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: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/8 -3/4 -5/8 -7/12 -1/2 -7/16 -5/12 -3/8 -5/14 -17/48 -5/16 -3/10 -19/64 -1/4 -7/32 -3/14 -3/16 -1/6 -5/32 -1/8 0/1 1/8 1/7 1/6 1/5 1/4 2/7 3/10 1/3 3/8 2/5 5/12 1/2 7/12 5/8 2/3 3/4 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -7/8 1/0 -6/7 -5/2 -5/6 -2/1 -4/5 1/0 -3/4 -1/1 -8/11 -1/2 -13/18 -2/1 -5/7 -3/4 -17/24 -1/2 -12/17 -1/2 -7/10 0/1 -2/3 -1/2 -5/8 0/1 -8/13 1/0 -19/31 1/0 -11/18 -2/1 -3/5 -1/2 -13/22 0/1 -10/17 -1/2 -7/12 -1/3 -18/31 -1/4 -11/19 -1/4 -15/26 0/1 -4/7 -1/6 -1/2 0/1 -4/9 -1/4 -7/16 0/1 -3/7 1/6 -8/19 1/4 -5/12 1/3 -12/29 1/2 -7/17 1/2 -2/5 1/2 -3/8 0/1 -4/11 1/4 -5/14 2/5 -11/31 1/2 -17/48 1/2 -6/17 1/2 -1/3 1/2 -5/16 1/1 -4/13 1/0 -3/10 0/1 -11/37 1/2 -19/64 1/3 1/1 -8/27 1/2 -5/17 1/2 -7/24 1/2 -2/7 3/4 -1/4 1/1 -2/9 1/2 -7/32 1/1 -5/23 9/8 -3/14 4/3 -1/5 1/0 -3/16 1/1 -2/11 3/2 -1/6 2/1 -3/19 1/0 -5/32 1/1 3/1 -2/13 1/0 -1/7 5/2 -1/8 1/0 0/1 1/0 1/8 1/0 1/7 -5/2 1/6 -2/1 1/5 1/0 1/4 -1/1 3/11 -1/2 5/18 -2/1 2/7 -3/4 7/24 -1/2 5/17 -1/2 3/10 0/1 1/3 -1/2 3/8 0/1 5/13 1/0 12/31 1/0 7/18 -2/1 2/5 -1/2 9/22 0/1 7/17 -1/2 5/12 -1/3 13/31 -1/4 8/19 -1/4 11/26 0/1 3/7 -1/6 1/2 0/1 5/9 -1/4 9/16 0/1 4/7 1/6 11/19 1/4 7/12 1/3 17/29 1/2 10/17 1/2 3/5 1/2 5/8 0/1 7/11 1/4 9/14 2/5 20/31 1/2 31/48 1/2 11/17 1/2 2/3 1/2 11/16 1/1 9/13 1/0 7/10 0/1 26/37 1/2 45/64 1/3 1/1 19/27 1/2 12/17 1/2 17/24 1/2 5/7 3/4 3/4 1/1 7/9 1/2 25/32 1/1 18/23 9/8 11/14 4/3 4/5 1/0 13/16 1/1 9/11 3/2 5/6 2/1 16/19 1/0 27/32 1/1 3/1 11/13 1/0 6/7 5/2 7/8 1/0 1/1 1/0 1/0 -1/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(47,42,160,143) (-1/1,-7/8) -> (7/24,5/17) Hyperbolic Matrix(65,56,224,193) (-7/8,-6/7) -> (2/7,7/24) Hyperbolic Matrix(129,110,-224,-191) (-6/7,-5/6) -> (-15/26,-4/7) Hyperbolic Matrix(17,14,-96,-79) (-5/6,-4/5) -> (-2/11,-1/6) Hyperbolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(113,82,288,209) (-8/11,-13/18) -> (7/18,2/5) Hyperbolic Matrix(111,80,-512,-369) (-13/18,-5/7) -> (-5/23,-3/14) Hyperbolic Matrix(31,22,224,159) (-5/7,-17/24) -> (1/8,1/7) Hyperbolic Matrix(17,12,160,113) (-17/24,-12/17) -> (0/1,1/8) Hyperbolic Matrix(321,226,-544,-383) (-12/17,-7/10) -> (-13/22,-10/17) Hyperbolic Matrix(49,34,-160,-111) (-7/10,-2/3) -> (-4/13,-3/10) Hyperbolic Matrix(79,50,-128,-81) (-2/3,-5/8) -> (-5/8,-8/13) Parabolic Matrix(417,256,992,609) (-8/13,-19/31) -> (13/31,8/19) Hyperbolic Matrix(353,216,-992,-607) (-19/31,-11/18) -> (-5/14,-11/31) Hyperbolic Matrix(79,48,288,175) (-11/18,-3/5) -> (3/11,5/18) Hyperbolic Matrix(257,152,-864,-511) (-3/5,-13/22) -> (-3/10,-11/37) Hyperbolic Matrix(335,196,-576,-337) (-10/17,-7/12) -> (-7/12,-18/31) Parabolic