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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 7/6 7/5 3/2 7/4 2/1 7/3 12/5 5/2 28/11 8/3 14/5 3/1 42/13 10/3 7/2 4/1 14/3 5/1 28/5 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/2 -6/1 -1/2 -2/5 -11/2 -3/8 -5/1 -3/8 -4/1 -1/3 -1/4 -7/2 -1/4 -10/3 -1/4 -2/9 -13/4 -5/24 -3/1 -1/4 -8/3 -1/5 -1/6 -21/8 -1/6 -13/5 -3/20 -18/7 -1/8 0/1 -5/2 -1/4 -17/7 -1/4 -29/12 -5/24 -12/5 -1/5 -1/6 -7/3 -1/6 -2/1 -1/6 0/1 -7/4 -1/6 -12/7 -1/6 -1/7 -17/10 -1/8 -5/3 -1/8 -28/17 0/1 -23/14 1/0 -18/11 -1/4 0/1 -13/8 -3/16 -21/13 -1/6 -8/5 -1/6 -1/7 -11/7 -3/20 -14/9 -1/7 -3/2 -1/8 -16/11 -1/6 -1/7 -29/20 -7/48 -42/29 -1/7 -13/9 -5/36 -10/7 -2/15 -1/8 -7/5 -1/8 -4/3 -1/8 -1/9 -9/7 -5/44 -14/11 -1/9 -5/4 -3/28 -11/9 -3/28 -28/23 -2/19 -17/14 -5/48 -6/5 -2/19 -1/10 -7/6 -1/10 -1/1 -1/12 0/1 0/1 1/1 1/12 7/6 1/10 6/5 1/10 2/19 11/9 3/28 5/4 3/28 4/3 1/9 1/8 7/5 1/8 10/7 1/8 2/15 13/9 5/36 3/2 1/8 8/5 1/7 1/6 21/13 1/6 13/8 3/16 18/11 0/1 1/4 5/3 1/8 17/10 1/8 29/17 5/36 12/7 1/7 1/6 7/4 1/6 2/1 0/1 1/6 7/3 1/6 12/5 1/6 1/5 17/7 1/4 5/2 1/4 28/11 0/1 23/9 1/12 18/7 0/1 1/8 13/5 3/20 21/8 1/6 8/3 1/6 1/5 11/4 3/16 14/5 1/5 3/1 1/4 16/5 1/6 1/5 29/9 7/36 42/13 1/5 13/4 5/24 10/3 2/9 1/4 7/2 1/4 4/1 1/4 1/3 9/2 5/16 14/3 1/3 5/1 3/8 11/2 3/8 28/5 2/5 17/3 5/12 6/1 2/5 1/2 7/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,112,8,69) (-7/1,1/0) -> (21/13,13/8) Hyperbolic Matrix(13,84,2,13) (-7/1,-6/1) -> (6/1,7/1) Hyperbolic Matrix(29,168,-24,-139) (-6/1,-11/2) -> (-17/14,-6/5) Hyperbolic Matrix(27,140,16,83) (-11/2,-5/1) -> (5/3,17/10) Hyperbolic Matrix(13,56,-10,-43) (-5/1,-4/1) -> (-4/3,-9/7) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(111,364,68,223) (-10/3,-13/4) -> (13/8,18/11) Hyperbolic Matrix(97,308,-40,-127) (-13/4,-3/1) -> (-17/7,-29/12) Hyperbolic Matrix(41,112,-26,-71) (-3/1,-8/3) -> (-8/5,-11/7) Hyperbolic Matrix(127,336,48,127) (-8/3,-21/8) -> (21/8,8/3) Hyperbolic Matrix(43,112,38,99) (-21/8,-13/5) -> (1/1,7/6) Hyperbolic Matrix(141,364,98,253) (-13/5,-18/7) -> (10/7,13/9) Hyperbolic Matrix(197,504,-120,-307) (-18/7,-5/2) -> (-23/14,-18/11) Hyperbolic Matrix(57,140,46,113) (-5/2,-17/7) -> (11/9,5/4) Hyperbolic Matrix(337,812,-232,-559) (-29/12,-12/5) -> (-16/11,-29/20) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(181,308,-124,-211) (-12/7,-17/10) -> (-3/2,-16/11) Hyperbolic Matrix(83,140,16,27) (-17/10,-5/3) -> (5/1,11/2) Hyperbolic Matrix(475,784,186,307) (-5/3,-28/17) -> (28/11,23/9) Hyperbolic Matrix(477,784,188,309) (-28/17,-23/14) -> (5/2,28/11) Hyperbolic Matrix(223,364,68,111) (-18/11,-13/8) -> (13/4,10/3) Hyperbolic Matrix(69,112,8,13) (-13/8,-21/13) -> (7/1,1/0) Hyperbolic Matrix(209,336,130,209) (-21/13,-8/5) -> (8/5,21/13) Hyperbolic Matrix(125,196,44,69) (-11/7,-14/9) -> (14/5,3/1) Hyperbolic Matrix(127,196,46,71) (-14/9,-3/2) -> (11/4,14/5) Hyperbolic Matrix(1217,1764,376,545) (-29/20,-42/29) -> (42/13,13/4) Hyperbolic Matrix(1219,1764,378,547) (-42/29,-13/9) -> (29/9,42/13) Hyperbolic Matrix(253,364,98,141) (-13/9,-10/7) -> (18/7,13/5) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(153,196,32,41) (-9/7,-14/11) -> (14/3,5/1) Hyperbolic Matrix(155,196,34,43) (-14/11,-5/4) -> (9/2,14/3) Hyperbolic