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 -7/4 -3/2 -5/4 -1/1 -5/6 -3/4 -1/2 -7/16 -3/7 -1/3 -1/4 -1/5 -1/6 -1/7 -1/8 0/1 1/6 3/16 1/5 1/4 2/7 5/16 1/3 3/8 2/5 1/2 4/7 5/8 2/3 3/4 4/5 5/6 7/8 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 -1/1 0/1 1/0 -11/6 -1/2 1/0 -9/5 -1/1 0/1 -7/4 0/1 -12/7 0/1 1/2 1/1 -17/10 1/2 1/0 -5/3 1/0 -8/5 -1/1 0/1 1/0 -11/7 0/1 1/0 -3/2 -1/2 1/0 -13/9 0/1 1/0 -10/7 -2/1 -1/1 1/0 -7/5 1/0 -4/3 -1/1 0/1 1/0 -13/10 -1/2 1/0 -22/17 -1/1 0/1 1/0 -9/7 1/0 -5/4 -1/1 -6/5 -1/1 -1/2 0/1 -7/6 -1/2 1/0 -8/7 -1/1 0/1 1/0 -1/1 -1/1 0/1 -5/6 -1/2 1/0 -9/11 -1/2 -13/16 0/1 -4/5 -1/1 0/1 1/0 -3/4 -1/1 -8/11 -1/1 -2/3 -1/2 -5/7 -1/2 -7/10 -1/2 1/0 -9/13 -1/1 1/0 -11/16 -1/1 -2/3 -1/1 -1/2 0/1 -5/8 -1/1 -8/13 -1/1 -2/3 -1/2 -3/5 -1/2 -7/12 -1/1 -4/7 -1/1 -2/3 -1/2 -1/2 -1/2 1/0 -4/9 -1/1 -2/3 -1/2 -7/16 -1/2 -10/23 -1/2 -2/5 -1/3 -3/7 -1/2 0/1 -5/12 -1/2 -2/5 -1/1 -1/2 0/1 -3/8 -1/2 -4/11 -1/2 -1/3 0/1 -1/3 -1/2 -5/16 -1/3 -4/13 -1/3 -2/7 -1/4 -3/10 -1/2 -1/4 -5/17 -1/3 0/1 -12/41 -1/2 -1/3 0/1 -7/24 -1/3 -2/7 -1/3 -1/4 0/1 -1/4 0/1 -2/9 0/1 1/1 1/0 -1/5 -1/1 0/1 -3/16 0/1 -2/11 0/1 1/1 1/0 -1/6 -1/2 1/0 -1/7 -1/2 -2/15 -1/2 -1/3 0/1 -1/8 0/1 0/1 -1/1 -1/2 0/1 1/6 -1/2 1/0 2/11 -1/1 -1/2 0/1 3/16 0/1 1/5 1/0 1/4 -1/1 3/11 -1/2 2/7 -1/1 -1/2 0/1 3/10 -1/2 1/0 4/13 -1/1 0/1 1/0 5/16 -1/1 1/3 -1/1 -1/2 3/8 -1/2 2/5 -1/1 -1/2 0/1 5/12 -1/1 3/7 -1/2 1/2 -1/2 1/0 5/9 -1/2 9/16 -1/2 4/7 -1/2 -1/3 0/1 7/12 0/1 3/5 -1/1 0/1 5/8 -1/1 2/3 -1/1 -1/2 0/1 11/16 -1/1 9/13 -3/4 7/10 -3/4 -1/2 12/17 -1/1 -3/4 -2/3 17/24 -2/3 5/7 -2/3 -1/2 3/4 -1/2 7/9 -1/2 0/1 4/5 -1/2 -1/3 0/1 13/16 0/1 9/11 -1/1 0/1 5/6 -1/2 1/0 6/7 -1/1 -1/2 0/1 7/8 -1/1 1/1 -1/2 7/6 -1/2 -1/4 6/5 -1/1 -1/2 0/1 5/4 -1/2 9/7 -1/2 -2/5 13/10 -1/2 -3/8 4/3 -1/2 -1/3 0/1 11/8 -1/3 7/5 -1/3 0/1 10/7 -1/1 -1/2 0/1 3/2 -1/2 -1/4 14/9 -1/1 -1/2 0/1 25/16 -1/2 11/7 -1/2 8/5 -1/2 -1/3 0/1 13/8 -1/2 5/3 -1/2 -1/3 17/10 -1/2 -1/4 29/17 -1/2 41/24 -1/3 12/7 -1/2 -1/3 0/1 7/4 -1/3 9/5 -1/4 11/6 -1/2 -1/4 13/7 -1/4 0/1 15/8 0/1 2/1 -1/2 -1/3 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(145,268,-112,-207) (-2/1,-11/6) -> (-13/10,-22/17) Hyperbolic Matrix(79,144,96,175) (-11/6,-9/5) -> (9/11,5/6) Hyperbolic Matrix(47,84,80,143) (-9/5,-7/4) -> (7/12,3/5) Hyperbolic