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): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 8 Genus: 15 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 17/10 2/1 34/13 17/5 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 3/85 -5/1 5/136 -9/2 9/238 -4/1 2/51 -7/2 7/170 -17/5 1/24 -10/3 5/119 -3/1 3/68 -8/3 4/85 -13/5 13/272 -5/2 5/102 -17/7 1/20 -12/5 6/119 -7/3 7/136 -9/4 9/170 -2/1 1/17 -11/6 11/170 -9/5 9/136 -7/4 7/102 -12/7 6/85 -17/10 1/14 -5/3 5/68 -13/8 13/170 -34/21 1/13 -21/13 21/272 -8/5 4/51 -3/2 3/34 -10/7 5/51 -17/12 1/10 -7/5 7/68 -4/3 2/17 -9/7 9/68 -5/4 5/34 -6/5 3/17 -7/6 7/34 -1/1 1/0 0/1 0/1 1/1 1/68 6/5 3/187 5/4 5/306 9/7 9/544 4/3 2/119 7/5 7/408 17/12 1/58 10/7 5/289 3/2 3/170 8/5 4/221 13/8 13/714 5/3 5/272 17/10 1/54 12/7 6/323 7/4 7/374 9/5 9/476 2/1 1/51 11/5 11/544 9/4 9/442 7/3 7/340 12/5 6/289 17/7 1/48 5/2 5/238 13/5 13/612 34/13 1/47 21/8 21/986 8/3 4/187 3/1 3/136 10/3 5/221 17/5 1/44 7/2 7/306 4/1 2/85 9/2 9/374 5/1 5/204 6/1 3/119 7/1 7/272 1/0 1/34 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,68,-6,-37) (-6/1,1/0) -> (-2/1,-11/6) Hyperbolic Matrix(13,68,4,21) (-6/1,-5/1) -> (3/1,10/3) Hyperbolic Matrix(29,136,-16,-75) (-5/1,-9/2) -> (-11/6,-9/5) Hyperbolic Matrix(31,136,18,79) (-9/2,-4/1) -> (12/7,7/4) Hyperbolic Matrix(19,68,12,43) (-4/1,-7/2) -> (3/2,8/5) Hyperbolic Matrix(59,204,24,83) (-7/2,-17/5) -> (17/7,5/2) Hyperbolic Matrix(101,340,30,101) (-17/5,-10/3) -> (10/3,17/5) Hyperbolic Matrix(21,68,4,13) (-10/3,-3/1) -> (5/1,6/1) Hyperbolic Matrix(25,68,18,49) (-3/1,-8/3) -> (4/3,7/5) Hyperbolic Matrix(103,272,-64,-169) (-8/3,-13/5) -> (-21/13,-8/5) Hyperbolic Matrix(53,136,30,77) (-13/5,-5/2) -> (7/4,9/5) Hyperbolic Matrix(83,204,24,59) (-5/2,-17/7) -> (17/5,7/2) Hyperbolic Matrix(169,408,70,169) (-17/7,-12/5) -> (12/5,17/7) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(59,136,36,83) (-7/3,-9/4) -> (13/8,5/3) Hyperbolic Matrix(31,68,-26,-57) (-9/4,-2/1) -> (-6/5,-7/6) Hyperbolic Matrix(77,136,30,53) (-9/5,-7/4) -> (5/2,13/5) Hyperbolic Matrix(79,136,18,31) (-7/4,-12/7) -> (4/1,9/2) Hyperbolic Matrix(239,408,140,239) (-12/7,-17/10) -> (17/10,12/7) Hyperbolic Matrix(121,204,86,145) (-17/10,-5/3) -> (7/5,17/12) Hyperbolic Matrix(83,136,36,59) (-5/3,-13/8) -> (9/4,7/3) Hyperbolic Matrix(713,1156,272,441) (-13/8,-34/21) -> (34/13,21/8) Hyperbolic Matrix(715,1156,274,443) (-34/21,-21/13) -> (13/5,34/13) Hyperbolic Matrix(43,68,12,19) (-8/5,-3/2) -> (7/2,4/1) Hyperbolic Matrix(47,68,38,55) (-3/2,-10/7) -> (6/5,5/4) Hyperbolic Matrix(239,340,168,239) (-10/7,-17/12) -> (17/12,10/7) Hyperbolic Matrix(145,204,86,121) (-17/12,-7/5) -> (5/3,17/10) Hyperbolic Matrix(49,68,18,25) (-7/5,-4/3) -> (8/3,3/1) Hyperbolic Matrix(105,136,44,57) (-4/3,-9/7) -> (7/3,12/5) Hyperbolic Matrix(53,68,-46,-59) (-9/7,-5/4) -> (-7/6,-1/1) Hyperbolic