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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/2 -1/1 -1/2 -1/3 -2/13 -2/15 0/1 1/8 2/11 2/9 1/4 2/7 1/3 3/8 2/5 7/16 1/2 9/16 4/7 5/8 2/3 3/4 4/5 1/1 5/4 4/3 11/8 3/2 13/8 7/4 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/2 -7/4 1/2 -12/7 1/0 -5/3 1/2 -18/11 -1/2 0/1 -13/8 0/1 -8/5 1/4 -11/7 1/2 -14/9 0/1 1/4 -3/2 1/2 -7/5 -1/2 -18/13 -1/6 0/1 -11/8 0/1 -4/3 1/4 -13/10 1/2 -22/17 1/3 1/2 -9/7 1/2 -14/11 2/5 1/2 -5/4 1/2 -1/1 1/2 -5/6 1/2 -4/5 1/0 -7/9 1/2 -10/13 0/1 1/2 -3/4 1/2 -2/3 1/2 1/1 -5/8 1/1 -8/13 9/8 -11/18 7/6 -3/5 3/2 -10/17 1/1 3/2 -7/12 3/2 -4/7 1/0 -9/16 1/1 -5/9 3/2 -6/11 3/2 2/1 -1/2 1/0 -6/13 -1/2 0/1 -5/11 1/2 -4/9 1/0 -7/16 0/1 -10/23 0/1 1/4 -3/7 1/2 -5/12 1/2 -2/5 1/2 1/1 -3/8 1/1 -4/11 5/4 -5/14 3/2 -6/17 1/1 3/2 -1/3 3/2 -3/10 5/2 -5/17 5/2 -7/24 3/1 -2/7 3/1 1/0 -1/4 1/0 -2/9 -2/1 1/0 -3/14 -3/2 -1/5 -1/2 -2/11 0/1 1/2 -1/6 1/0 -2/13 0/1 1/2 -1/7 1/2 -2/15 1/2 1/1 -1/8 1/1 0/1 1/0 1/8 -1/1 1/7 -1/2 1/6 1/0 2/11 -1/2 0/1 1/5 1/2 3/14 3/2 2/9 2/1 1/0 1/4 1/0 2/7 -3/1 1/0 1/3 -3/2 4/11 -5/4 3/8 -1/1 2/5 -1/1 -1/2 3/7 -1/2 7/16 0/1 4/9 1/0 1/2 1/0 6/11 -2/1 -3/2 5/9 -3/2 9/16 -1/1 4/7 1/0 7/12 -3/2 10/17 -3/2 -1/1 3/5 -3/2 11/18 -7/6 8/13 -9/8 5/8 -1/1 2/3 -1/1 -1/2 3/4 -1/2 10/13 -1/2 0/1 17/22 1/0 7/9 -1/2 4/5 1/0 9/11 -1/2 14/17 -1/1 -1/2 5/6 -1/2 1/1 -1/2 6/5 -1/1 -1/2 5/4 -1/2 14/11 -1/2 -2/5 9/7 -1/2 31/24 -1/3 22/17 -1/2 -1/3 13/10 -1/2 4/3 -1/4 11/8 0/1 18/13 0/1 1/6 7/5 1/2 10/7 -1/1 1/0 23/16 -1/1 13/9 -1/2 3/2 -1/2 14/9 -1/4 0/1 25/16 0/1 11/7 -1/2 8/5 -1/4 13/8 0/1 18/11 0/1 1/2 5/3 -1/2 22/13 0/1 1/2 17/10 1/0 12/7 1/0 7/4 -1/2 2/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(65,112,112,193) (-7/4,-12/7) -> (4/7,7/12) Hyperbolic Matrix(47,80,-104,-177) (-12/7,-5/3) -> (-5/11,-4/9) Hyperbolic Matrix(17,28,88,145) (-5/3,-18/11) -> (2/11,1/5) Hyperbolic Matrix(287,468,176,287) (-18/11,-13/8) -> (13/8,18/11) Hyperbolic Matrix(129,208,80,129) (-13/8,-8/5) -> (8/5,13/8) Hyperbolic Matrix(63,100,80,127) (-8/5,-11/7) -> (7/9,4/5) Hyperbolic Matrix(97,152,-224,-351) (-11/7,-14/9) -> (-10/23,-3/7) Hyperbolic Matrix(31,48,144,223) (-14/9,-3/2) -> (3/14,2/9) Hyperbolic Matrix(17,24,-56,-79) (-3/2,-7/5) -> (-1/3,-3/10) Hyperbolic Matrix(95,132,-208,-289) (-7/5,-18/13) -> (-6/13,-5/11) Hyperbolic Matrix(287,396,208,287) (-18/13,-11/8) -> (11/8,18/13) Hyperbolic Matrix(65,88,48,65) (-11/8,-4/3) -> (4/3,11/8) Hyperbolic Matrix(113,148,184,241) (-4/3,-13/10) -> (11/18,8/13) Hyperbolic Matrix(225,292,272,353) (-13/10,-22/17) -> (14/17,5/6) Hyperbolic Matrix(31,40,-224,-289) (-22/17,-9/7) -> (-1/7,-2/15) Hyperbolic