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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -5/12 -1/3 -3/10 -1/8 0/1 3/11 1/3 2/5 1/2 5/9 3/4 1/1 4/3 3/2 13/8 9/5 2/1 5/2 3/1 7/2 11/3 4/1 57/13 9/2 5/1 11/2 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/4 -5/11 1/5 -4/9 0/1 1/3 -7/16 0/1 1/7 -3/7 1/5 -5/12 1/4 -7/17 1/3 -2/5 0/1 1/4 -7/18 1/4 3/11 -5/13 1/3 -3/8 1/4 1/3 -1/3 1/3 -3/10 1/4 1/2 -5/17 1/3 -2/7 0/1 1/3 -5/18 0/1 1/2 -3/11 1/3 -4/15 0/1 1/2 -1/4 0/1 1/3 -1/5 1/3 -2/11 0/1 1/3 -1/6 1/3 1/2 -1/7 1/3 -1/8 1/2 0/1 0/1 1/2 1/4 1/2 1/1 3/11 1/1 5/18 1/1 1/0 2/7 0/1 1/1 1/3 1/1 5/14 0/1 1/3 4/11 0/1 1/2 3/8 1/2 1/1 2/5 1/1 5/12 1/1 1/0 3/7 1/1 1/2 0/1 1/1 5/9 0/1 9/16 0/1 1/3 4/7 0/1 1/2 7/12 0/1 1/1 3/5 1/1 8/13 0/1 5/8 0/1 1/2 12/19 0/1 1/2 7/11 1/1 2/3 0/1 1/1 3/4 1/2 1/0 4/5 0/1 1/1 9/11 1/3 5/6 1/2 1/1 6/7 0/1 1/2 1/1 1/1 6/5 0/1 1/0 11/9 1/1 5/4 0/1 1/1 4/3 1/1 11/8 1/1 2/1 18/13 2/1 1/0 25/18 2/1 1/0 7/5 1/1 3/2 1/1 1/0 8/5 2/1 1/0 13/8 1/0 18/11 0/1 1/0 23/14 0/1 1/1 5/3 1/1 7/4 3/1 1/0 9/5 1/0 11/6 -3/1 1/0 2/1 0/1 1/0 5/2 1/0 8/3 -2/1 1/0 19/7 -1/1 30/11 0/1 1/0 41/15 1/0 11/4 -2/1 1/0 14/5 -2/1 -1/1 3/1 -1/1 13/4 -1/1 -1/2 10/3 -1/2 0/1 27/8 -1/2 17/5 -1/3 7/2 -1/3 0/1 11/3 0/1 15/4 0/1 1/5 4/1 0/1 1/1 13/3 1/1 35/8 2/1 1/0 57/13 1/0 22/5 0/1 1/0 9/2 1/1 1/0 5/1 -1/1 11/2 -1/2 0/1 6/1 0/1 7/1 1/1 1/0 0/1 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(25,12,2,1) (-1/2,-5/11) -> (7/1,1/0) Hyperbolic Matrix(125,56,154,69) (-5/11,-4/9) -> (4/5,9/11) Hyperbolic Matrix(199,88,52,23) (-4/9,-7/16) -> (15/4,4/1) Hyperbolic Matrix(51,22,146,63) (-7/16,-3/7) -> (1/3,5/14) Hyperbolic Matrix(119,50,-288,-121) (-3/7,-5/12) -> (-5/12,-7/17) Parabolic Matrix(239,98,378,155) (-7/17,-2/5) -> (12/19,7/11) Hyperbolic Matrix(91,36,48,19) (-2/5,-7/18) -> (11/6,2/1) Hyperbolic Matrix(227,88,276,107) (-7/18,-5/13) -> (9/11,5/6) Hyperbolic Matrix(21,8,-134,-51) (-5/13,-3/8) -> (-1/6,-1/7) Hyperbolic Matrix(65,24,46,17) (-3/8,-1/3) -> (7/5,3/2) Hyperbolic Matrix(59,18,-200,-61) (-1/3,-3/10) -> (-3/10,-5/17) Parabolic Matrix(159,46,38,11) (-5/17,-2/7) -> (4/1,13/3) Hyperbolic Matrix(329,92,118,33) (-2/7,-5/18) -> (11/4,14/5) Hyperbolic Matrix(211,58,40,11) (-5/18,-3/11) -> (5/1,11/2) Hyperbolic Matrix(231,62,190,51) (-3/11,-4/15) -> (6/5,11/9) Hyperbolic Matrix(151,40,268,71) (-4/15,-1/4) -> (9/16,4/7) Hyperbolic Matrix(17,4,38,9) (-1/4,-1/5) -> (3/7,1/2) Hyperbolic Matrix(103,20,36,7) (-1/5,-2/11) -> (14/5,3/1) Hyperbolic Matrix(67,12,240,43) (-2/11,-1/6) -> (5/18,2/7) Hyperbolic Matrix(325,44,96,13) (-1/7,-1/8) -> (27/8,17/5) Hyperbolic Matrix(107,10,32,3) (-1/8,0/1) -> (10/3,27/8) Hyperbolic Matrix(35,-8,22,-5) (0/1,1/4) -> (3/2,8/5) Hyperbolic Matrix(67,-18,242,-65) (1/4,3/11) -> (3/11,5/18) Parabolic