Matrix(55,68,38,47) (-5/4,-6/5) -> (10/7,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-68,26,-31) (1/1,6/5) -> (2/1,11/5) Hyperbolic Matrix(107,-136,48,-61) (5/4,9/7) -> (11/5,9/4) Hyperbolic Matrix(169,-272,64,-103) (8/5,13/8) -> (21/8,8/3) Hyperbolic Matrix(37,-68,6,-11) (9/5,2/1) -> (6/1,7/1) Hyperbolic Matrix(15,-68,2,-9) (9/2,5/1) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,68,-6,-37) -> Matrix(57,-2,884,-31) Matrix(13,68,4,21) -> Matrix(55,-2,2448,-89) Matrix(29,136,-16,-75) -> Matrix(107,-4,1632,-61) Matrix(31,136,18,79) -> Matrix(105,-4,5644,-215) Matrix(19,68,12,43) -> Matrix(49,-2,2720,-111) Matrix(59,204,24,83) -> Matrix(145,-6,6936,-287) Matrix(101,340,30,101) -> Matrix(239,-10,10540,-441) Matrix(21,68,4,13) -> Matrix(47,-2,1904,-81) Matrix(25,68,18,49) -> Matrix(43,-2,2516,-117) Matrix(103,272,-64,-169) -> Matrix(169,-8,2176,-103) Matrix(53,136,30,77) -> Matrix(83,-4,4420,-213) Matrix(83,204,24,59) -> Matrix(121,-6,5304,-263) Matrix(169,408,70,169) -> Matrix(239,-12,11492,-577) Matrix(57,136,44,105) -> Matrix(79,-4,4760,-241) Matrix(59,136,36,83) -> Matrix(77,-4,4216,-219) Matrix(31,68,-26,-57) -> Matrix(37,-2,204,-11) Matrix(77,136,30,53) -> Matrix(59,-4,2788,-189) Matrix(79,136,18,31) -> Matrix(57,-4,2380,-167) Matrix(239,408,140,239) -> Matrix(169,-12,9112,-647) Matrix(121,204,86,145) -> Matrix(83,-6,4828,-349) Matrix(83,136,36,59) -> Matrix(53,-4,2584,-195) Matrix(713,1156,272,441) -> Matrix(443,-34,20808,-1597) Matrix(715,1156,274,443) -> Matrix(441,-34,20740,-1599) Matrix(43,68,12,19) -> Matrix(25,-2,1088,-87) Matrix(47,68,38,55) -> Matrix(21,-2,1292,-123) Matrix(239,340,168,239) -> Matrix(101,-10,5848,-579) Matrix(145,204,86,121) -> Matrix(59,-6,3196,-325) Matrix(49,68,18,25) -> Matrix(19,-2,884,-93) Matrix(105,136,44,57) -> Matrix(31,-4,1496,-193) Matrix(53,68,-46,-59) -> Matrix(15,-2,68,-9) Matrix(55,68,38,47) -> Matrix(13,-2,748,-115) Matrix(1,0,2,1) -> Matrix(1,0,68,1) Matrix(57,-68,26,-31) -> Matrix(125,-2,6188,-99) Matrix(107,-136,48,-61) -> Matrix(243,-4,11968,-197) Matrix(169,-272,64,-103) -> Matrix(441,-8,20672,-375) Matrix(37,-68,6,-11) -> Matrix(105,-2,4148,-79) Matrix(15,-68,2,-9) -> Matrix(83,-2,3196,-77) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 8 Genus: 15 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 36 Degree of the the map Y: 36 Permutation triple for Y: ((2,6,22,28,30,26,29,13,4,3,12,33,19,35,17,23,7)(5,18,32,24,16,15,25,20,10,9,21,34,27,8,14,31,11); (1,4,16,33,24,23,14,13,34,19,11,3,10,28,27,22,31,36,25,29,9,26,8,7,15,35,21,6,20,12,18,17,5,2)(30,32); (1,2,8,28,27,13,25,7,24,30,10,6,5,19,16,33,20,36,31,23,18,17,15,4,14,26,32,12,11,22,21,29,9,3)(34,35)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. 