Matrix(97,124,176,225) (-9/7,-14/11) -> (6/11,5/9) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(17,20,-40,-47) (-5/4,-1/1) -> (-3/7,-5/12) Hyperbolic Matrix(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(49,40,-136,-111) (-5/6,-4/5) -> (-4/11,-5/14) Hyperbolic Matrix(127,100,80,63) (-4/5,-7/9) -> (11/7,8/5) Hyperbolic Matrix(31,24,-208,-161) (-7/9,-10/13) -> (-2/13,-1/7) Hyperbolic Matrix(79,60,104,79) (-10/13,-3/4) -> (3/4,10/13) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(241,148,184,113) (-8/13,-11/18) -> (13/10,4/3) Hyperbolic Matrix(33,20,160,97) (-11/18,-3/5) -> (1/5,3/14) Hyperbolic Matrix(47,28,-136,-81) (-3/5,-10/17) -> (-6/17,-1/3) Hyperbolic Matrix(239,140,408,239) (-10/17,-7/12) -> (7/12,10/17) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(127,72,224,127) (-4/7,-9/16) -> (9/16,4/7) Hyperbolic Matrix(143,80,-488,-273) (-9/16,-5/9) -> (-5/17,-7/24) Hyperbolic Matrix(225,124,176,97) (-5/9,-6/11) -> (14/11,9/7) Hyperbolic Matrix(15,8,88,47) (-6/11,-1/2) -> (1/6,2/11) Hyperbolic Matrix(241,112,312,145) (-1/2,-6/13) -> (10/13,17/22) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(799,348,512,223) (-7/16,-10/23) -> (14/9,25/16) Hyperbolic Matrix(97,40,80,33) (-5/12,-2/5) -> (6/5,5/4) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(65,24,176,65) (-3/8,-4/11) -> (4/11,3/8) Hyperbolic Matrix(529,188,408,145) (-5/14,-6/17) -> (22/17,13/10) Hyperbolic Matrix(401,116,280,81) (-7/24,-2/7) -> (10/7,23/16) Hyperbolic Matrix(15,4,56,15) (-2/7,-1/4) -> (1/4,2/7) Hyperbolic Matrix(17,4,72,17) (-1/4,-2/9) -> (2/9,1/4) Hyperbolic Matrix(223,48,144,31) (-2/9,-3/14) -> (3/2,14/9) Hyperbolic Matrix(97,20,160,33) (-3/14,-1/5) -> (3/5,11/18) Hyperbolic Matrix(145,28,88,17) (-1/5,-2/11) -> (18/11,5/3) Hyperbolic Matrix(47,8,88,15) (-2/11,-1/6) -> (1/2,6/11) Hyperbolic Matrix(353,56,208,33) (-1/6,-2/13) -> (22/13,17/10) Hyperbolic Matrix(641,84,496,65) (-2/15,-1/8) -> (31/24,22/17) Hyperbolic Matrix(1,0,16,1) (-1/8,0/1) -> (0/1,1/8) Parabolic Matrix(289,-40,224,-31) (1/8,1/7) -> (9/7,31/24) Hyperbolic Matrix(161,-24,208,-31) (1/7,1/6) -> (17/22,7/9) Hyperbolic Matrix(79,-24,56,-17) (2/7,1/3) -> (7/5,10/7) Hyperbolic Matrix(111,-40,136,-49) (1/3,4/11) -> (4/5,9/11) Hyperbolic Matrix(47,-20,40,-17) (2/5,3/7) -> (1/1,6/5) Hyperbolic Matrix(351,-152,224,-97) (3/7,7/16) -> (25/16,11/7) Hyperbolic Matrix(177,-80,104,-47) (4/9,1/2) -> (17/10,12/7) Hyperbolic Matrix(415,-232,288,-161) (5/9,9/16) -> (23/16,13/9) Hyperbolic Matrix(223,-132,272,-161) (10/17,3/5) -> (9/11,14/17) Hyperbolic Matrix(81,-68,56,-47) (5/6,1/1) -> (13/9,3/2) Hyperbolic