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 -6/1 -5/1 -4/1 -10/3 -3/1 -2/1 -4/3 -8/7 -1/1 -4/5 -2/3 -4/7 0/1 1/2 4/7 2/3 3/4 4/5 1/1 5/4 4/3 3/2 8/5 7/4 2/1 16/7 7/3 5/2 8/3 3/1 16/5 10/3 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 -11/2 -1/2 -16/3 0/1 -5/1 -1/1 0/1 1/0 -4/1 -1/1 -11/3 -1/1 -4/5 -3/4 -7/2 -3/4 -10/3 -1/2 -13/4 -1/1 -1/2 -16/5 -1/1 -3/1 -1/1 -2/3 -1/2 -8/3 -1/2 -5/2 -1/2 -12/5 -1/1 -7/3 -1/1 -2/3 -1/2 -2/1 -1/2 -9/5 -1/2 -1/3 0/1 -16/9 -1/3 -7/4 -1/3 0/1 -12/7 0/1 -5/3 -1/1 -1/2 0/1 -8/5 -1/2 -3/2 -1/2 -16/11 -1/3 -13/9 -1/3 -1/4 0/1 -10/7 -1/2 -17/12 -2/5 -1/3 -24/17 -1/3 -7/5 -1/3 -1/4 0/1 -4/3 0/1 -9/7 -1/1 -1/2 0/1 -5/4 -1/2 0/1 -16/13 0/1 -11/9 -1/1 -1/2 0/1 -6/5 -1/2 -7/6 -1/4 -8/7 0/1 -1/1 -1/1 -1/2 0/1 -6/7 1/0 -5/6 1/0 -4/5 -1/1 -7/9 -1/1 -1/2 0/1 -10/13 -1/2 -3/4 -1/1 -1/2 -8/11 -1/2 -5/7 -1/1 -1/2 0/1 -7/10 -1/2 -2/3 -1/2 -9/14 -1/2 -16/25 -1/3 -7/11 -1/2 -1/3 0/1 -5/8 -1/2 0/1 -8/13 -1/2 -3/5 -1/2 -1/3 0/1 -10/17 -1/4 -17/29 -1/5 -1/6 0/1 -24/41 0/1 -7/12 -1/3 0/1 -4/7 0/1 -5/9 -1/1 0/1 1/0 -6/11 1/0 -7/13 -2/1 -1/1 1/0 -8/15 -1/1 -1/2 -1/2 0/1 0/1 1/2 1/0 6/11 -1/2 5/9 -1/1 -1/2 0/1 4/7 0/1 7/12 0/1 1/1 10/17 1/2 3/5 0/1 1/1 1/0 5/8 0/1 1/0 7/11 0/1 1/1 1/0 2/3 1/0 9/13 -2/1 -1/1 1/0 7/10 1/0 5/7 -1/1 0/1 1/0 3/4 -1/1 1/0 10/13 1/0 17/22 -3/2 7/9 -1/1 0/1 1/0 4/5 -1/1 5/6 -1/2 6/7 -1/2 7/8 -1/3 0/1 1/1 -1/1 0/1 1/0 6/5 1/0 11/9 -1/1 0/1 1/0 16/13 0/1 5/4 0/1 1/0 4/3 0/1 11/8 0/1 1/2 7/5 0/1 1/2 1/1 10/7 1/0 13/9 0/1 1/2 1/1 16/11 1/1 3/2 1/0 8/5 1/0 13/8 -1/1 1/0 5/3 -1/1 0/1 1/0 12/7 0/1 7/4 0/1 1/1 2/1 1/0 9/4 -2/1 -1/1 16/7 -1/1 23/10 1/0 7/3 -2/1 -1/1 1/0 12/5 -1/1 5/2 1/0 8/3 1/0 11/4 -2/1 1/0 3/1 -2/1 -1/1 1/0 16/5 -1/1 13/4 -1/1 1/0 10/3 1/0 17/5 -3/1 -2/1 1/0 41/12 -7/3 -2/1 24/7 -2/1 7/2 -3/2 4/1 -1/1 9/2 -1/2 5/1 -1/1 -1/2 0/1 16/3 0/1 11/2 1/0 6/1 1/0 7/1 -2/1 -3/2 -1/1 15/2 -5/4 8/1 -1/1 1/0 -1/1 0/1 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(17,96,20,113) (-6/1,-11/2) -> (5/6,6/7) Hyperbolic Matrix(65,352,12,65) (-11/2,-16/3) -> (16/3,11/2) Hyperbolic Matrix(49,256,40,209) (-16/3,-5/1) -> (11/9,16/13) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(31,112,44,159) (-11/3,-7/2) -> (7/10,5/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(49,160,64,209) (-10/3,-13/4) -> (3/4,10/13) Hyperbolic Matrix(129,416,40,129) (-13/4,-16/5) -> (16/5,13/4) Hyperbolic Matrix(81,256,56,177) (-16/5,-3/1) -> (13/9,16/11) Hyperbolic