2/25 -13/11 -13/165 -7/6 -7/90 -1/1 -1/15 0/1 0/1 1/1 1/15 4/3 4/45 15/11 1/11 11/8 11/120 7/5 7/75 10/7 2/21 3/2 1/10 5/3 1/9 2/1 2/15 7/3 7/45 19/8 19/120 12/5 4/25 5/2 1/6 3/1 1/5 10/3 2/9 17/5 17/75 7/2 7/30 11/3 11/45 15/4 1/4 4/1 4/15 5/1 1/3 6/1 2/5 13/2 13/30 7/1 7/15 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,30,-4,-17) (-4/1,1/0) -> (-2/1,-7/4) Hyperbolic Matrix(31,120,8,31) (-4/1,-15/4) -> (15/4,4/1) Hyperbolic Matrix(89,330,24,89) (-15/4,-11/3) -> (11/3,15/4) Hyperbolic Matrix(59,210,-34,-121) (-11/3,-7/2) -> (-7/4,-19/11) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(13,30,-10,-23) (-5/2,-2/1) -> (-4/3,-5/4) Hyperbolic Matrix(157,270,-132,-227) (-19/11,-12/7) -> (-6/5,-13/11) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(191,270,-162,-229) (-17/12,-7/5) -> (-13/11,-7/6) Hyperbolic Matrix(43,60,-38,-53) (-7/5,-11/8) -> (-7/6,-1/1) Hyperbolic Matrix(241,330,176,241) (-11/8,-15/11) -> (15/11,11/8) Hyperbolic Matrix(89,120,66,89) (-15/11,-4/3) -> (4/3,15/11) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(23,-30,10,-13) (1/1,4/3) -> (2/1,7/3) Hyperbolic Matrix(151,-210,64,-89) (11/8,7/5) -> (7/3,19/8) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(17,-30,4,-7) (5/3,2/1) -> (4/1,5/1) Hyperbolic Matrix(113,-270,18,-43) (19/8,12/5) -> (6/1,13/2) Hyperbolic Matrix(79,-270,12,-41) (17/5,7/2) -> (13/2,7/1) Hyperbolic Matrix(17,-60,2,-7) (7/2,11/3) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,30,-4,-17) -> Matrix(7,2,-60,-17) Matrix(31,120,8,31) -> Matrix(31,8,120,31) Matrix(89,330,24,89) -> Matrix(89,22,360,89) Matrix(59,210,-34,-121) -> Matrix(59,14,-510,-121) Matrix(71,240,-50,-169) -> Matrix(71,16,-750,-169) Matrix(19,60,6,19) -> Matrix(19,4,90,19) Matrix(11,30,4,11) -> Matrix(11,2,60,11) Matrix(13,30,-10,-23) -> Matrix(13,2,-150,-23) Matrix(157,270,-132,-227) -> Matrix(157,18,-1980,-227) Matrix(53,90,10,17) -> Matrix(53,6,150,17) Matrix(19,30,12,19) -> Matrix(19,2,180,19) Matrix(41,60,28,41) -> Matrix(41,4,420,41) Matrix(191,270,-162,-229) -> Matrix(191,18,-2430,-229) Matrix(43,60,-38,-53) -> Matrix(43,4,-570,-53) Matrix(241,330,176,241) -> Matrix(241,22,2640,241) Matrix(89,120,66,89) -> Matrix(89,8,990,89) Matrix(73,90,30,37) -> Matrix(73,6,450,37) Matrix(1,0,2,1) -> Matrix(1,0,30,1) Matrix(23,-30,10,-13) -> Matrix(23,-2,150,-13) Matrix(151,-210,64,-89) -> Matrix(151,-14,960,-89) Matrix(169,-240,50,-71) -> Matrix(169,-16,750,-71) Matrix(17,-30,4,-7) -> Matrix(17,-2,60,-7) Matrix(113,-270,18,-43) -> Matrix(113,-18,270,-43) Matrix(79,-270,12,-41) -> Matrix(79,-18,180,-41) Matrix(17,-60,2,-7) -> Matrix(17,-4,30,-7) 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: ((2,6,8,18,20,10,9,4,3,11,12,23,14,5,7)(13,24,16,19,22)(15,17,21); (1,4,11,22,12,21,19,20,18,24,15,14,13,5,2)(3,10,17,8,7)(6,9,16); (1,2,8,16,18,17,24,14,23,22,21,10,19,9,3)(4,6,5,15,12)(7,13,11)) ----------------------------------------------------------------------- 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 0/1 3/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/15 3/2 1/10 5/3 1/9 2/1 2/15 5/2 1/6 3/1 1/5 4/1 4/15 5/1 1/3 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(11,-15,3,-4) (1/1,3/2) -> (3/1,4/1) Hyperbolic Matrix(19,-30,7,-11) (3/2,5/3) -> (5/2,3/1) Hyperbolic Matrix(17,-30,4,-7) (5/3,2/1) -> (4/1,5/1) Hyperbolic Matrix(7,-15,1,-2) (2/1,5/2) -> (5/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,15,1) Matrix(11,-15,3,-4) -> Matrix(11,-1,45,-4) Matrix(19,-30,7,-11) -> Matrix(19,-2,105,-11) Matrix(17,-30,4,-7) -> Matrix(17,-2,60,-7) Matrix(7,-15,1,-2) -> 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 0/1 0/1 15 1 2/1 2/15 1 15 5/2 1/6 5 3 3/1 1/5 3 5 5/1 1/3 5 3 1/0 1/0 1 15 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,-15,1,-2) (2/1,5/2) -> (5/1,1/0) Hyperbolic Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(4,-15,1,-4) (3/1,5/1) -> (3/1,5/1) 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(1,0,15,-1) (0/1,2/1) -> (0/1,2/15) Matrix(7,-15,1,-2) -> Matrix(7,-1,15,-2) Matrix(11,-30,4,-11) -> Matrix(11,-2,60,-11) (5/2,3/1) -> (1/6,1/5) Matrix(4,-15,1,-4) -> Matrix(4,-1,15,-4) (3/1,5/1) -> (1/5,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.