Matrix(113,140,46,57) (-5/4,-11/9) -> (17/7,5/2) Hyperbolic Matrix(643,784,114,139) (-11/9,-28/23) -> (28/5,17/3) Hyperbolic Matrix(645,784,116,141) (-28/23,-17/14) -> (11/2,28/5) Hyperbolic Matrix(71,84,60,71) (-6/5,-7/6) -> (7/6,6/5) Hyperbolic Matrix(99,112,38,43) (-7/6,-1/1) -> (13/5,21/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(211,-308,124,-181) (13/9,3/2) -> (17/10,29/17) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(475,-812,148,-253) (29/17,12/7) -> (16/5,29/9) Hyperbolic Matrix(127,-308,40,-97) (12/5,17/7) -> (3/1,16/5) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,112,8,69) -> Matrix(3,2,16,11) Matrix(13,84,2,13) -> Matrix(9,4,20,9) Matrix(29,168,-24,-139) -> Matrix(9,4,-88,-39) Matrix(27,140,16,83) -> Matrix(5,2,32,13) Matrix(13,56,-10,-43) -> Matrix(7,2,-60,-17) Matrix(15,56,4,15) -> Matrix(7,2,24,7) Matrix(41,140,12,41) -> Matrix(17,4,72,17) Matrix(111,364,68,223) -> Matrix(9,2,40,9) Matrix(97,308,-40,-127) -> Matrix(1,0,0,1) Matrix(41,112,-26,-71) -> Matrix(11,2,-72,-13) Matrix(127,336,48,127) -> Matrix(11,2,60,11) Matrix(43,112,38,99) -> Matrix(13,2,136,21) Matrix(141,364,98,253) -> Matrix(15,2,112,15) Matrix(197,504,-120,-307) -> Matrix(1,0,4,1) Matrix(57,140,46,113) -> Matrix(11,2,104,19) Matrix(337,812,-232,-559) -> Matrix(11,2,-72,-13) Matrix(71,168,30,71) -> Matrix(11,2,60,11) Matrix(13,28,6,13) -> Matrix(1,0,12,1) Matrix(15,28,8,15) -> Matrix(1,0,12,1) Matrix(97,168,56,97) -> Matrix(13,2,84,13) Matrix(181,308,-124,-211) -> Matrix(1,0,0,1) Matrix(83,140,16,27) -> Matrix(13,2,32,5) Matrix(475,784,186,307) -> Matrix(1,0,20,1) Matrix(477,784,188,309) -> Matrix(1,0,4,1) Matrix(223,364,68,111) -> Matrix(9,2,40,9) Matrix(69,112,8,13) -> Matrix(11,2,16,3) Matrix(209,336,130,209) -> Matrix(13,2,84,13) Matrix(125,196,44,69) -> Matrix(27,4,128,19) Matrix(127,196,46,71) -> Matrix(29,4,152,21) Matrix(1217,1764,376,545) -> Matrix(83,12,408,59) Matrix(1219,1764,378,547) -> Matrix(85,12,432,61) Matrix(253,364,98,141) -> Matrix(15,2,112,15) Matrix(99,140,70,99) -> Matrix(31,4,240,31) Matrix(41,56,30,41) -> Matrix(17,2,144,17) Matrix(153,196,32,41) -> Matrix(71,8,204,23) Matrix(155,196,34,43) -> Matrix(73,8,228,25) Matrix(113,140,46,57) -> Matrix(19,2,104,11) Matrix(643,784,114,139) -> Matrix(151,16,368,39) Matrix(645,784,116,141) -> Matrix(153,16,392,41) Matrix(71,84,60,71) -> Matrix(39,4,380,39) Matrix(99,112,38,43) -> Matrix(21,2,136,13) Matrix(1,0,2,1) -> Matrix(1,0,24,1) Matrix(139,-168,24,-29) -> Matrix(39,-4,88,-9) Matrix(43,-56,10,-13) -> Matrix(17,-2,60,-7) Matrix(211,-308,124,-181) -> Matrix(1,0,0,1) Matrix(71,-112,26,-41) -> Matrix(13,-2,72,-11) Matrix(307,-504,120,-197) -> Matrix(1,0,4,1) Matrix(475,-812,148,-253) -> Matrix(13,-2,72,-11) Matrix(127,-308,40,-97) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 12 Degree of the the map X: 24 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,13,4,3,12,7)(5,18,8,10,9,16,15)(11,14,37,21,20,24,32)(17,40,39,28,27,26,41)(19,38,43,30,25,36,33)(22,42,44,23,34,46,35); (1,4,16,40,24,38,46,48,42,33,32,17,5,2)(3,10,30,34,41,45,39,44,25,8,7,20,31,11)(6,21,43,47,36,14,13,35,26,18,29,9,28,22)(12,23)(15,19)(27,37); (1,2,8,26,37,36,44,48,46,30,21,27,9,3)(4,14,31,20,6,5,19,42,28,45,41,35,38,15)(7,23,39,16,29,18,17,34,12,11,33,47,43,24)(10,25)(13,22)(32,40)) ----------------------------------------------------------------------- 