Matrix(17,48,-28,-79) (-3/1,-8/3) -> (-8/13,-3/5) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(33,80,40,97) (-5/2,-12/5) -> (4/5,5/6) Hyperbolic Matrix(47,112,60,143) (-12/5,-7/3) -> (7/9,4/5) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,-224,-401) (-9/5,-16/9) -> (-16/25,-7/11) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(65,112,112,193) (-7/4,-12/7) -> (4/7,7/12) Hyperbolic Matrix(47,80,84,143) (-12/7,-5/3) -> (5/9,4/7) Hyperbolic Matrix(49,80,-68,-111) (-5/3,-8/5) -> (-8/11,-5/7) 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(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(111,160,188,271) (-13/9,-10/7) -> (10/17,3/5) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,-296,-417) (-24/17,-7/5) -> (-7/13,-8/15) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(63,80,100,127) (-9/7,-5/4) -> (5/8,7/11) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(79,96,144,175) (-11/9,-6/5) -> (6/11,5/9) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(143,160,-244,-273) (-8/7,-1/1) -> (-17/29,-24/41) Hyperbolic Matrix(129,112,-220,-191) (-1/1,-6/7) -> (-10/17,-17/29) Hyperbolic Matrix(113,96,20,17) (-6/7,-5/6) -> (11/2,6/1) Hyperbolic Matrix(97,80,40,33) (-5/6,-4/5) -> (12/5,5/2) Hyperbolic Matrix(143,112,60,47) (-4/5,-7/9) -> (7/3,12/5) Hyperbolic Matrix(145,112,-268,-207) (-7/9,-10/13) -> (-6/11,-7/13) Hyperbolic Matrix(209,160,64,49) (-10/13,-3/4) -> (13/4,10/3) Hyperbolic Matrix(175,128,108,79) (-3/4,-8/11) -> (8/5,13/8) Hyperbolic Matrix(113,80,24,17) (-5/7,-7/10) -> (9/2,5/1) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(799,512,348,223) (-9/14,-16/25) -> (16/7,23/10) Hyperbolic Matrix(177,112,128,81) (-7/11,-5/8) -> (11/8,7/5) Hyperbolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(271,160,188,111) (-3/5,-10/17) -> (10/7,13/9) Hyperbolic Matrix(1969,1152,576,337) (-24/41,-7/12) -> (41/12,24/7) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(143,80,84,47) (-4/7,-5/9) -> (5/3,12/7) Hyperbolic Matrix(175,96,144,79) (-5/9,-6/11) -> (6/5,11/9) Hyperbolic Matrix(241,128,32,17) (-8/15,-1/2) -> (15/2,8/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(207,-112,268,-145) (1/2,6/11) -> (10/13,17/22) Hyperbolic Matrix(191,-112,220,-129) (7/12,10/17) -> (6/7,7/8) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(369,-256,160,-111) (9/13,7/10) -> (23/10,7/3) Hyperbolic Matrix(111,-80,68,-49) (5/7,3/4) -> (13/8,5/3) Hyperbolic Matrix(289,-224,40,-31) (17/22,7/9) -> (7/1,15/2) Hyperbolic