Matrix(65,112,112,193) (-7/4,-12/7) -> (4/7,7/12) Hyperbolic Matrix(129,220,-112,-191) (-12/7,-17/10) -> (-7/6,-8/7) Hyperbolic Matrix(111,188,160,271) (-17/10,-5/3) -> (9/13,7/10) Hyperbolic Matrix(17,28,-48,-79) (-5/3,-8/5) -> (-4/11,-1/3) Hyperbolic Matrix(63,100,80,127) (-8/5,-11/7) -> (7/9,4/5) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(111,160,-256,-369) (-13/9,-10/7) -> (-10/23,-3/7) Hyperbolic Matrix(31,44,112,159) (-10/7,-7/5) -> (3/11,2/7) Hyperbolic Matrix(49,68,-80,-111) (-7/5,-4/3) -> (-8/13,-3/5) Hyperbolic Matrix(49,64,160,209) (-4/3,-13/10) -> (3/10,4/13) Hyperbolic Matrix(31,40,-224,-289) (-22/17,-9/7) -> (-1/7,-2/15) Hyperbolic Matrix(47,60,112,143) (-9/7,-5/4) -> (5/12,3/7) Hyperbolic Matrix(33,40,80,97) (-5/4,-6/5) -> (2/5,5/12) Hyperbolic Matrix(17,20,96,113) (-6/5,-7/6) -> (1/6,2/11) Hyperbolic Matrix(47,52,-160,-177) (-8/7,-1/1) -> (-5/17,-12/41) Hyperbolic Matrix(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(175,144,96,79) (-5/6,-9/11) -> (9/5,11/6) Hyperbolic Matrix(49,40,256,209) (-9/11,-13/16) -> (3/16,1/5) Hyperbolic Matrix(129,104,160,129) (-13/16,-4/5) -> (4/5,13/16) Hyperbolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(177,128,112,81) (-8/11,-5/7) -> (11/7,8/5) Hyperbolic Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(271,188,160,111) (-7/10,-9/13) -> (5/3,17/10) Hyperbolic Matrix(81,56,256,177) (-9/13,-11/16) -> (5/16,1/3) Hyperbolic Matrix(65,44,96,65) (-11/16,-2/3) -> (2/3,11/16) 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(143,84,80,47) (-3/5,-7/12) -> (7/4,9/5) 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(799,348,512,223) (-7/16,-10/23) -> (14/9,25/16) Hyperbolic Matrix(143,60,112,47) (-3/7,-5/12) -> (5/4,9/7) 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(177,56,256,81) (-1/3,-5/16) -> (11/16,9/13) Hyperbolic Matrix(129,40,416,129) (-5/16,-4/13) -> (4/13,5/16) Hyperbolic Matrix(209,64,160,49) (-4/13,-3/10) -> (13/10,4/3) Hyperbolic Matrix(1969,576,1152,337) (-12/41,-7/24) -> (41/24,12/7) 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(113,24,80,17) (-2/9,-1/5) -> (7/5,10/7) Hyperbolic Matrix(209,40,256,49) (-1/5,-3/16) -> (13/16,9/11) Hyperbolic Matrix(65,12,352,65) (-3/16,-2/11) -> (2/11,3/16) Hyperbolic Matrix(113,20,96,17) (-2/11,-1/6) -> (7/6,6/5) Hyperbolic Matrix(241,32,128,17) (-2/15,-1/8) -> (15/8,2/1) Hyperbolic