Matrix(55,-16,86,-25) (2/7,1/3) -> (7/11,2/3) Hyperbolic Matrix(505,-182,308,-111) (5/14,4/11) -> (18/11,23/14) Hyperbolic Matrix(103,-38,122,-45) (4/11,3/8) -> (5/6,6/7) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(127,-54,40,-17) (5/12,3/7) -> (3/1,13/4) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(293,-170,212,-123) (4/7,7/12) -> (11/8,18/13) Hyperbolic Matrix(239,-140,70,-41) (7/12,3/5) -> (17/5,7/2) Hyperbolic Matrix(121,-74,18,-11) (3/5,8/13) -> (6/1,7/1) Hyperbolic Matrix(191,-118,34,-21) (8/13,5/8) -> (11/2,6/1) Hyperbolic Matrix(619,-390,446,-281) (5/8,12/19) -> (18/13,25/18) Hyperbolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(103,-122,38,-45) (1/1,6/5) -> (8/3,19/7) Hyperbolic Matrix(227,-278,138,-169) (11/9,5/4) -> (23/14,5/3) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(409,-570,94,-131) (25/18,7/5) -> (13/3,35/8) Hyperbolic Matrix(209,-338,128,-207) (8/5,13/8) -> (13/8,18/11) Parabolic Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(957,-2612,218,-595) (30/11,41/15) -> (57/13,22/5) Hyperbolic Matrix(753,-2062,172,-471) (41/15,11/4) -> (35/8,57/13) Hyperbolic Matrix(115,-376,26,-85) (13/4,10/3) -> (22/5,9/2) Hyperbolic Matrix(67,-242,18,-65) (7/2,11/3) -> (11/3,15/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(25,12,2,1) -> Matrix(1,0,-4,1) Matrix(125,56,154,69) -> Matrix(1,0,-2,1) Matrix(199,88,52,23) -> Matrix(1,0,-2,1) Matrix(51,22,146,63) -> Matrix(1,0,-4,1) Matrix(119,50,-288,-121) -> Matrix(9,-2,32,-7) Matrix(239,98,378,155) -> Matrix(1,0,-2,1) Matrix(91,36,48,19) -> Matrix(1,0,-4,1) Matrix(227,88,276,107) -> Matrix(7,-2,18,-5) Matrix(21,8,-134,-51) -> Matrix(7,-2,18,-5) Matrix(65,24,46,17) -> Matrix(7,-2,4,-1) Matrix(59,18,-200,-61) -> Matrix(1,0,0,1) Matrix(159,46,38,11) -> Matrix(1,0,-2,1) Matrix(329,92,118,33) -> Matrix(5,-2,-2,1) Matrix(211,58,40,11) -> Matrix(1,0,-4,1) Matrix(231,62,190,51) -> Matrix(1,0,-2,1) Matrix(151,40,268,71) -> Matrix(1,0,0,1) Matrix(17,4,38,9) -> Matrix(1,0,-2,1) Matrix(103,20,36,7) -> Matrix(5,-2,-2,1) Matrix(67,12,240,43) -> Matrix(1,0,-2,1) Matrix(325,44,96,13) -> Matrix(5,-2,-12,5) Matrix(107,10,32,3) -> Matrix(1,0,-4,1) Matrix(35,-8,22,-5) -> Matrix(3,-2,2,-1) Matrix(67,-18,242,-65) -> Matrix(3,-2,2,-1) Matrix(55,-16,86,-25) -> Matrix(1,0,0,1) Matrix(505,-182,308,-111) -> Matrix(1,0,-2,1) Matrix(103,-38,122,-45) -> Matrix(1,0,0,1) Matrix(41,-16,100,-39) -> Matrix(3,-2,2,-1) Matrix(127,-54,40,-17) -> Matrix(1,0,-2,1) Matrix(91,-50,162,-89) -> Matrix(1,0,2,1) Matrix(293,-170,212,-123) -> Matrix(3,-2,2,-1) Matrix(239,-140,70,-41) -> Matrix(1,0,-4,1) Matrix(121,-74,18,-11) -> Matrix(1,0,0,1) Matrix(191,-118,34,-21) -> Matrix(1,0,-4,1) Matrix(619,-390,446,-281) -> Matrix(3,-2,2,-1) Matrix(25,-18,32,-23) -> Matrix(1,0,0,1) Matrix(163