Matrix(383,222,992,575) (-18/31,-11/19) -> (5/13,12/31) Hyperbolic Matrix(97,56,-608,-351) (-11/19,-15/26) -> (-1/6,-3/19) 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(143,62,256,111) (-7/16,-3/7) -> (5/9,9/16) Hyperbolic Matrix(33,14,-224,-95) (-3/7,-8/19) -> (-2/13,-1/7) Hyperbolic Matrix(239,100,-576,-241) (-8/19,-5/12) -> (-5/12,-12/29) Parabolic Matrix(353,146,544,225) (-12/29,-7/17) -> (11/17,2/3) Hyperbolic Matrix(161,66,-544,-223) (-7/17,-2/5) -> (-8/27,-5/17) Hyperbolic Matrix(47,18,-128,-49) (-2/5,-3/8) -> (-3/8,-4/11) Parabolic Matrix(177,64,224,81) (-4/11,-5/14) -> (11/14,4/5) Hyperbolic Matrix(1489,528,2304,817) (-11/31,-17/48) -> (31/48,11/17) Hyperbolic Matrix(1487,526,2304,815) (-17/48,-6/17) -> (20/31,31/48) Hyperbolic Matrix(319,112,544,191) (-6/17,-1/3) -> (17/29,10/17) Hyperbolic Matrix(177,56,256,81) (-1/3,-5/16) -> (11/16,9/13) Hyperbolic Matrix(175,54,256,79) (-5/16,-4/13) -> (2/3,11/16) Hyperbolic Matrix(2881,856,4096,1217) (-11/37,-19/64) -> (45/64,19/27) Hyperbolic Matrix(2879,854,4096,1215) (-19/64,-8/27) -> (26/37,45/64) Hyperbolic Matrix(143,42,160,47) (-5/17,-7/24) -> (7/8,1/1) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(15,4,-64,-17) (-2/7,-1/4) -> (-1/4,-2/9) Parabolic Matrix(801,176,1024,225) (-2/9,-7/32) -> (25/32,18/23) Hyperbolic Matrix(799,174,1024,223) (-7/32,-5/23) -> (7/9,25/32) Hyperbolic Matrix(143,30,224,47) (-3/14,-1/5) -> (7/11,9/14) Hyperbolic Matrix(209,40,256,49) (-1/5,-3/16) -> (13/16,9/11) Hyperbolic Matrix(207,38,256,47) (-3/16,-2/11) -> (4/5,13/16) Hyperbolic Matrix(865,136,1024,161) (-3/19,-5/32) -> (27/32,11/13) Hyperbolic Matrix(863,134,1024,159) (-5/32,-2/13) -> (16/19,27/32) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(113,12,160,17) (-1/8,0/1) -> (12/17,17/24) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(79,-14,96,-17) (1/6,1/5) -> (9/11,5/6) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(401,-112,512,-143) (5/18,2/7) -> (18/23,11/14) Hyperbolic Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(111,-34,160,-49) (3/10,1/3) -> (9/13,7/10) Hyperbolic Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(607,-248,864,-353) (2/5,9/22) -> (7/10,26/37) Hyperbolic Matrix(241,-100,576,-239) (7/17,5/12) -> (5/12,13/31) Parabolic Matrix(511,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(337,-196,576,-335) (11/19,7/12) -> (7/12,17/29) Parabolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(81,-50,128,-79) (3/5,5/8) -> (5/8,7/11) Parabolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(47,42,160,143) -> Matrix(1,2,-2,-3) Matrix(65,56,224,193) -> Matrix(1,4,-2,-7) Matrix(129,110,-224,-191) -> Matrix(1,2,-4,-7) Matrix(17,14,-96,-79) -> Matrix(3,4,2,3) Matrix(47,36,-64,-49) -> Matrix(1,2,-2,-3) Matrix(