Matrix(113,12,160,17) (-1/8,0/1) -> (12/17,17/24) Hyperbolic Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(79,-28,48,-17) (1/3,3/8) -> (13/8,5/3) 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(111,-68,80,-49) (3/5,5/8) -> (11/8,7/5) Hyperbolic Matrix(417,-296,224,-159) (17/24,5/7) -> (13/7,15/8) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(273,-244,160,-143) (7/8,1/1) -> (29/17,41/24) Hyperbolic Matrix(191,-220,112,-129) (1/1,7/6) -> (17/10,29/17) Hyperbolic Matrix(207,-268,112,-145) (9/7,13/10) -> (11/6,13/7) 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,-2,1) Matrix(145,268,-112,-207) -> Matrix(1,0,0,1) Matrix(79,144,96,175) -> Matrix(1,0,0,1) Matrix(47,84,80,143) -> Matrix(1,0,0,1) Matrix(65,112,112,193) -> Matrix(1,0,-4,1) Matrix(129,220,-112,-191) -> Matrix(1,0,-2,1) Matrix(111,188,160,271) -> Matrix(3,-2,-4,3) Matrix(17,28,-48,-79) -> Matrix(1,0,-2,1) Matrix(63,100,80,127) -> Matrix(1,0,-2,1) Matrix(47,72,-32,-49) -> Matrix(1,0,0,1) Matrix(111,160,-256,-369) -> Matrix(1,0,-2,1) Matrix(31,44,112,159) -> Matrix(1,2,-2,-3) Matrix(49,68,-80,-111) -> Matrix(1,2,-2,-3) Matrix(49,64,160,209) -> Matrix(1,0,0,1) Matrix(31,40,-224,-289) -> Matrix(1,0,-2,1) Matrix(47,60,112,143) -> Matrix(1,2,-2,-3) Matrix(33,40,80,97) -> Matrix(1,0,0,1) Matrix(17,20,96,113) -> Matrix(1,0,0,1) Matrix(47,52,-160,-177) -> Matrix(1,0,-2,1) Matrix(33,28,-112,-95) -> Matrix(1,0,-2,1) Matrix(175,144,96,79) -> Matrix(1,0,-2,1) Matrix(49,40,256,209) -> Matrix(1,0,2,1) Matrix(129,104,160,129) -> Matrix(1,0,-2,1) Matrix(47,36,-64,-49) -> Matrix(1,2,-2,-3) Matrix(177,128,112,81) -> Matrix(3,2,-8,-5) Matrix(17,12,-112,-79) -> Matrix(1,0,0,1) Matrix(271,188,160,111) -> Matrix(1,0,-2,1) Matrix(81,56,256,177) -> Matrix(1,2,-2,-3) Matrix(65,44,96,65) -> Matrix(1,0,0,1) Matrix(31,20,48,31) -> Matrix(1,0,0,1) Matrix(175,108,128,79) -> Matrix(3,2,-8,-5) Matrix(143,84,80,47) -> Matrix(1,0,-2,1) Matrix(193,112,112,65) -> Matrix(3,2,-8,-5) Matrix(15,8,-32,-17) -> Matrix(1,0,0,1) Matrix(145,64,256,113) -> Matrix(3,2,-8,-5) Matrix(799,348,512,223) -> Matrix(5,2,-8,-3) Matrix(143,60,112,47) -> Matrix(3,2,-8,-5) Matrix(97,40,80,33) -> Matrix(1,0,0,1) Matrix(31,12,80,31) -> Matrix(1,0,0,1) Matrix(207,76,128,47) -> Matrix(1,0,0,1) Matrix(177,56,256,81) -> Matrix(11