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): 108 Minimal number of generators: 19 Number of equivalence classes of cusps: 6 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -10/17 -5/17 0/1 1/2 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 -5/17 -3/4 -4/17 -2/3 -3/17 -3/5 -5/34 -10/17 -1/7 -7/12 -12/85 -4/7 -7/51 -1/2 -2/17 -2/5 -5/51 -1/3 -3/34 -3/10 -10/119 -5/17 -1/12 -2/7 -7/85 -1/4 -4/51 -2/9 -9/119 -1/5 -5/68 -1/6 -6/85 0/1 -1/17 1/5 -5/102 1/4 -4/85 1/3 -3/68 2/5 -5/119 7/17 -1/24 5/12 -12/289 3/7 -7/170 1/2 -2/51 3/5 -5/136 2/3 -3/85 7/10 -10/289 12/17 -1/29 5/7 -7/204 3/4 -4/119 7/9 -9/272 4/5 -5/153 5/6 -6/187 1/1 -1/34 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(7,6,-34,-29) (-1/1,-4/5) -> (-2/9,-1/5) Hyperbolic Matrix(5,4,-34,-27) (-4/5,-3/4) -> (-1/6,0/1) Hyperbolic Matrix(19,14,-34,-25) (-3/4,-2/3) -> (-4/7,-1/2) Hyperbolic Matrix(13,8,34,21) (-2/3,-3/5) -> (1/3,2/5) Hyperbolic Matrix(145,86,204,121) (-3/5,-10/17) -> (12/17,5/7) Hyperbolic Matrix(263,154,374,219) (-10/17,-7/12) -> (7/10,12/17) Hyperbolic Matrix(83,48,102,59) (-7/12,-4/7) -> (4/5,5/6) Hyperbolic Matrix(9,4,-34,-15) (-1/2,-2/5) -> (-2/7,-1/4) Hyperbolic Matrix(21,8,34,13) (-2/5,-1/3) -> (3/5,2/3) Hyperbolic Matrix(79,24,102,31) (-1/3,-3/10) -> (3/4,7/9) Hyperbolic Matrix(155,46,374,111) (-3/10,-5/17) -> (7/17,5/12) Hyperbolic Matrix(83,24,204,59) (-5/17,-2/7) -> (2/5,7/17) Hyperbolic Matrix(71,16,102,23) (-1/4,-2/9) -> (2/3,7/10) Hyperbolic Matrix(43,8,102,19) (-1/5,-1/6) -> (5/12,3/7) Hyperbolic Matrix(27,-4,34,-5) (0/1,1/5) -> (7/9,4/5) Hyperbolic Matrix(29,-6,34,-7) (1/5,1/4) -> (5/6,1/1) Hyperbolic Matrix(15,-4,34,-9) (1/4,1/3) -> (3/7,1/2) Hyperbolic Matrix(25,-14,34,-19) (1/2,3/5) -> (5/7,3/4) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-34,1) Matrix(7,6,-34,-29) -> Matrix(5,2,-68,-27) Matrix(5,4,-34,-27) -> Matrix(7,2,-102,-29) Matrix(19,14,-34,-25) -> Matrix(9,2,-68,-15) Matrix(13,8,34,21) -> Matrix(13,2,-306,-47) Matrix(145,86,204,121) -> Matrix(83,12,-2414,-349) Matrix(263,154,374,219) -> Matrix(155,22,-4488,-637) Matrix(83,48,102,59) -> Matrix(43,6,-1326,-185) Matrix(9,4,-34,-15) -> Matrix(19,2,-238,-25) Matrix(21,8,34,13) -> Matrix(21,2,-578,-55) Matrix(79,24,102,31) -> Matrix(71,6,-2142,-181) Matrix(155,46,374,111) -> Matrix(263,22,-6324,-529) Matrix(83,24,204,59) -> Matrix(145,12,-3468,-287) Matrix(71,16,102,23) -> Matrix(79,6,-2278,-173) Matrix(43,8,102,19) -> Matrix(83,6,-2006,-145) Matrix(27,-4,34,-5) -> Matrix(39,2,-1190,-61) Matrix(29,-6,34,-7) -> Matrix(41,2,-1292,-63) Matrix(15,-4,34,-9) -> Matrix(43,2,-1054,-49) Matrix(25,-14,34,-19) -> Matrix(53,2,-1564,-59) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 108 Minimal number of generators: 19 Number of equivalence classes of cusps: 6 Genus: 7 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 18 Degree of the the map Y: 18 Permutation triple for Y: ((1,4,12,9,16,8,7,11,18,15,6,3,10,14,13,5,2); (1,2,8,14,13,11,4,7,16,17,10,6,5,15,12,9,3); (2,6,5,14,17,16,12,11,4,3,10,9,15,18,13,8,7)) ----------------------------------------------------------------------- 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): 18 Minimal number of generators: 5 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/17 1/3 -3/68 1/2 -2/51 3/5 -5/136 2/3 -3/85 3/4 -4/119 1/1 -1/34 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(10,-3,17,-5) (0/1,1/3) -> (1/2,3/5) Hyperbolic Matrix(12,-5,17,-7) (1/3,1/2) -> (2/3,3/4) Hyperbolic Matrix(21,-13,34,-21) (3/5,2/3) -> (3/5,2/3) Elliptic Matrix(13,-10,17,-13) (3/4,1/1) -> (3/4,1/1) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-17,1) Matrix(10,-3,17,-5) -> Matrix(22,1,-595,-27) Matrix(12,-5,17,-7) -> Matrix(24,1,-697,-29) Matrix(21,-13,34,-21) -> Matrix(55,2,-1513,-55) Matrix(13,-10,17,-13) -> Matrix(30,1,-901,-30) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 18 Minimal number of generators: 5 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 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 -1/17 1 17 1/3 -3/68 1 17 2/5 -5/119 1 17 1/2 -2/51 1 17 1/0 0/1 17 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(7,-2,17,-5) (0/1,1/3) -> (2/5,1/2) Glide Reflection Matrix(13,-5,34,-13) (1/3,2/5) -> (1/3,2/5) Elliptic Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,34,1) (0/1,1/0) -> (-1/17,0/1) Matrix(7,-2,17,-5) -> Matrix(22,1,-527,-24) Matrix(13,-5,34,-13) -> Matrix(47,2,-1105,-47) (-1/23,-1/24).(-2/47,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,51,1) (1/2,1/0) -> (-2/51,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.