Matrix(177,-160,52,-47) (7/8,1/1) -> (17/5,41/12) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) 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(3,2,-8,-5) Matrix(17,96,20,113) -> Matrix(1,0,0,1) Matrix(65,352,12,65) -> Matrix(1,0,2,1) Matrix(49,256,40,209) -> Matrix(1,0,0,1) Matrix(15,64,-4,-17) -> Matrix(3,4,-4,-5) Matrix(31,112,44,159) -> Matrix(5,4,-4,-3) Matrix(33,112,-28,-95) -> Matrix(3,2,-8,-5) Matrix(49,160,64,209) -> Matrix(3,2,-2,-1) Matrix(129,416,40,129) -> Matrix(3,2,-2,-1) Matrix(81,256,56,177) -> Matrix(3,2,4,3) Matrix(17,48,-28,-79) -> Matrix(3,2,-8,-5) Matrix(31,80,12,31) -> Matrix(3,2,-2,-1) Matrix(33,80,40,97) -> Matrix(1,0,0,1) Matrix(47,112,60,143) -> Matrix(3,2,-2,-1) Matrix(15,32,-8,-17) -> Matrix(3,2,-8,-5) Matrix(143,256,-224,-401) -> Matrix(1,0,0,1) Matrix(145,256,64,113) -> Matrix(7,2,-4,-1) Matrix(65,112,112,193) -> Matrix(1,0,4,1) Matrix(47,80,84,143) -> Matrix(1,0,0,1) Matrix(49,80,-68,-111) -> Matrix(1,0,0,1) Matrix(31,48,20,31) -> Matrix(1,0,2,1) Matrix(65,96,44,65) -> Matrix(5,2,2,1) Matrix(177,256,56,81) -> Matrix(7,2,-4,-1) Matrix(111,160,188,271) -> Matrix(1,0,4,1) Matrix(113,160,12,17) -> Matrix(5,2,-8,-3) Matrix(159,224,-296,-417) -> Matrix(7,2,-4,-1) Matrix(47,64,-36,-49) -> Matrix(1,0,2,1) Matrix(63,80,100,127) -> Matrix(1,0,2,1) Matrix(129,160,104,129) -> Matrix(1,0,2,1) Matrix(209,256,40,49) -> Matrix(1,0,0,1) Matrix(79,96,144,175) -> Matrix(1,0,0,1) Matrix(193,224,56,65) -> Matrix(11,2,-6,-1) Matrix(143,160,-244,-273) -> Matrix(1,0,-4,1) Matrix(129,112,-220,-191) -> Matrix(1,0,-4,1) Matrix(113,96,20,17) -> Matrix(1,0,0,1) Matrix(97,80,40,33) -> Matrix(1,0,0,1) Matrix(143,112,60,47) -> Matrix(3,2,-2,-1) Matrix(145,112,-268,-207) -> Matrix(3,2,-2,-1) Matrix(209,160,64,49) -> Matrix(3,2,-2,-1) Matrix(175,128,108,79) -> Matrix(3,2,-2,-1) Matrix(113,80,24,17) -> Matrix(1,0,0,1) Matrix(47,32,-72,-49) -> Matrix(3,2,-8,-5) Matrix(799,512,348,223) -> Matrix(1,0,2,1) Matrix(177,112,128,81) -> Matrix(1,0,4,1) Matrix(207,128,76,47) -> Matrix(3,2,-2,-1) Matrix(271,160,188,111) -> Matrix(1,0,4,1) Matrix(1969,1152,576,337) -> Matrix(1,-2,0,1) Matrix(193,112,112,65) -> Matrix(1,0,4,1) Matrix(143,80,84,47) -> Matrix(1,0,0,1) Matrix(175,96,144,79) -> Matrix(1,0,0,1) Matrix(241,128,32,17) -> Matrix(7,6,-6,-5) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(207,-112,268,-145) -> Matrix(3,2,-2,-1) Matrix(191,-112,220,-129) -> Matrix(1,0,-4,1) Matrix(79,-48,28,-17) -> Matrix(1,-2,0,1) Matrix(49,-32,72,-47) -> Matrix(1,-2,0,1) Matrix(369