,-144,60,-53) -> Matrix(1,0,-2,1) Matrix(103,-122,38,-45) -> Matrix(1,-2,0,1) Matrix(227,-278,138,-169) -> Matrix(1,0,0,1) Matrix(49,-64,36,-47) -> Matrix(3,-2,2,-1) Matrix(409,-570,94,-131) -> Matrix(1,0,0,1) Matrix(209,-338,128,-207) -> Matrix(1,-2,0,1) Matrix(47,-80,10,-17) -> Matrix(1,-2,0,1) Matrix(91,-162,50,-89) -> Matrix(1,-6,0,1) Matrix(21,-50,8,-19) -> Matrix(1,-2,0,1) Matrix(957,-2612,218,-595) -> Matrix(1,0,0,1) Matrix(753,-2062,172,-471) -> Matrix(1,4,0,1) Matrix(115,-376,26,-85) -> Matrix(1,0,2,1) Matrix(67,-242,18,-65) -> Matrix(1,0,8,1) 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 9 Degree of the the map X: 9 Degree of the the map Y: 48 Permutation triple for Y: ((1,6,13,37,44,40,48,47,33,32,46,22,7,2)(3,11,35,39,43,18,38,42,26,8,25,36,12,4)(5,16,41,27,31,10,9,30,24,23,34,45,21,17)(14,15)(19,20)(28,29); (1,4,15,35,40,16,5)(3,10,33,19,22,34,11)(6,20,44,27,26,42,21)(7,24,30,47,38,14,8)(9,29,23,39,37,13,12)(17,28,41,25,46,32,18); (1,2,8,27,28,9,3)(4,13,21,45,22,25,14)(5,18,43,23,7,19,6)(10,31,44,39,15,38,32)(12,36,41,40,20,33,30)(17,42,47,48,35,34,29)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 2 1 0/1 (0/1,1/2) 0 14 1/4 (1/2,1/1) 0 14 3/11 1/1 1 1 2/7 (0/1,1/1) 0 14 1/3 1/1 1 7 4/11 (0/1,1/2) 0 14 3/8 (1/2,1/1) 0 14 2/5 1/1 1 2 1/2 (0/1,1/1) 0 14 5/9 0/1 1 1 4/7 (0/1,1/2) 0 14 7/12 (0/1,1/1) 0 14 3/5 1/1 1 7 8/13 0/1 2 2 5/8 (0/1,1/2) 0 14 7/11 1/1 1 7 2/3 (0/1,1/1) 0 14 3/4 0 2 4/5 (0/1,1/1) 0 14 5/6 (1/2,1/1) 0 14 6/7 (0/1,1/2) 0 14 1/1 1/1 1 7 6/5 (0/1,1/0) 0 14 5/4 (0/1,1/1) 0 14 4/3 1/1 1 2 3/2 (1/1,1/0) 0 14 8/5 (2/1,1/0) 0 14 13/8 1/0 2 2 18/11 (0/1,1/0) 0 14 5/3 1/1 1 7 7/4 (3/1,1/0) 0 14 9/5 1/0 3 1 2/1 (0/1,1/0) 0 14 5/2 1/0 2 2 8/3 (-2/1,1/0) 0 14 19/7 -1/1 1 7 30/11 (0/1,1/0) 0 14 41/15 1/0 2 1 11/4 (-2/1,1/0) 0 14 3/1 -1/1 1 7 10/3 (-1/2,0/1) 0 14 17/5 -1/3 1 7 7/2 (-1/3,0/1) 0 14 11/3 0/1 4 1 4/1 (0/1,1/1) 0 14 9/2 (1/1,1/0) 0 14 5/1 -1/1 1 7 6/1 0/1 2 2 1/0 (0/1,1/0) 0 14 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(35,-8,22,-5) (0/1,1/4) -> (3/2,8/5) Hyperbolic Matrix(23,-6,88,-23) (1/4,3/11) -> (1/4,3/11) Reflection Matrix(43,-12,154,-43) (3/11,2/7) -> (3/11,2/7) Reflection Matrix(55,-16,86,-25) (2/7,1/3) -> (7/11,2/3) Hyperbolic Matrix(63,-22,20,-7) (1/3,4/11) -> (3/1,10/3) Glide Reflection Matrix(103,-38,122,-45) (4/11,3/8) -> (5/6,6/7) Hyperbolic Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(9,-4,20,-9) (2/5,1/2) -> (2/5,1/2) Reflection Matrix(19,-10,36,-19) (1/2,5/9) -> (1/2,5/9) Reflection Matrix(71,-40,126,-71) (5/9,4/7) -> (5/9,4/7) Reflection Matrix(107,-62,88,-51) (4/7,7/12) -> (6/5,5/4) Glide Reflection Matrix(239,-140,70,-41) (7/12,3/5) -> (17/5,7/2) Hyperbolic Matrix(95,-58,18,-11) (3/5,8/13) -> (5/1,6/1) Glide Reflection