Matrix(175,-244,104,-145) (18/13,7/5) -> (5/3,22/13) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-4,1) Matrix(15,28,8,15) -> Matrix(1,0,-4,1) Matrix(65,112,112,193) -> Matrix(1,-2,0,1) Matrix(47,80,-104,-177) -> Matrix(1,0,0,1) Matrix(17,28,88,145) -> Matrix(1,0,0,1) Matrix(287,468,176,287) -> Matrix(1,0,4,1) Matrix(129,208,80,129) -> Matrix(1,0,-8,1) Matrix(63,100,80,127) -> Matrix(1,0,-4,1) Matrix(97,152,-224,-351) -> Matrix(1,0,0,1) Matrix(31,48,144,223) -> Matrix(7,-2,4,-1) Matrix(17,24,-56,-79) -> Matrix(1,2,0,1) Matrix(95,132,-208,-289) -> Matrix(1,0,4,1) Matrix(287,396,208,287) -> Matrix(1,0,12,1) Matrix(65,88,48,65) -> Matrix(1,0,-8,1) Matrix(113,148,184,241) -> Matrix(23,-8,-20,7) Matrix(225,292,272,353) -> Matrix(1,0,-4,1) Matrix(31,40,-224,-289) -> Matrix(5,-2,8,-3) Matrix(97,124,176,225) -> Matrix(11,-4,-8,3) Matrix(111,140,88,111) -> Matrix(9,-4,-20,9) Matrix(17,20,-40,-47) -> Matrix(1,0,0,1) Matrix(33,28,-112,-95) -> Matrix(1,2,0,1) Matrix(49,40,-136,-111) -> Matrix(5,-4,4,-3) Matrix(127,100,80,63) -> Matrix(1,0,-4,1) Matrix(31,24,-208,-161) -> Matrix(1,0,0,1) Matrix(79,60,104,79) -> Matrix(1,0,-4,1) Matrix(17,12,24,17) -> Matrix(3,-2,-4,3) Matrix(31,20,48,31) -> Matrix(3,-2,-4,3) Matrix(129,80,208,129) -> Matrix(17,-18,-16,17) Matrix(241,148,184,113) -> Matrix(7,-8,-20,23) Matrix(33,20,160,97) -> Matrix(3,-4,4,-5) Matrix(47,28,-136,-81) -> Matrix(1,0,0,1) Matrix(239,140,408,239) -> Matrix(5,-6,-4,5) Matrix(193,112,112,65) -> Matrix(1,-2,0,1) Matrix(127,72,224,127) -> Matrix(1,-2,0,1) Matrix(143,80,-488,-273) -> Matrix(11,-14,4,-5) Matrix(225,124,176,97) -> Matrix(3,-4,-8,11) Matrix(15,8,88,47) -> Matrix(1,-2,0,1) Matrix(241,112,312,145) -> Matrix(1,0,0,1) Matrix(127,56,288,127) -> Matrix(1,0,0,1) Matrix(799,348,512,223) -> Matrix(1,0,-8,1) Matrix(97,40,80,33) -> Matrix(3,-2,-4,3) Matrix(31,12,80,31) -> Matrix(3,-2,-4,3) Matrix(65,24,176,65) -> Matrix(9,-10,-8,9) Matrix(529,188,408,145) -> Matrix(3,-4,-8,11) Matrix(401,116,280,81) -> Matrix(1,-4,0,1) Matrix(15,4,56,15) -> Matrix(1,-6,0,1) Matrix(17,4,72,17) -> Matrix(1,4,0,1) Matrix(223,48,144,31) -> Matrix(1,2,-4,-7) Matrix(97,20,160,33) -> Matrix(5,4,-4,-3) Matrix(145,28,88,17) -> Matrix(1,0,0,1) Matrix(47,8,88,15) -> Matrix(1,-2,0,1) Matrix(353,56,208,33) -> Matrix(1,0,0,1) Matrix(641,84,496,65) -> Matrix(1,0,-4,1) Matrix(1,0,16,1) -> Matrix(1,-2,0,1) Matrix(289,-40,224,-31) -> Matrix(3,2,-8,-5) Matrix(161,-24,208,-31) -> Matrix(1,0,0,1) Matrix(79,-24,56,-17) -> Matrix(1,2,0,1) Matrix(111,-40,136,-49) -> Matrix(3,4,-4,-5) Matrix(47,-20,40,-17) -> Matrix(1,0,0,1) Matrix(351,-152,224,-97) -> Matrix(1,0,0,1) Matrix(177,-80,104,-47) -> Matrix(1,0,0,1) Matrix(415,-232,288,-161) -> Matrix(3,4,-4,-5) Matrix(223,-132,272,-161) -> Matrix(3,4,-4,-5) Matrix(81,-68,56,-47) -> Matrix(1,0,0,1) Matrix(175,-244,104,-145) -> Matrix(1,0,-4,1) 