,-256,160,-111) -> Matrix(1,0,0,1) Matrix(111,-80,68,-49) -> Matrix(1,0,0,1) Matrix(289,-224,40,-31) -> Matrix(3,2,-2,-1) Matrix(177,-160,52,-47) -> Matrix(1,-2,0,1) Matrix(95,-112,28,-33) -> Matrix(1,-2,0,1) Matrix(49,-64,36,-47) -> Matrix(1,0,2,1) Matrix(79,-112,12,-17) -> Matrix(1,-2,0,1) Matrix(17,-32,8,-15) -> Matrix(1,-2,0,1) Matrix(17,-64,4,-15) -> Matrix(3,4,-4,-5) 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): 14 Degree of the the map X: 14 Degree of the the map Y: 64 Permutation triple for Y: ((1,2)(3,10,37,11)(4,16,17,5)(6,22,51,23)(7,28,29,8)(9,34)(12,40)(13,14)(15,27)(18,30)(19,20)(21,38)(24,36)(25,26)(31,32)(33,39,50,53)(35,41,49,52)(42,47,55,43)(44,48,54,45)(46,56)(57,64)(58,62)(59,63)(60,61); (1,5,20,44,64,49,21,6)(2,8,32,55,57,33,9,3)(4,14,45,60,52,24,23,15)(7,26,47,61,39,12,11,27)(10,30,29,13,43,58,53,36)(16,31,54,59,35,34,51,46)(17,25,48,62,41,40,22,18)(19,42,63,50,38,37,56,28); (1,3,12,41,59,42,13,4)(2,6,24,53,63,54,25,7)(5,18,10,9,35,60,47,19)(8,30,22,21,50,61,45,31)(11,38,49,62,43,32,16,15)(14,29,56,51,40,39,57,44)(17,46,37,36,52,64,55,26)(20,28,27,23,34,33,58,48)) ----------------------------------------------------------------------- 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 -10/3 -3/1 -2/1 -4/3 -1/1 -2/3 0/1 1/2 2/3 3/4 1/1 4/3 3/2 2/1 8/3 3/1 10/3 4/1 16/3 6/1 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 -1/1 0/1 1/0 -4/1 -1/1 -7/2 -3/4 -10/3 -1/2 -3/1 -1/1 -2/3 -1/2 -2/1 -1/2 -5/3 -1/1 -1/2 0/1 -8/5 -1/2 -3/2 -1/2 -16/11 -1/3 -13/9 -1/3 -1/4 0/1 -10/7 -1/2 -17/12 -2/5 -1/3 -24/17 -1/3 -7/5 -1/3 -1/4 0/1 -4/3 0/1 -5/4 -1/2 0/1 -16/13 0/1 -11/9 -1/1 -1/2 0/1 -6/5 -1/2 -7/6 -1/4 -8/7 0/1 -1/1 -1/1 -1/2 0/1 -4/5 -1/1 -3/4 -1/1 -1/2 -2/3 -1/2 -5/8 -1/2 0/1 -8/13 -1/2 -3/5 -1/2 -1/3 0/1 -4/7 0/1 -1/2 -1/2 0/1 0/1 1/2 1/0 4/7 0/1 3/5 0/1 1/1 1/0 2/3 1/0 5/7 -1/1 0/1 1/0 3/4 -1/1 1/0 4/5 -1/1 1/1 -1/1 0/1 1/0 6/5 1/0 5/4 0/1 1/0 4/3 0/1 7/5 0/1 1/2 1/1 10/7 1/0 3/2 1/0 2/1 1/0 5/2 1/0 8/3 1/0 11/4 -2/1 1/0 3/1 -2/1 -1/1 1/0 16/5 -1/1 13/4 -1/1 1/0 10/3 1/0 17/5 -3/1 -2/1 1/0 24/7 -2/1 7/2 -3/2 4/1 -1/1 5/1 -1/1 -1/2 0/1 16/3 0/1 11/2 1/0 6/1 1/0 7/1 -2/1 -3/2 -1/1 8/1 -1/1 1/0 -1/1 0/1 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(23,128,-16,-89) (-6/1,-5/1) -> (-13/9,-10/7) Hyperbolic Matrix(7,32,12,55) (-5/1,-4/1) -> (4/7,3/5) Hyperbolic