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 -6/1 -4/1 -2/1 -4/3 -8/7 0/1 1/1 16/13 4/3 16/11 3/2 8/5 16/9 2/1 16/7 5/2 8/3 48/17 3/1 16/5 7/2 15/4 4/1 9/2 5/1 16/3 11/2 6/1 13/2 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 1/0 -7/1 -1/2 -6/1 -1/2 -11/2 -1/6 0/1 -16/3 0/1 -5/1 1/2 -9/2 0/1 1/2 -13/3 1/2 -4/1 1/0 -7/2 -1/1 -1/2 -10/3 -1/2 -13/4 -1/1 1/0 -16/5 -1/1 -3/1 -1/2 -17/6 -1/1 1/0 -14/5 -1/2 -11/4 -2/3 -1/2 -8/3 -1/2 -5/2 -1/2 -1/3 -7/3 -1/6 -16/7 0/1 -9/4 0/1 1/2 -11/5 1/2 -2/1 -1/2 -9/5 -5/14 -16/9 -1/3 -7/4 -1/3 -3/10 -12/7 -1/4 -29/17 -1/2 -17/10 -1/3 -1/4 -5/3 -3/10 -18/11 -5/18 -49/30 -11/40 -3/11 -80/49 -3/11 -31/19 -7/26 -13/8 -3/11 -1/4 -8/5 -1/4 -3/2 -1/4 0/1 -16/11 0/1 -13/9 -1/2 -23/16 -1/2 0/1 -10/7 -1/2 -17/12 -1/3 -1/4 -24/17 -1/4 -7/5 -1/2 -18/13 -3/10 -11/8 -3/10 -2/7 -26/19 -5/18 -41/30 -2/7 -5/18 -15/11 -5/18 -4/3 -1/4 -17/13 -7/30 -13/10 -1/4 -3/13 -22/17 -5/22 -9/7 -5/22 -5/4 -3/14 -1/5 -16/13 -1/5 -11/9 -5/26 -17/14 -1/5 -3/16 -6/5 -1/6 -7/6 -1/5 -1/6 -8/7 -1/6 -9/8 -1/6 0/1 -1/1 -1/6 0/1 0/1 1/1 1/6 6/5 1/6 11/9 5/26 16/13 1/5 5/4 1/5 3/14 4/3 1/4 15/11 5/18 11/8 2/7 3/10 7/5 1/2 24/17 1/4 17/12 1/4 1/3 10/7 1/2 13/9 1/2 16/11 0/1 3/2 0/1 1/4 8/5 1/4 13/8 1/4 3/11 31/19 7/26 18/11 5/18 5/3 3/10 7/4 3/10 1/3 16/9 1/3 9/5 5/14 11/6 2/5 1/2 2/1 1/2 9/4 -1/2 0/1 16/7 0/1 7/3 1/6 19/8 1/5 1/4 31/13 3/14 12/5 1/4 5/2 1/3 1/2 8/3 1/2 11/4 1/2 2/3 14/5 1/2 31/11 5/6 48/17 1/1 17/6 1/1 1/0 3/1 1/2 16/5 1/1 13/4 1/1 1/0 10/3 1/2 17/5 1/2 41/12 0/1 1/2 24/7 1/2 7/2 1/2 1/1 11/3 1/2 26/7 3/2 15/4 1/1 1/0 4/1 1/0 17/4 -1/1 1/0 13/3 -1/2 22/5 -1/2 9/2 -1/2 0/1 14/3 1/2 5/1 -1/2 16/3 0/1 11/2 0/1 1/6 17/3 1/2 23/4 0/1 1/2 6/1 1/2 13/2 1/1 1/0 7/1 1/2 8/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,16,0,1) (-8/1,1/0) -> (8/1,1/0) Parabolic Matrix(31,224,22,159) (-8/1,-7/1) -> (7/5,24/17) Hyperbolic Matrix(31,208,-24,-161) (-7/1,-6/1) -> (-22/17,-9/7) Hyperbolic Matrix(63,352,-46,-257) (-6/1,-11/2) -> (-11/8,-26/19) Hyperbolic Matrix(65,352,12,65) (-11/2,-16/3) -> (16/3,11/2) Hyperbolic Matrix(31,160,6,31) (-16/3,-5/1) -> (5/1,16/3) Hyperbolic Matrix(31,144,-14,-65) (-5/1,-9/2) -> (-9/4,-11/5) Hyperbolic Matrix(95,416,-66,-289) (-9/2,-13/3) -> (-13/9,-23/16) Hyperbolic Matrix(65,272,-38,-159) (-13/3,-4/1) -> (-12/7,-29/17) Hyperbolic Matrix(31,112,-18,-65) (-4/1,-7/2) -> (-7/4,-12/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(127,416,-98,-321) (-10/3,-13/4) -> (-13/10,-22/17) Hyperbolic