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): 11 Degree of the the map X: 22 Degree of the the map Y: 64 Permutation triple for Y: ((1,6,23,56,57,24,7,2)(3,12,13,4)(5,10,9,18)(8,21,20,28)(11,35,49,32,45,16,15,36)(14,38,37,17,47,27,34,33)(19,39,29,53,40,31,30,51)(22,43,59,26)(25,42,61,54)(41,52,46,55)(44,58,48,62)(50,63,64,60); (1,4,16,20,53,42,33,58,57,63,35,52,51,43,17,5)(2,10,34,59,30,46,45,64,56,48,37,61,29,28,11,3)(6,21,47,50,19,18,49,25,24,41,14,13,40,44,15,22)(7,26,36,62,39,12,38,55,23,54,32,9,31,60,27,8); (1,3)(2,8,29,62,56,55,30,9)(4,14,42,49,63,47,43,15)(5,19,52,24,58,40,20,6)(7,25,23,22)(10,17,48,33)(11,26,34,60,45,54,37,12)(13,39,50,31)(16,46,35,28)(18,32)(21,27)(36,44)(38,41)(51,59)(53,61)(57,64)) ----------------------------------------------------------------------- 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): 192 Minimal number of generators: 33 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: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/2 -1/1 -1/2 -7/16 -1/3 0/1 1/4 2/7 1/3 3/8 2/5 1/2 4/7 5/8 2/3 3/4 4/5 1/1 5/4 4/3 3/2 7/4 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/2 -7/4 1/2 -5/3 1/2 -8/5 1/4 -3/2 1/2 -7/5 -1/2 -4/3 1/4 -5/4 1/2 -1/1 1/2 -5/6 1/2 -4/5 1/0 -3/4 1/2 -2/3 1/2 1/1 -5/8 1/1 -8/13 9/8 -3/5 3/2 -7/12 3/2 -4/7 1/0 -1/2 1/0 -4/9 1/0 -7/16 0/1 -3/7 1/2 -5/12 1/2 -2/5 1/2 1/1 -3/8 1/1 -4/11 5/4 -1/3 3/2 -3/10 5/2 -5/17 5/2 -7/24 3/1 -2/7 3/1 1/0 -1/4 1/0 0/1 1/0 1/4 1/0 2/7 -3/1 1/0 1/3 -3/2 3/8 -1/1 2/5 -1/1 -1/2 3/7 -1/2 1/2 1/0 5/9 -3/2 9/16 -1/1 4/7 1/0 3/5 -3/2 5/8 -1/1 2/3 -1/1 -1/2 3/4 -1/2 4/5 1/0 1/1 -1/2 6/5 -1/1 -1/2 5/4 -1/2 4/3 -1/4 11/8 0/1 7/5 1/2 10/7 -1/1 1/0 3/2 -1/2 14/9 -1/4 0/1 25/16 0/1 11/7 -1/2 8/5 -1/4 13/8 0/1 5/3 -1/2 12/7 1/0 7/4 -1/2 2/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(33,56,-56,-95) (-7/4,-5/3) -> (-3/5,-7/12) Hyperbolic Matrix(17,28,-48,-79) (-5/3,-8/5) -> (-4/11,-1/3) Hyperbolic Matrix(33,52,-40,-63) (-8/5,-3/2) -> (-5/6,-4/5) Hyperbolic Matrix(17,24,-56,-79) (-3/2,-7/5) -> (-1/3,-3/10) Hyperbolic Matrix(49,68,-80,-111) (-7/5,-4/3) -> (-8/13,-3/5) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(17,20,-40,-47) (-5/4,-1/1) -> (-3/7,-5/12) Hyperbolic Matrix(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(175,108,128,79) (-5/8,-8/13) -> (4/3,11/8) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(129,56,-440,-191) (-7/16,-3/7) -> (-5/17,-7/24) Hyperbolic