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. ----------------------------------------------------------------------- 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): 54 Minimal number of generators: 11 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: 4 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 17/5 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/68 5/4 5/306 4/3 2/119 3/2 3/170 5/3 5/272 17/10 1/54 12/7 6/323 7/4 7/374 2/1 1/51 5/2 5/238 3/1 3/136 10/3 5/221 17/5 1/44 7/2 7/306 4/1 2/85 9/2 9/374 5/1 5/204 6/1 3/119 1/0 1/34 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(29,-34,6,-7) (1/1,5/4) -> (9/2,5/1) Hyperbolic Matrix(27,-34,4,-5) (5/4,4/3) -> (6/1,1/0) Hyperbolic Matrix(25,-34,14,-19) (4/3,3/2) -> (7/4,2/1) Hyperbolic Matrix(21,-34,13,-21) (3/2,5/3) -> (3/2,5/3) Elliptic Matrix(121,-204,35,-59) (5/3,17/10) -> (17/5,7/2) Hyperbolic Matrix(219,-374,65,-111) (17/10,12/7) -> (10/3,17/5) Hyperbolic Matrix(59,-102,11,-19) (12/7,7/4) -> (5/1,6/1) Hyperbolic Matrix(15,-34,4,-9) (2/1,5/2) -> (7/2,4/1) Hyperbolic Matrix(13,-34,5,-13) (5/2,3/1) -> (5/2,3/1) Elliptic Matrix(31,-102,7,-23) (3/1,10/3) -> (4/1,9/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,34,1) Matrix(29,-34,6,-7) -> Matrix(63,-1,2584,-41) Matrix(27,-34,4,-5) -> Matrix(61,-1,2380,-39) Matrix(25,-34,14,-19) -> Matrix(59,-1,3128,-53) Matrix(21,-34,13,-21) -> Matrix(55,-1,3026,-55) Matrix(121,-204,35,-59) -> Matrix(325,-6,14246,-263) Matrix(219,-374,65,-111) -> Matrix(593,-11,26146,-485) Matrix(59,-102,11,-19) -> Matrix(161,-3,6494,-121) Matrix(15,-34,4,-9) -> Matrix(49,-1,2108,-43) Matrix(13,-34,5,-13) -> Matrix(47,-1,2210,-47) Matrix(31,-102,7,-23) -> Matrix(133,-3,5542,-125) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 54 Minimal number of generators: 11 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: 4 Genus: 3 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 34 1 2/1 1/51 2 17 5/2 5/238 1 34 3/1 3/136 1 34 10/3 5/221 2 17 17/5 1/44 17 2 7/2 7/306 1 34 4/1 2/85 2 17 5/1 5/204 1 34 6/1 3/119 2 17 1/0 1/34 1 34 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(15,-34,4,-9) (2/1,5/2) -> (7/2,4/1) Hyperbolic Matrix(13,-34,5,-13) (5/2,3/1) -> (5/2,3/1) Elliptic Matrix(21,-68,4,-13) (3/1,10/3) -> (5/1,6/1) Glide Reflection Matrix(101,-340,30,-101) (10/3,17/5) -> (10/3,17/5) Reflection Matrix(69,-238,20,-69) (17/5,7/2) -> (17/5,7/2) Reflection Matrix(7,-34,1,-5) (4/1,5/1) -> (6/1,1/0) Glide Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,68,-1) (0/1,1/0) -> (0/1,1/34) Matrix(1,0,1,-1) -> Matrix(1,0,102,-1) (0/1,2/1) -> (0/1,1/51) Matrix(15,-34,4,-9) -> Matrix(49,-1,2108,-43) Matrix(13,-34,5,-13) -> Matrix(47,-1,2210,-47) (0/1,1/47).(1/48,1/46) Matrix(21,-68,4,-13) -> Matrix(89,-2,3604,-81) Matrix(101,-340,30,-101) -> Matrix(441,-10,19448,-441) (10/3,17/5) -> (5/221,1/44) Matrix(69,-238,20,-69) -> Matrix(307,-7,13464,-307) (17/5,7/2) -> (1/44,7/306) Matrix(7,-34,1,-5) -> Matrix(41,-1,1598,-39) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.