Matrix(129,416,40,129) (-13/4,-16/5) -> (16/5,13/4) Hyperbolic Matrix(31,96,10,31) (-16/5,-3/1) -> (3/1,16/5) Hyperbolic Matrix(95,272,22,63) (-3/1,-17/6) -> (17/4,13/3) Hyperbolic Matrix(353,992,-216,-607) (-17/6,-14/5) -> (-18/11,-49/30) Hyperbolic Matrix(63,176,34,95) (-14/5,-11/4) -> (11/6,2/1) Hyperbolic Matrix(65,176,24,65) (-11/4,-8/3) -> (8/3,11/4) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(33,80,-26,-63) (-5/2,-7/3) -> (-9/7,-5/4) Hyperbolic Matrix(97,224,42,97) (-7/3,-16/7) -> (16/7,7/3) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(95,208,58,127) (-11/5,-2/1) -> (18/11,5/3) Hyperbolic Matrix(97,176,-70,-127) (-2/1,-9/5) -> (-7/5,-18/13) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(127,224,72,127) (-16/9,-7/4) -> (7/4,16/9) Hyperbolic Matrix(319,544,112,191) (-29/17,-17/10) -> (17/6,3/1) Hyperbolic Matrix(161,272,-132,-223) (-17/10,-5/3) -> (-11/9,-17/14) Hyperbolic Matrix(97,160,20,33) (-5/3,-18/11) -> (14/3,5/1) Hyperbolic Matrix(2273,3712,804,1313) (-49/30,-80/49) -> (48/17,17/6) Hyperbolic Matrix(2431,3968,862,1407) (-80/49,-31/19) -> (31/11,48/17) Hyperbolic Matrix(609,992,256,417) (-31/19,-13/8) -> (19/8,31/13) Hyperbolic Matrix(129,208,80,129) (-13/8,-8/5) -> (8/5,13/8) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(65,96,44,65) (-3/2,-16/11) -> (16/11,3/2) Hyperbolic Matrix(287,416,198,287) (-16/11,-13/9) -> (13/9,16/11) Hyperbolic Matrix(703,1008,-514,-737) (-23/16,-10/7) -> (-26/19,-41/30) Hyperbolic Matrix(191,272,-158,-225) (-10/7,-17/12) -> (-17/14,-6/5) Hyperbolic Matrix(577,816,408,577) (-17/12,-24/17) -> (24/17,17/12) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(255,352,92,127) (-18/13,-11/8) -> (11/4,14/5) Hyperbolic Matrix(961,1312,282,385) (-41/30,-15/11) -> (17/5,41/12) Hyperbolic Matrix(95,128,-72,-97) (-15/11,-4/3) -> (-4/3,-17/13) Parabolic Matrix(479,624,294,383) (-17/13,-13/10) -> (13/8,31/19) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(287,352,234,287) (-16/13,-11/9) -> (11/9,16/13) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(479,544,140,159) (-8/7,-9/8) -> (41/12,24/7) Hyperbolic Matrix(159,176,28,31) (-9/8,-1/1) -> (17/3,23/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(225,-304,94,-127) (4/3,15/11) -> (31/13,12/5) Hyperbolic Matrix(257,-352,46,-63) (15/11,11/8) -> (11/2,17/3) Hyperbolic