Matrix(97,40,80,33) (-5/12,-2/5) -> (6/5,5/4) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(207,76,128,47) (-3/8,-4/11) -> (8/5,13/8) Hyperbolic Matrix(511,148,328,95) (-7/24,-2/7) -> (14/9,25/16) Hyperbolic Matrix(15,4,56,15) (-2/7,-1/4) -> (1/4,2/7) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(79,-24,56,-17) (2/7,1/3) -> (7/5,10/7) Hyperbolic Matrix(79,-28,48,-17) (1/3,3/8) -> (13/8,5/3) Hyperbolic Matrix(47,-20,40,-17) (2/5,3/7) -> (1/1,6/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(401,-224,256,-143) (5/9,9/16) -> (25/16,11/7) Hyperbolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(111,-68,80,-49) (3/5,5/8) -> (11/8,7/5) Hyperbolic Matrix(63,-52,40,-33) (4/5,1/1) -> (11/7,8/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-4,1) Matrix(15,28,8,15) -> Matrix(1,0,-4,1) Matrix(33,56,-56,-95) -> Matrix(1,1,0,1) Matrix(17,28,-48,-79) -> Matrix(1,1,0,1) Matrix(33,52,-40,-63) -> Matrix(3,-1,4,-1) Matrix(17,24,-56,-79) -> Matrix(1,2,0,1) Matrix(49,68,-80,-111) -> Matrix(5,1,4,1) Matrix(31,40,24,31) -> Matrix(3,-1,-8,3) Matrix(17,20,-40,-47) -> Matrix(1,0,0,1) Matrix(33,28,-112,-95) -> Matrix(1,2,0,1) Matrix(31,24,40,31) -> Matrix(1,-1,0,1) Matrix(17,12,24,17) -> Matrix(3,-2,-4,3) Matrix(31,20,48,31) -> Matrix(3,-2,-4,3) Matrix(175,108,128,79) -> Matrix(1,-1,-12,13) Matrix(193,112,112,65) -> Matrix(1,-2,0,1) Matrix(15,8,-32,-17) -> Matrix(1,-1,0,1) Matrix(145,64,256,113) -> Matrix(1,-1,0,1) Matrix(129,56,-440,-191) -> Matrix(11,-3,4,-1) Matrix(97,40,80,33) -> Matrix(3,-2,-4,3) Matrix(31,12,80,31) -> Matrix(3,-2,-4,3) Matrix(207,76,128,47) -> Matrix(1,-1,-8,9) Matrix(511,148,328,95) -> Matrix(1,-3,-4,13) Matrix(15,4,56,15) -> Matrix(1,-6,0,1) Matrix(1,0,8,1) -> Matrix(1,-1,0,1) Matrix(79,-24,56,-17) -> Matrix(1,2,0,1) Matrix(79,-28,48,-17) -> Matrix(1,1,0,1) Matrix(47,-20,40,-17) -> Matrix(1,0,0,1) Matrix(17,-8,32,-15) -> Matrix(1,-1,0,1) Matrix(401,-224,256,-143) -> Matrix(1,1,0,1) Matrix(95,-56,56,-33) -> Matrix(1,1,0,1) Matrix(111,-68,80,-49) -> Matrix(1,1,4,5) Matrix(63,-52,40,-33) -> Matrix(1,1,-4,-3) Matrix(49,-72,32,-47) -> Matrix(1,1,-4,-3) 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): 11 ----------------------------------------------------------------------- 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 1/0 2 16 1/4 1/0 5 4 2/7 (-3/1,1/0) 0 16 1/3 -3/2 1 16 3/8 -1/1 6 2 2/5 (-1/1,-1/2) 0 16 3/7 -1/2 1 16 1/2 1/0 1 8 5/9 -3/2 1 16 9/16 -1/1 4 2 4/7 1/0 2 16 3/5 -3/2 1 16 5/8 -1/1 10 2 2/3 (-1/1,-1/2) 0 16 3/4 -1/2 1 4 4/5 1/0 2 16 1/1 -1/2 1 16 6/5 (-1/1,-1/2) 0 16 5/4 -1/2 3 4 4/3 -1/4 2 16 11/8 0/1 10 2 7/5 1/2 1 16 10/7 (-1/1,1/0) 0 16 3/2 -1/2 1 8 14/9 (-1/4,0/1) 0 16 25/16 0/1 4 2 11/7 -1/2 1 16 8/5 -1/4 2 16 13/8 0/1 6 2 5/3 -1/2 1 16 12/7 1/0 2 16 7/4 -1/2 1 4 2/1 (-1/2,0/1) 0 16 1/0 0/1 2 2 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(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(79,-24,56,-17) (2/7,1/3) -> (7/5,10/7) Hyperbolic Matrix(79,-28,48,-17) (1/3,3/8) -> (13/8,5/3) Hyperbolic Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(47,-20,40,-17) (2/5,3/7) -> (1/1,6/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(401,-224,256,-143) (5/9,9/16) -> (25/16,11/7) Hyperbolic Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(111,-68,80,-49) (3/5,5/8) -> (11/8,7/5) Hyperbolic Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(63,-52,40,-33) (4/5,1/1) -> (11/7,8/5) Hyperbolic Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(65,-88,48,-65) (4/3,11/8) -> (4/3,11/8) Reflection Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(449,-700,288,-449) (14/9,25/16) -> (14/9,25/16) Reflection Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) 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(1,0,8,-1) -> Matrix(1,1,0,-1) (0/1,1/4) -> (-1/2,1/0) Matrix(15,-4,56,-15) -> Matrix(1,6,0,-1) (1/4,2/7) -> (-3/1,1/0) Matrix(79,-24,56,-17) -> Matrix(1,2,0,1) 1/0 Matrix(79,-28,48,-17) -> Matrix(1,1,0,1) 1/0 Matrix(31,-12,80,-31) -> Matrix(3,2,-4,-3) (3/8,2/5) -> (-1/1,-1/2) Matrix(47,-20,40,-17) -> Matrix(1,0,0,1) Matrix(17,-8,32,-15) -> Matrix(1,-1,0,1) 1/0 Matrix(401,-224,256,-143) -> Matrix(1,1,0,1) 1/0 Matrix(127,-72,224,-127) -> Matrix(1,2,0,-1) (9/16,4/7) -> (-1/1,1/0) Matrix(95,-56,56,-33) -> Matrix(1,1,0,1) 1/0 Matrix(111,-68,80,-49) -> Matrix(1,1,4,5) Matrix(31,-20,48,-31) -> Matrix(3,2,-4,-3) (5/8,2/3) -> (-1/1,-1/2) Matrix(17,-12,24,-17) -> Matrix(3,2,-4,-3) (2/3,3/4) -> (-1/1,-1/2) Matrix(31,-24,40,-31) -> Matrix(1,1,0,-1) (3/4,4/5) -> (-1/2,1/0) Matrix(63,-52,40,-33) -> Matrix(1,1,-4,-3) -1/2 Matrix(49,-60,40,-49) -> Matrix(3,2,-4,-3) (6/5,5/4) -> (-1/1,-1/2) Matrix(31,-40,24,-31) -> Matrix(3,1,-8,-3) (5/4,4/3) -> (-1/2,-1/4) Matrix(65,-88,48,-65) -> Matrix(-1,0,8,1) (4/3,11/8) -> (-1/4,0/1) Matrix(49,-72,32,-47) -> Matrix(1,1,-4,-3) -1/2 Matrix(449,-700,288,-449) -> Matrix(-1,0,8,1) (14/9,25/16) -> (-1/4,0/1) Matrix(129,-208,80,-129) -> Matrix(-1,0,8,1) (8/5,13/8) -> (-1/4,0/1) Matrix(97,-168,56,-97) -> Matrix(1,1,0,-1) (12/7,7/4) -> (-1/2,1/0) Matrix(15,-28,8,-15) -> Matrix(-1,0,4,1) (7/4,2/1) -> (-1/2,0/1) Matrix(-1,4,0,1) -> Matrix(-1,0,4,1) (2/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.