113,82,288,209) -> Matrix(1,0,0,1) Matrix(111,80,-512,-369) -> Matrix(5,6,4,5) Matrix(31,22,224,159) -> Matrix(7,4,-2,-1) Matrix(17,12,160,113) -> Matrix(3,2,-2,-1) Matrix(321,226,-544,-383) -> Matrix(1,0,0,1) Matrix(49,34,-160,-111) -> Matrix(1,0,2,1) Matrix(79,50,-128,-81) -> Matrix(1,0,2,1) Matrix(417,256,992,609) -> Matrix(1,0,-4,1) Matrix(353,216,-992,-607) -> Matrix(1,4,2,9) Matrix(79,48,288,175) -> Matrix(1,0,0,1) Matrix(257,152,-864,-511) -> Matrix(1,0,4,1) Matrix(335,196,-576,-337) -> Matrix(5,2,-18,-7) Matrix(383,222,992,575) -> Matrix(1,0,4,1) Matrix(97,56,-608,-351) -> Matrix(9,2,4,1) Matrix(15,8,-32,-17) -> Matrix(1,0,2,1) Matrix(145,64,256,113) -> Matrix(1,0,10,1) Matrix(143,62,256,111) -> Matrix(1,0,-10,1) Matrix(33,14,-224,-95) -> Matrix(7,-2,4,-1) Matrix(239,100,-576,-241) -> Matrix(7,-2,18,-5) Matrix(353,146,544,225) -> Matrix(5,-2,8,-3) Matrix(161,66,-544,-223) -> Matrix(1,0,0,1) Matrix(47,18,-128,-49) -> Matrix(1,0,2,1) Matrix(177,64,224,81) -> Matrix(7,-2,4,-1) Matrix(1489,528,2304,817) -> Matrix(21,-10,40,-19) Matrix(1487,526,2304,815) -> Matrix(19,-10,40,-21) Matrix(319,112,544,191) -> Matrix(3,-2,8,-5) Matrix(177,56,256,81) -> Matrix(3,-2,2,-1) Matrix(175,54,256,79) -> Matrix(1,-2,2,-3) Matrix(2881,856,4096,1217) -> Matrix(1,0,0,1) Matrix(2879,854,4096,1215) -> Matrix(1,0,0,1) Matrix(143,42,160,47) -> Matrix(3,-2,2,-1) Matrix(193,56,224,65) -> Matrix(7,-4,2,-1) Matrix(15,4,-64,-17) -> Matrix(3,-2,2,-1) Matrix(801,176,1024,225) -> Matrix(11,-10,10,-9) Matrix(799,174,1024,223) -> Matrix(9,-10,10,-11) Matrix(143,30,224,47) -> Matrix(1,-2,4,-7) Matrix(209,40,256,49) -> Matrix(3,-2,2,-1) Matrix(207,38,256,47) -> Matrix(1,-2,2,-3) Matrix(865,136,1024,161) -> Matrix(1,0,0,1) Matrix(863,134,1024,159) -> Matrix(1,0,0,1) Matrix(159,22,224,31) -> Matrix(1,-4,2,-7) Matrix(113,12,160,17) -> Matrix(1,-2,2,-3) Matrix(95,-14,224,-33) -> Matrix(1,2,-4,-7) Matrix(79,-14,96,-17) -> Matrix(3,4,2,3) Matrix(17,-4,64,-15) -> Matrix(1,2,-2,-3) Matrix(401,-112,512,-143) -> Matrix(5,6,4,5) Matrix(223,-66,544,-161) -> Matrix(1,0,0,1) Matrix(111,-34,160,-49) -> Matrix(1,0,2,1) Matrix(49,-18,128,-47) -> Matrix(1,0,2,1) Matrix(639,-248,992,-385) -> Matrix(1,4,2,9) Matrix(607,-248,864,-353) -> Matrix(1,0,4,1) Matrix(241,-100,576,-239) -> Matrix(5,2,-18,-7) Matrix(511,-216,608,-257) -> Matrix(9,2,4,1) Matrix(17,-8,32,-15) -> Matrix(1,0,2,1) Matrix(191,-110,224,-129) -> Matrix(7,-2,4,-1) Matrix(337,-196,576,-335) -> Matrix(7,-2,18,-5) Matrix(383,-226,544,-321) -> Matrix(1,0,0,1) Matrix(81,-50,128,-79) -> Matrix(1,0,2,1) Matrix(49,-36,64,-47) -> Matrix(3,-2,2,-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): 22 Degree of the the map X: 22 Degree of the the map Y: 64 Permutation triple for Y: ((1,4,16,42,56,54,53,61,64,50,34,31,43,17,5,2)(3,10,32,49,25,8,7,24,55,37,58,62,44,51,33,11)(6,20,30,9,29,45,27,26,39,14,13,38,60,63,48,21)(12,35,28,47,19,18,46,59,41,15,40,57,52,23,22,36); (1,2,8,27,57,40,38,55,64,61,51,48,47,28,9,3)(4,14,39,53,52,62,58,35,50,20,6,5,19,49,32,15)(7,23,54,63,60,34,12,11,33,18,17,45,29,16,41,24)(10,30,59,46,26,25,56,42,37,13,36,22,21,44,43,31); (2,6,22,7)(3,12,13,4)(5,18)(8,26)(9,10)(15,16)(17,44,52,27)(19,48,54,25)(21,51)(23,53)(28,58,42,29)(30,50,55,41)(31,60,40,32)(33,61,39,46)(34,35)(37,38)) ----------------------------------------------------------------------- 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 0/1 1/8 1/7 1/6 1/5 1/4 3/10 1/3 3/8 5/12 1/2 9/16 7/12 5/8 9/14 31/48 11/16 45/64 3/4 25/32 13/16 27/32 7/8 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/8 1/0 1/7 -5/2 1/6 -2/1 1/5 1/0 1/4 -1/1 3/11 -1/2 5/18 -2/1 2/7 -3/4 7/24 -1/2 5/17 -1/2 3/10 0/1 1/3 -1/2 3/8 0/1 5/13 1/0 12/31 1/0 7/18 -2/1 2/5 -1/2 9/22 0/1 7/17 -1/2 5/12 -1/3 13/31 -1/4 8/19 -1/4 11/26 0/1 3/7 -1/6 1/2 0/1 5/9 -1/4 9/16 0/1 4/7 1/6 11/19 1/4 7/12 1/3 17/29 1/2 10/17 1/2 3/5 1/2 5/8 0/1 7/11 1/4 9/14 2/5 20/31 1/2 31/48 1/2 11/17 1/2 2/3 1/2 11/16 1/1 9/13 1/0 7/10 0/1 26/37 1/2 45/64 1/3 1/1 19/27 1/2 12/17 1/2 17/24 1/2 5/7 3/4 3/4 1/1 7/9 1/2 25/32 1/1 18/23 9/8 11/14 4/3 4/5 1/0 13/16 1/1 9/11 3/2 5/6 2/1 16/19 1/0 27/32 1/1 3/1 11/13 1/0 6/7 5/2 7/8 1/0 1/1 1/0 1/0 -1/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(47,-5,160,-17) (0/1,1/8) -> (7/24,5/17) Hyperbolic Matrix(65,-9,224,-31) (1/8,1/7) -> (2/7,7/24) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(79,-14,96,-17) (1/6,1/5) -> (9/11,5/6) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(113,-31,288,-79) (3/11,5/18) -> (7/18,2/5) Hyperbolic Matrix(401,-112,512,-143) (5/18,2/7) -> (18/23,11/14) Hyperbolic Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(111,-34,160,-49) (3/10,1/3) -> (9/13,7/10) Hyperbolic Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(417,-161,992,-383) (5/13,12/31) -> (13/31,8/19) Hyperbolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(607,-248,864,-353) (2/5,9/22) -> (7/10,26/37) Hyperbolic Matrix(241,-100,576,-239) (7/17,5/12) -> (5/12,13/31) Parabolic Matrix(511,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(145,-81,256,-143) (5/9,9/16) -> (9/16,4/7) Parabolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(337,-196,576,-335) (11/19,7/12) -> (7/12,17/29) Parabolic Matrix(353,-207,544,-319) (17/29,10/17) -> (11/17,2/3) Hyperbolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(81,-50,128,-79) (3/5,5/8) -> (5/8,7/11) Parabolic Matrix(177,-113,224,-143) (7/11,9/14) -> (11/14,4/5) Hyperbolic Matrix(1489,-961,2304,-1487) (20/31,31/48) -> (31/48,11/17) Parabolic