Matrix(127,-176,70,-97) (11/8,7/5) -> (9/5,11/6) Hyperbolic Matrix(417,-592,112,-159) (17/12,10/7) -> (26/7,15/4) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(65,-112,18,-31) (5/3,7/4) -> (7/2,11/3) Hyperbolic Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(63,-208,10,-33) (13/4,10/3) -> (6/1,13/2) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(127,-560,22,-97) (22/5,9/2) -> (23/4,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,16,0,1) -> Matrix(1,0,0,1) Matrix(31,224,22,159) -> Matrix(1,0,4,1) Matrix(31,208,-24,-161) -> Matrix(1,-2,-4,9) Matrix(63,352,-46,-257) -> Matrix(9,2,-32,-7) Matrix(65,352,12,65) -> Matrix(1,0,12,1) Matrix(31,160,6,31) -> Matrix(1,0,-4,1) Matrix(31,144,-14,-65) -> Matrix(1,0,0,1) Matrix(95,416,-66,-289) -> Matrix(1,0,-4,1) Matrix(65,272,-38,-159) -> Matrix(1,0,-4,1) Matrix(31,112,-18,-65) -> Matrix(1,2,-4,-7) Matrix(33,112,-28,-95) -> Matrix(1,0,-4,1) Matrix(127,416,-98,-321) -> Matrix(1,-2,-4,9) Matrix(129,416,40,129) -> Matrix(1,2,0,1) Matrix(31,96,10,31) -> Matrix(3,2,4,3) Matrix(95,272,22,63) -> Matrix(1,0,0,1) Matrix(353,992,-216,-607) -> Matrix(11,8,-40,-29) Matrix(63,176,34,95) -> Matrix(1,0,4,1) Matrix(65,176,24,65) -> Matrix(7,4,12,7) Matrix(31,80,12,31) -> Matrix(5,2,12,5) Matrix(33,80,-26,-63) -> Matrix(7,2,-32,-9) Matrix(97,224,42,97) -> Matrix(1,0,12,1) Matrix(127,288,56,127) -> Matrix(1,0,-4,1) Matrix(95,208,58,127) -> Matrix(1,-2,4,-7) Matrix(97,176,-70,-127) -> Matrix(11,4,-36,-13) Matrix(161,288,90,161) -> Matrix(29,10,84,29) Matrix(127,224,72,127) -> Matrix(19,6,60,19) Matrix(319,544,112,191) -> Matrix(1,0,4,1) Matrix(161,272,-132,-223) -> Matrix(5,2,-28,-11) Matrix(97,160,20,33) -> Matrix(7,2,-4,-1) Matrix(2273,3712,804,1313) -> Matrix(51,14,40,11) Matrix(2431,3968,862,1407) -> Matrix(81,22,92,25) Matrix(609,992,256,417) -> Matrix(7,2,24,7) Matrix(129,208,80,129) -> Matrix(23,6,88,23) Matrix(31,48,20,31) -> Matrix(1,0,8,1) Matrix(65,96,44,65) -> Matrix(1,0,8,1) Matrix(287,416,198,287) -> Matrix(1,0,4,1) Matrix(703,1008,-514,-737) -> Matrix(9,2,-32,-7) Matrix(191,272,-158,-225) -> Matrix(5,2,-28,-11) Matrix(577,816,408,577) -> Matrix(7,2,24,7) Matrix(159,224,22,31) -> Matrix(1,0,4,1) Matrix(255,352,92,127) -> Matrix(13,4,16,5) Matrix(961,1312,282,385) -> Matrix(7,2,-4,-1) Matrix(95,128,-72,-97) -> Matrix(23,6,-96,-25) Matrix(479,624,294,383) -> Matrix(1,0,8,1) Matrix(129,