Matrix(9,32,16,57) (-4/1,-7/2) -> (1/2,4/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(39,128,-32,-105) (-10/3,-3/1) -> (-11/9,-6/5) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,-64,-105) (-5/3,-8/5) -> (-8/13,-3/5) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(153,224,28,41) (-3/2,-16/11) -> (16/3,11/2) 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(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(23,32,28,39) (-7/5,-4/3) -> (4/5,1/1) Hyperbolic Matrix(25,32,32,41) (-4/3,-5/4) -> (3/4,4/5) Hyperbolic Matrix(233,288,72,89) (-5/4,-16/13) -> (16/5,13/4) 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(57,64,8,9) (-8/7,-1/1) -> (7/1,8/1) Hyperbolic Matrix(39,32,28,23) (-1/1,-4/5) -> (4/3,7/5) Hyperbolic Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3) Hyperbolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(55,32,12,7) (-3/5,-4/7) -> (4/1,5/1) Hyperbolic Matrix(57,32,16,9) (-4/7,-1/2) -> (7/2,4/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(89,-64,32,-23) (5/7,3/4) -> (11/4,3/1) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(105,-128,32,-39) (6/5,5/4) -> (13/4,10/3) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(3,2,-8,-5) Matrix(23,128,-16,-89) -> Matrix(1,1,-4,-3) Matrix(7,32,12,55) -> Matrix(1,1,0,1) Matrix(9,32,16,57) -> Matrix(1,1,-4,-3) Matrix(33,112,-28,-95) -> Matrix(3,2,-8,-5) Matrix(39,128,-32,-105) -> Matrix(1,1,-4,-3) Matrix(7,16,-4,-9) -> Matrix(1,1,-4,-3) Matrix(39,64,-64,-105) -> Matrix(1,1,-4,-3) Matrix(41,64,16,25) -> Matrix(1,1,-2,-1) Matrix(153,224,28,41) -> Matrix(3,1,2,1) Matrix(177,256,56,81) -> Matrix(7,2,-4,-1) Matrix(113,160,12,17) -> Matrix(5,2,-8,-3) Matrix(409,576,120,169) -> Matrix(11,3,-4,-1) Matrix(23,32,28,39) -> Matrix(3,1,-4,-1) Matrix(25,32,32,41) -> Matrix(1,1,-2,-1) Matrix(233,288,72,89) -> Matrix(1,1,-2,-1) Matrix(209,256,40,49) -> Matrix(1,0,0,1) Matrix(193,224,56,65) -> Matrix(11,2,-6,-1) Matrix(57,64,8,9) -> Matrix(1,-1,0,1) Matrix(39,32,28,23) -> Matrix(1,1,0,1) Matrix(41,32,32,25) -> Matrix(1,1,-2,-1) Matrix(23,16,-36,-25) -> Matrix(1,1,-4,-3) Matrix(207,128,76,47) -> Matrix(3,2,-2,-1) Matrix(55,32,12,7) -> Matrix(3,1,-4,-1) Matrix(57,32,16,9) -> Matrix(1,-1,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(25,-16,36,-23) -> Matrix(1,-1,0,1) Matrix(89,-64,32,-23) -> Matrix(1,-1,0,1) Matrix(95,-112,28,-33) -> Matrix(1,-2,0,1) Matrix(105,-128,32,-39) -> Matrix(1,-1,0,1) Matrix(79,-112,12,-17) -> Matrix(1,-2,0,1) Matrix(89,-128,16,-23) -> Matrix(1,-1,0,1) Matrix(9,-16,4,-7) -> Matrix(1,-1,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 14 ----------------------------------------------------------------------- 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 1 2 1/2 1/0 1 8 4/7 0/1 2 2 3/5 0 8 2/3 1/0 1 4 5/7 0 8 3/4 (-1/1,1/0) 0 8 4/5 -1/1 1 2 1/1 0 8 6/5 1/0 1 4 5/4 (0/1,1/0) 0 8 4/3 0/1 1 2 7/5 0 8 10/7 1/0 1 4 3/2 1/0 1 8 2/1 1/0 1 4 5/2 1/0 1 8 8/3 1/0 1 2 11/4 (-2/1,1/0) 0 8 3/1 0 8 16/5 -1/1 2 2 13/4 (-1/1,1/0) 0 8 10/3 1/0 1 4 17/5 0 8 24/7 -2/1 5 2 7/2 -3/2 1 8 4/1 -1/1 2 2 5/1 0 8 16/3 0/1 2 2 11/2 1/0 1 8 6/1 1/0 1 4 7/1 0 8 8/1 -1/1 5 2 1/0 (-1/1,0/1) 0 8 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(57,-32,16,-9) (1/2,4/7) -> (7/2,4/1) Glide Reflection Matrix(55,-32,12,-7) (4/7,3/5) -> (4/1,5/1) Glide Reflection Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(89,-64,32,-23) (5/7,3/4) -> (11/4,3/1) Hyperbolic Matrix(41,-32,32,-25) (3/4,4/5) -> (5/4,4/3) Glide Reflection Matrix(39,-32,28,-23) (4/5,1/1) -> (4/3,7/5) Glide Reflection Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(105,-128,32,-39) (6/5,5/4) -> (13/4,10/3) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(41,-128,8,-25) (3/1,16/5) -> (5/1,16/3) Glide Reflection Matrix(129,-416,40,-129) (16/5,13/4) -> (16/5,13/4) Reflection Matrix(89,-304,12,-41) (17/5,24/7) -> (7/1,8/1) Glide Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(65,-352,12,-65) (16/3,11/2) -> (16/3,11/2) 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,4,-1) -> Matrix(1,0,0,-1) (0/1,1/2) -> (0/1,1/0) Matrix(57,-32,16,-9) -> Matrix(3,1,-2,-1) Matrix(55,-32,12,-7) -> Matrix(1,-1,-2,1) Matrix(25,-16,36,-23) -> Matrix(1,-1,0,1) 1/0 Matrix(89,-64,32,-23) -> Matrix(1,-1,0,1) 1/0 Matrix(41,-32,32,-25) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(39,-32,28,-23) -> Matrix(1,1,2,1) Matrix(95,-112,28,-33) -> Matrix(1,-2,0,1) 1/0 Matrix(105,-128,32,-39) -> Matrix(1,-1,0,1) 1/0 Matrix(79,-112,12,-17) -> Matrix(1,-2,0,1) 1/0 Matrix(89,-128,16,-23) -> Matrix(1,-1,0,1) 1/0 Matrix(9,-16,4,-7) -> Matrix(1,-1,0,1) 1/0 Matrix(31,-80,12,-31) -> Matrix(1,2,0,-1) (5/2,8/3) -> (-1/1,1/0) Matrix(65,-176,24,-65) -> Matrix(1,4,0,-1) (8/3,11/4) -> (-2/1,1/0) Matrix(41,-128,8,-25) -> Matrix(1,1,-2,-3) Matrix(129,-416,40,-129) -> Matrix(1,2,0,-1) (16/5,13/4) -> (-1/1,1/0) Matrix(89,-304,12,-41) -> Matrix(3,7,-2,-5) Matrix(97,-336,28,-97) -> Matrix(7,12,-4,-7) (24/7,7/2) -> (-2/1,-3/2) Matrix(65,-352,12,-65) -> Matrix(1,0,0,-1) (16/3,11/2) -> (0/1,1/0) Matrix(-1,16,0,1) -> Matrix(-1,0,2,1) (8/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.