Matrix(177,-121,256,-175) (2/3,11/16) -> (11/16,9/13) Parabolic Matrix(2881,-2025,4096,-2879) (26/37,45/64) -> (45/64,19/27) Parabolic Matrix(143,-101,160,-113) (12/17,17/24) -> (7/8,1/1) Hyperbolic Matrix(193,-137,224,-159) (17/24,5/7) -> (6/7,7/8) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(801,-625,1024,-799) (7/9,25/32) -> (25/32,18/23) Parabolic Matrix(209,-169,256,-207) (4/5,13/16) -> (13/16,9/11) Parabolic Matrix(865,-729,1024,-863) (16/19,27/32) -> (27/32,11/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,0,1) Matrix(47,-5,160,-17) -> Matrix(1,2,-2,-3) Matrix(65,-9,224,-31) -> Matrix(1,4,-2,-7) Matrix(95,-14,224,-33) -> Matrix(1,2,-4,-7) Matrix(79,-14,96,-17) -> Matrix(3,4,2,3) Matrix(17,-4,64,-15) -> Matrix(1,2,-2,-3) Matrix(113,-31,288,-79) -> Matrix(1,0,0,1) Matrix(401,-112,512,-143) -> Matrix(5,6,4,5) Matrix(223,-66,544,-161) -> Matrix(1,0,0,1) Matrix(111,-34,160,-49) -> Matrix(1,0,2,1) Matrix(49,-18,128,-47) -> Matrix(1,0,2,1) Matrix(417,-161,992,-383) -> Matrix(1,0,-4,1) Matrix(639,-248,992,-385) -> Matrix(1,4,2,9) Matrix(607,-248,864,-353) -> Matrix(1,0,4,1) Matrix(241,-100,576,-239) -> Matrix(5,2,-18,-7) Matrix(511,-216,608,-257) -> Matrix(9,2,4,1) Matrix(17,-8,32,-15) -> Matrix(1,0,2,1) Matrix(145,-81,256,-143) -> Matrix(1,0,10,1) Matrix(191,-110,224,-129) -> Matrix(7,-2,4,-1) Matrix(337,-196,576,-335) -> Matrix(7,-2,18,-5) Matrix(353,-207,544,-319) -> Matrix(5,-2,8,-3) Matrix(383,-226,544,-321) -> Matrix(1,0,0,1) Matrix(81,-50,128,-79) -> Matrix(1,0,2,1) Matrix(177,-113,224,-143) -> Matrix(7,-2,4,-1) Matrix(1489,-961,2304,-1487) -> Matrix(21,-10,40,-19) Matrix(177,-121,256,-175) -> Matrix(3,-2,2,-1) Matrix(2881,-2025,4096,-2879) -> Matrix(1,0,0,1) Matrix(143,-101,160,-113) -> Matrix(3,-2,2,-1) Matrix(193,-137,224,-159) -> Matrix(7,-4,2,-1) Matrix(49,-36,64,-47) -> Matrix(3,-2,2,-1) Matrix(801,-625,1024,-799) -> Matrix(11,-10,10,-9) Matrix(209,-169,256,-207) -> Matrix(3,-2,2,-1) Matrix(865,-729,1024,-863) -> Matrix(1,0,0,1) 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): 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 1 16 1/8 1/0 2 2 1/7 -5/2 1 16 1/6 -2/1 2 8 3/16 -1/1 2 1 1/5 1/0 1 16 1/4 -1/1 2 4 3/11 -1/2 1 16 5/18 -2/1 2 8 9/32 -1/1 10 1 2/7 -3/4 1 16 7/24 -1/2 2 2 5/17 -1/2 1 16 3/10 0/1 2 8 5/16 -1/1 2 1 1/3 -1/2 1 16 3/8 0/1 2 2 5/13 1/0 1 16 12/31 1/0 1 16 31/80 1/0 10 1 7/18 -2/1 2 8 2/5 -1/2 1 16 13/32 (-1/2,0/1) 0 1 9/22 0/1 2 8 7/17 -1/2 1 16 5/12 -1/3 2 4 13/31 -1/4 1 16 8/19 -1/4 1 16 27/64 (-1/4,0/1) 0 1 11/26 0/1 2 8 3/7 -1/6 1 16 7/16 0/1 10 1 1/2 0/1 2 8 1/0 (0/1,1/0) 0 1 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(47,-5,160,-17) (0/1,1/8) -> (7/24,5/17) Hyperbolic