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 0/1 2/1 7/3 14/5 3/1 7/2 4/1 14/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/12 5/4 3/28 4/3 1/9 1/8 7/5 1/8 3/2 1/8 8/5 1/7 1/6 5/3 1/8 7/4 1/6 2/1 0/1 1/6 7/3 1/6 5/2 1/4 13/5 3/20 21/8 1/6 8/3 1/6 1/5 11/4 3/16 14/5 1/5 3/1 1/4 7/2 1/4 4/1 1/4 1/3 9/2 5/16 14/3 1/3 5/1 3/8 6/1 2/5 1/2 7/1 1/2 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(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(41,-56,11,-15) (4/3,7/5) -> (7/2,4/1) Hyperbolic Matrix(29,-42,9,-13) (7/5,3/2) -> (3/1,7/2) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic Matrix(41,-70,17,-29) (5/3,7/4) -> (7/3,5/2) Hyperbolic Matrix(15,-28,7,-13) (7/4,2/1) -> (2/1,7/3) Parabolic Matrix(43,-112,5,-13) (13/5,21/8) -> (7/1,1/0) Hyperbolic Matrix(69,-182,11,-29) (21/8,8/3) -> (6/1,7/1) Hyperbolic Matrix(71,-196,25,-69) (11/4,14/5) -> (14/5,3/1) Parabolic Matrix(43,-196,9,-41) (9/2,14/3) -> (14/3,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,12,1) Matrix(57,-70,22,-27) -> Matrix(9,-1,64,-7) Matrix(43,-56,10,-13) -> Matrix(17,-2,60,-7) Matrix(41,-56,11,-15) -> Matrix(17,-2,60,-7) Matrix(29,-42,9,-13) -> Matrix(7,-1,36,-5) Matrix(71,-112,26,-41) -> Matrix(13,-2,72,-11) Matrix(43,-70,8,-13) -> Matrix(5,-1,16,-3) Matrix(41,-70,17,-29) -> Matrix(7,-1,36,-5) Matrix(15,-28,7,-13) -> Matrix(1,0,0,1) Matrix(43,-112,5,-13) -> Matrix(13,-2,20,-3) Matrix(69,-182,11,-29) -> Matrix(17,-3,40,-7) Matrix(71,-196,25,-69) -> Matrix(21,-4,100,-19) Matrix(43,-196,9,-41) -> Matrix(25,-8,72,-23) 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): 6 ----------------------------------------------------------------------- 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 12 1 2/1 (0/1,1/6) 0 7 7/3 1/6 1 2 5/2 1/4 1 14 8/3 (1/6,1/5) 0 7 14/5 1/5 4 1 3/1 1/4 1 14 7/2 1/4 3 2 4/1 (1/4,1/3) 0 7 14/3 1/3 8 1 5/1 3/8 1 14 6/1 (2/5,1/2) 0 7 7/1 1/2 5 2 1/0 1/0 1 14 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(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(27,-70,5,-13) (5/2,8/3) -> (5/1,6/1) Glide Reflection Matrix(41,-112,15,-41) (8/3,14/5) -> (8/3,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(13,-56,3,-13) (4/1,14/3) -> (4/1,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,12,-1) (0/1,2/1) -> (0/1,1/6) Matrix(13,-28,6,-13) -> Matrix(1,0,12,-1) (2/1,7/3) -> (0/1,1/6) Matrix(29,-70,12,-29) -> Matrix(5,-1,24,-5) (7/3,5/2) -> (1/6,1/4) Matrix(27,-70,5,-13) -> Matrix(7,-1,20,-3) Matrix(41,-112,15,-41) -> Matrix(11,-2,60,-11) (8/3,14/5) -> (1/6,1/5) Matrix(29,-84,10,-29) -> Matrix(9,-2,40,-9) (14/5,3/1) -> (1/5,1/4) Matrix(13,-42,4,-13) -> Matrix(5,-1,24,-5) (3/1,7/2) -> (1/6,1/4) Matrix(15,-56,4,-15) -> Matrix(7,-2,24,-7) (7/2,4/1) -> (1/4,1/3) Matrix(13,-56,3,-13) -> Matrix(7,-2,24,-7) (4/1,14/3) -> (1/4,1/3) Matrix(29,-140,6,-29) -> Matrix(17,-6,48,-17) (14/3,5/1) -> (1/3,3/8) Matrix(13,-84,2,-13) -> Matrix(9,-4,20,-9) (6/1,7/1) -> (2/5,1/2) Matrix(-1,14,0,1) -> Matrix(-1,1,0,1) (7/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.