Matrix(129,-80,208,-129) (8/13,5/8) -> (8/13,5/8) Reflection Matrix(145,-92,52,-33) (5/8,7/11) -> (11/4,3/1) Glide Reflection Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(69,-56,16,-13) (4/5,5/6) -> (4/1,9/2) Glide Reflection Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(103,-122,38,-45) (1/1,6/5) -> (8/3,19/7) Hyperbolic Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(209,-338,128,-207) (8/5,13/8) -> (13/8,18/11) Parabolic Matrix(217,-356,64,-105) (18/11,5/3) -> (10/3,17/5) Glide Reflection Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(901,-2460,330,-901) (30/11,41/15) -> (30/11,41/15) Reflection Matrix(329,-902,120,-329) (41/15,11/4) -> (41/15,11/4) Reflection Matrix(43,-154,12,-43) (7/2,11/3) -> (7/2,11/3) Reflection Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(-1,0,2,1) -> Matrix(1,0,4,-1) (-1/1,0/1) -> (0/1,1/2) Matrix(35,-8,22,-5) -> Matrix(3,-2,2,-1) 1/1 Matrix(23,-6,88,-23) -> Matrix(3,-2,4,-3) (1/4,3/11) -> (1/2,1/1) Matrix(43,-12,154,-43) -> Matrix(1,0,2,-1) (3/11,2/7) -> (0/1,1/1) Matrix(55,-16,86,-25) -> Matrix(1,0,0,1) Matrix(63,-22,20,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(103,-38,122,-45) -> Matrix(1,0,0,1) Matrix(31,-12,80,-31) -> Matrix(3,-2,4,-3) (3/8,2/5) -> (1/2,1/1) Matrix(9,-4,20,-9) -> Matrix(1,0,2,-1) (2/5,1/2) -> (0/1,1/1) Matrix(19,-10,36,-19) -> Matrix(1,0,2,-1) (1/2,5/9) -> (0/1,1/1) Matrix(71,-40,126,-71) -> Matrix(1,0,4,-1) (5/9,4/7) -> (0/1,1/2) Matrix(107,-62,88,-51) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(239,-140,70,-41) -> Matrix(1,0,-4,1) 0/1 Matrix(95,-58,18,-11) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(129,-80,208,-129) -> Matrix(1,0,4,-1) (8/13,5/8) -> (0/1,1/2) Matrix(145,-92,52,-33) -> Matrix(3,-2,-2,1) Matrix(25,-18,32,-23) -> Matrix(1,0,0,1) Matrix(69,-56,16,-13) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(163,-144,60,-53) -> Matrix(1,0,-2,1) 0/1 Matrix(103,-122,38,-45) -> Matrix(1,-2,0,1) 1/0 Matrix(31,-40,24,-31) -> Matrix(1,0,2,-1) (5/4,4/3) -> (0/1,1/1) Matrix(17,-24,12,-17) -> Matrix(-1,2,0,1) (4/3,3/2) -> (1/1,1/0) Matrix(209,-338,128,-207) -> Matrix(1,-2,0,1) 1/0 Matrix(217,-356,64,-105) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(47,-80,10,-17) -> Matrix(1,-2,0,1) 1/0 Matrix(71,-126,40,-71) -> Matrix(-1,6,0,1) (7/4,9/5) -> (3/1,1/0) Matrix(19,-36,10,-19) -> Matrix(1,0,0,-1) (9/5,2/1) -> (0/1,1/0) Matrix(21,-50,8,-19) -> Matrix(1,-2,0,1) 1/0 Matrix(901,-2460,330,-901) -> Matrix(1,0,0,-1) (30/11,41/15) -> (0/1,1/0) Matrix(329,-902,120,-329) -> Matrix(1,4,0,-1) (41/15,11/4) -> (-2/1,1/0) Matrix(43,-154,12,-43) -> Matrix(-1,0,6,1) (7/2,11/3) -> (-1/3,0/1) Matrix(23,-88,6,-23) -> Matrix(1,0,2,-1) (11/3,4/1) -> (0/1,1/1) Matrix(-1,12,0,1) -> Matrix(1,0,0,-1) (6/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.