160,104,129) -> Matrix(29,6,140,29) Matrix(287,352,234,287) -> Matrix(51,10,260,51) Matrix(193,224,56,65) -> Matrix(11,2,16,3) Matrix(479,544,140,159) -> Matrix(1,0,8,1) Matrix(159,176,28,31) -> Matrix(1,0,8,1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(95,-112,28,-33) -> Matrix(1,0,-4,1) Matrix(289,-352,78,-95) -> Matrix(21,-4,16,-3) Matrix(63,-80,26,-33) -> Matrix(9,-2,32,-7) Matrix(225,-304,94,-127) -> Matrix(15,-4,64,-17) Matrix(257,-352,46,-63) -> Matrix(7,-2,32,-9) Matrix(127,-176,70,-97) -> Matrix(13,-4,36,-11) Matrix(417,-592,112,-159) -> Matrix(7,-2,4,-1) Matrix(289,-416,66,-95) -> Matrix(1,0,-4,1) Matrix(607,-992,216,-353) -> Matrix(29,-8,40,-11) Matrix(65,-112,18,-31) -> Matrix(7,-2,4,-1) Matrix(65,-144,14,-31) -> Matrix(1,0,0,1) Matrix(95,-224,14,-33) -> Matrix(1,0,-4,1) Matrix(63,-208,10,-33) -> Matrix(1,0,0,1) Matrix(33,-128,8,-31) -> Matrix(1,-2,0,1) Matrix(127,-560,22,-97) -> 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): 12 Degree of the the map X: 24 Degree of the the map Y: 64 Permutation triple for Y: ((2,6,23,36,63,41,24,7)(3,12,31,55,57,39,13,4)(5,18,50,19)(8,22,56,27)(9,32,33,10)(11,25)(14,21)(15,44,37,16)(17,26,49,28,51,53,20,47)(29,35,34,45,43,42,61,30)(46,62)(52,60); (1,4,16,34,62,51,50,63,64,55,44,61,46,17,5,2)(3,10,35,58,26,8,7,25,57,32,42,54,53,56,36,11)(6,21,31,9,30,59,28,27,41,14,13,33,45,48,47,22)(12,37,29,52,20,19,23,40,39,15,43,60,49,18,24,38); (1,2,8,28,60,29,9,3)(4,14,6,5,20,54,42,15)(7,18,17,48,45,16,12,11)(10,13,40,23,22,26,46,34)(19,51,59,30,44,39,25,36)(21,41,50,49,58,35,37,55)(24,27,53,62,61,32,31,38)(33,57,64,63,56,47,52,43)) ----------------------------------------------------------------------- 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 -6/1 -4/1 -2/1 -4/3 -8/7 0/1 1/1 4/3 3/2 8/5 2/1 16/7 5/2 8/3 3/1 16/5 7/2 4/1 5/1 16/3 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 -5/1 1/2 -4/1 1/0 -7/2 -1/1 -1/2 -10/3 -1/2 -3/1 -1/2 -8/3 -1/2 -5/2 -1/2 -1/3 -7/3 -1/6 -2/1 -1/2 -9/5 -5/14 -16/9 -1/3 -7/4 -1/3 -3/10 -12/7 -1/4 -5/3 -3/10 -8/5 -1/4 -3/2 -1/4 0/1 -16/11 0/1 -13/9 -1/2 -10/7 -1/2 -17/12 -1/3 -1/4 -24/17 -1/4 -7/5 -1/2 -4/3 -1/4 -9/7 -5/22 -5/4 -3/14 -1/5 -16/13 -1/5 -11/9 -5/26 -6/5 -1/6 -7/6 -1/5 -1/6 -8/7 -1/6 -1/1 -1/6 0/1 0/1 1/1 1/6 6/5 1/6 5/4 1/5 3/14 4/3 1/4 7/5 1/2 10/7 1/2 3/2 0/1 1/4 8/5 1/4 5/3 3/10 7/4 3/10 1/3 2/1 1/2 9/4 -1/2 0/1 16/7 0/1 7/3 1/6 12/5 1/4 5/2 