Matrix(65,-9,224,-31) (1/8,1/7) -> (2/7,7/24) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(17,-3,96,-17) (1/6,3/16) -> (1/6,3/16) Reflection Matrix(31,-6,160,-31) (3/16,1/5) -> (3/16,1/5) Reflection Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(113,-31,288,-79) (3/11,5/18) -> (7/18,2/5) Hyperbolic Matrix(161,-45,576,-161) (5/18,9/32) -> (5/18,9/32) Reflection Matrix(127,-36,448,-127) (9/32,2/7) -> (9/32,2/7) Reflection Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(49,-15,160,-49) (3/10,5/16) -> (3/10,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(417,-161,992,-383) (5/13,12/31) -> (13/31,8/19) Hyperbolic Matrix(1921,-744,4960,-1921) (12/31,31/80) -> (12/31,31/80) Reflection Matrix(559,-217,1440,-559) (31/80,7/18) -> (31/80,7/18) Reflection Matrix(129,-52,320,-129) (2/5,13/32) -> (2/5,13/32) Reflection Matrix(287,-117,704,-287) (13/32,9/22) -> (13/32,9/22) Reflection Matrix(241,-100,576,-239) (7/17,5/12) -> (5/12,13/31) Parabolic Matrix(1025,-432,2432,-1025) (8/19,27/64) -> (8/19,27/64) Reflection Matrix(703,-297,1664,-703) (27/64,11/26) -> (27/64,11/26) Reflection Matrix(97,-42,224,-97) (3/7,7/16) -> (3/7,7/16) Reflection Matrix(15,-7,32,-15) (7/16,1/2) -> (7/16,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(47,-5,160,-17) -> Matrix(1,2,-2,-3) -1/1 Matrix(65,-9,224,-31) -> Matrix(1,4,-2,-7) Matrix(95,-14,224,-33) -> Matrix(1,2,-4,-7) Matrix(17,-3,96,-17) -> Matrix(3,4,-2,-3) (1/6,3/16) -> (-2/1,-1/1) Matrix(31,-6,160,-31) -> Matrix(1,2,0,-1) (3/16,1/5) -> (-1/1,1/0) Matrix(17,-4,64,-15) -> Matrix(1,2,-2,-3) -1/1 Matrix(113,-31,288,-79) -> Matrix(1,0,0,1) Matrix(161,-45,576,-161) -> Matrix(3,4,-2,-3) (5/18,9/32) -> (-2/1,-1/1) Matrix(127,-36,448,-127) -> Matrix(7,6,-8,-7) (9/32,2/7) -> (-1/1,-3/4) Matrix(223,-66,544,-161) -> Matrix(1,0,0,1) Matrix(49,-15,160,-49) -> Matrix(-1,0,2,1) (3/10,5/16) -> (-1/1,0/1) Matrix(31,-10,96,-31) -> Matrix(3,2,-4,-3) (5/16,1/3) -> (-1/1,-1/2) Matrix(49,-18,128,-47) -> Matrix(1,0,2,1) 0/1 Matrix(417,-161,992,-383) -> Matrix(1,0,-4,1) 0/1 Matrix(1921,-744,4960,-1921) -> Matrix(-1,6,0,1) (12/31,31/80) -> (3/1,1/0) Matrix(559,-217,1440,-559) -> Matrix(1,4,0,-1) (31/80,7/18) -> (-2/1,1/0) Matrix(129,-52,320,-129) -> Matrix(-1,0,4,1) (2/5,13/32) -> (-1/2,0/1) Matrix(287,-117,704,-287) -> Matrix(-1,0,4,1) (13/32,9/22) -> (-1/2,0/1) Matrix(241,-100,576,-239) -> Matrix(5,2,-18,-7) -1/3 Matrix(1025,-432,2432,-1025) -> Matrix(-1,0,8,1) (8/19,27/64) -> (-1/4,0/1) Matrix(703,-297,1664,-703) -> Matrix(-1,0,8,1) (27/64,11/26) -> (-1/4,0/1) Matrix(97,-42,224,-97) -> Matrix(-1,0,12,1) (3/7,7/16) -> (-1/6,0/1) Matrix(15,-7,32,-15) -> Matrix(-1,0,2,1) (7/16,1/2) -> (-1/1,0/1) Matrix(-1,1,0,1) -> Matrix(1,0,0,-1) (1/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.