1/3 1/2 8/3 1/2 3/1 1/2 16/5 1/1 13/4 1/1 1/0 10/3 1/2 17/5 1/2 24/7 1/2 7/2 1/2 1/1 4/1 1/0 9/2 -1/2 0/1 5/1 -1/2 16/3 0/1 11/2 0/1 1/6 6/1 1/2 7/1 1/2 8/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,112,-12,-79) (-6/1,1/0) -> (-10/7,-17/12) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(17,80,-10,-47) (-5/1,-4/1) -> (-12/7,-5/3) Hyperbolic Matrix(31,112,-18,-65) (-4/1,-7/2) -> (-7/4,-12/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(33,80,-26,-63) (-5/2,-7/3) -> (-9/7,-5/4) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(175,256,54,79) (-3/2,-16/11) -> (16/5,13/4) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(143,160,42,47) (-8/7,-1/1) -> (17/5,24/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(1,1,-4,-3) Matrix(17,96,-14,-79) -> Matrix(3,1,-16,-5) Matrix(17,80,-10,-47) -> Matrix(1,1,-4,-3) Matrix(31,112,-18,-65) -> Matrix(1,2,-4,-7) Matrix(33,112,-28,-95) -> Matrix(1,0,-4,1) Matrix(49,160,-34,-111) -> Matrix(1,1,-4,-3) Matrix(17,48,6,17) -> Matrix(1,1,0,1) Matrix(31,80,12,31) -> Matrix(5,2,12,5) Matrix(33,80,-26,-63) -> Matrix(7,2,-32,-9) Matrix(15,32,-8,-17) -> Matrix(1,1,-4,-3) Matrix(143,256,62,111) -> Matrix(3,1,32,11) Matrix(145,256,64,113) -> Matrix(3,1,-16,-5) Matrix(49,80,30,49) -> Matrix(11,3,40,11) Matrix(31,48,20,31) -> Matrix(1,0,8,1) Matrix(175,256,54,79) -> Matrix(5,1,4,1) Matrix(177,256,56,81) -> Matrix(1,1,0,1) Matrix(113,160,12,17) -> Matrix(3,1,-4,-1) Matrix(159,224,22,31) -> Matrix(1,0,4,1) Matrix(47,64,-36,-49) -> Matrix(11,3,-48,-13) Matrix(207,256,38,47) -> Matrix(5,1,44,9) Matrix(209,256,40,49) -> Matrix(5,1,-36,-7) Matrix(193,224,56,65) -> Matrix(11,2,16,3) Matrix(143,160,42,47) -> Matrix(5,1,4,1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(95,-112,28,-33) -> Matrix(1,0,-4,1) Matrix(79,-96,14,-17) -> Matrix(5,-1,16,-3) Matrix(63,-80,26,-33) -> Matrix(9,-2,32,-7) Matrix(81,-112,34,-47) -> Matrix(3,-1,16,-5) Matrix(79,-112,12,-17) -> Matrix(3,-1,4,-1) Matrix(111,-160,34,-49) -> Matrix(3,-1,4,-1) Matrix(47,-80,10,-17) -> Matrix(3,-1,4,-1) Matrix(17,-32,8,-15) -> Matrix(3,-1,4,-1) Matrix(17,-64,4,-15) -> Matrix(1,-1,0,1) 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 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): 12 ----------------------------------------------------------------------- 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 6 1 1/1 1/6 1 16 6/5 1/6 1 8 5/4 (1/5,3/14) 0 16 4/3 1/4 3 4 7/5 1/2 1 16 10/7 1/2 1 8 3/2 (0/1,1/4) 0 16 8/5 1/4 3 2 5/3 3/10 1 16 7/4 (3/10,1/3) 0 16 2/1 1/2 1 8 9/4 (-1/2,0/1) 0 16 16/7 0/1 8 1 7/3 1/6 1 16 12/5 1/4 3 4 5/2 (1/3,1/2) 0 16 8/3 1/2 3 2 3/1 1/2 1 16 16/5 1/1 2 1 13/4 (1/1,1/0) 0 16 10/3 1/2 1 8 17/5 1/2 1 16 24/7 1/2 1 2 7/2 (1/2,1/1) 0 16 4/1 1/0 1 4 9/2 (-1/2,0/1) 0 16 5/1 -1/2 1 16 16/3 0/1 8 1 11/2 (0/1,1/6) 0 16 6/1 1/2 1 8 7/1 1/2 1 16 8/1 1/0 1 2 1/0 (0/1,1/0) 0 16 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(31,-96,10,-31) (3/1,16/5) -> (3/1,16/5) Reflection Matrix(129,-416,40,-129) (16/5,13/4) -> (16/5,13/4) Reflection Matrix(239,-816,70,-239) (17/5,24/7) -> (17/5,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(65,-352,12,-65) (16/3,11/2) -> (16/3,11/2) Reflection Matrix(15,-112,2,-15) (7/1,8/1) -> (7/1,8/1) Reflection Matrix(-1,16,0,1) (8/1,1/0) -> (8/1,1/0) 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(1,0,2,-1) -> Matrix(1,0,12,-1) (0/1,1/1) -> (0/1,1/6) Matrix(95,-112,28,-33) -> Matrix(1,0,-4,1) 0/1 Matrix(79,-96,14,-17) -> Matrix(5,-1,16,-3) 1/4 Matrix(63,-80,26,-33) -> Matrix(9,-2,32,-7) 1/4 Matrix(81,-112,34,-47) -> Matrix(3,-1,16,-5) 1/4 Matrix(79,-112,12,-17) -> Matrix(3,-1,4,-1) 1/2 Matrix(111,-160,34,-49) -> Matrix(3,-1,4,-1) 1/2 Matrix(31,-48,20,-31) -> Matrix(1,0,8,-1) (3/2,8/5) -> (0/1,1/4) Matrix(49,-80,30,-49) -> Matrix(11,-3,40,-11) (8/5,5/3) -> (1/4,3/10) Matrix(47,-80,10,-17) -> Matrix(3,-1,4,-1) 1/2 Matrix(17,-32,8,-15) -> Matrix(3,-1,4,-1) 1/2 Matrix(127,-288,56,-127) -> Matrix(-1,0,4,1) (9/4,16/7) -> (-1/2,0/1) Matrix(97,-224,42,-97) -> Matrix(1,0,12,-1) (16/7,7/3) -> (0/1,1/6) Matrix(31,-80,12,-31) -> Matrix(5,-2,12,-5) (5/2,8/3) -> (1/3,1/2) Matrix(17,-48,6,-17) -> Matrix(-1,1,0,1) (8/3,3/1) -> (1/2,1/0) Matrix(31,-96,10,-31) -> Matrix(3,-2,4,-3) (3/1,16/5) -> (1/2,1/1) Matrix(129,-416,40,-129) -> Matrix(-1,2,0,1) (16/5,13/4) -> (1/1,1/0) Matrix(239,-816,70,-239) -> Matrix(-1,1,0,1) (17/5,24/7) -> (1/2,1/0) Matrix(97,-336,28,-97) -> Matrix(3,-2,4,-3) (24/7,7/2) -> (1/2,1/1) Matrix(17,-64,4,-15) -> Matrix(1,-1,0,1) 1/0 Matrix(31,-160,6,-31) -> Matrix(-1,0,4,1) (5/1,16/3) -> (-1/2,0/1) Matrix(65,-352,12,-65) -> Matrix(1,0,12,-1) (16/3,11/2) -> (0/1,1/6) Matrix(15,-112,2,-15) -> Matrix(-1,1,0,1) (7/1,8/1) -> (1/2,1/0) Matrix(-1,16,0,1) -> Matrix(1,0,0,-1) (8/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.