,4,-14,-5) Matrix(129,40,416,129) -> Matrix(7,2,-4,-1) Matrix(209,64,160,49) -> Matrix(7,2,-18,-5) Matrix(1969,576,1152,337) -> Matrix(1,0,0,1) Matrix(193,56,224,65) -> Matrix(1,0,2,1) Matrix(15,4,-64,-17) -> Matrix(1,0,4,1) Matrix(113,24,80,17) -> Matrix(1,0,-2,1) Matrix(209,40,256,49) -> Matrix(1,0,0,1) Matrix(65,12,352,65) -> Matrix(1,0,-2,1) Matrix(113,20,96,17) -> Matrix(1,0,-2,1) Matrix(241,32,128,17) -> Matrix(1,0,0,1) Matrix(113,12,160,17) -> Matrix(1,2,-2,-3) Matrix(79,-12,112,-17) -> Matrix(1,2,-2,-3) Matrix(17,-4,64,-15) -> Matrix(1,2,-2,-3) Matrix(95,-28,112,-33) -> Matrix(1,0,0,1) Matrix(79,-28,48,-17) -> Matrix(3,2,-8,-5) Matrix(17,-8,32,-15) -> Matrix(1,0,0,1) Matrix(401,-224,256,-143) -> Matrix(3,2,-8,-5) Matrix(111,-68,80,-49) -> Matrix(1,0,-2,1) Matrix(417,-296,224,-159) -> Matrix(3,2,-14,-9) Matrix(49,-36,64,-47) -> Matrix(3,2,-8,-5) Matrix(273,-244,160,-143) -> Matrix(3,2,-8,-5) Matrix(191,-220,112,-129) -> Matrix(1,0,0,1) Matrix(207,-268,112,-145) -> Matrix(5,2,-18,-7) Matrix(49,-72,32,-47) -> 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 10 Degree of the the map X: 10 Degree of the the map Y: 64 Permutation triple for Y: ((1,6,24,49,54,25,7,2)(3,12,41,62,63,42,13,4)(5,18,50,59,33,10,9,19)(8,22,21,36,53,55,46,29)(11,37,52,57,45,16,15,32)(14,28,39,17,48,60,35,34)(20,40,61,64,43,31,30,51)(23,47,56,27,26,44,58,38); (1,4,16,46,64,47,17,5)(2,10,35,56,61,36,11,3)(6,22,14,13,43,59,52,23)(7,27,57,50,40,12,28,8)(9,31,42,60,53,24,38,32)(15,26,25,55,48,41,20,19)(18,39,58,30,29,45,63,49)(21,51,44,34,33,54,62,37); (1,3)(2,8,30,9)(4,14,44,15)(5,20,21,6)(7,26)(10,34)(11,38,39,12)(13,31)(16,29)(17,18)(19,32)(22,28)(23,24)(25,33,43,46)(27,35,42,45)(36,37)(40,41)(47,52,62,48)(49,53,61,50)(51,58)(54,63)(55,60)(56,64)(57,59)) ----------------------------------------------------------------------- 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 (-1/1,0/1) 0 8 -5/6 0 4 -4/5 0 8 -3/4 -1/1 1 2 -5/7 -1/2 1 8 -7/10 0 4 -2/3 0 8 -5/8 -1/1 1 2 -3/5 -1/2 1 8 -4/7 0 8 -1/2 0 4 -4/9 0 8 -7/16 -1/2 3 2 -3/7 (-1/2,0/1) 0 8 -2/5 0 8 -3/8 -1/2 1 2 -1/3 -1/2 1 8 -3/10 0 4 -5/17 (-1/3,0/1) 0 8 -7/24 -1/3 1 2 -2/7 0 8 -1/4 0/1 2 2 -1/5 (-1/1,0/1) 0 8 -1/6 0 4 -1/7 -1/2 1 8 -1/8 0/1 2 2 0/1 0 8 1/6 0 4 2/11 0 8 3/16 0/1 3 2 1/5 1/0 1 8 1/4 -1/1 1 2 3/11 -1/2 1 8 2/7 0 8 3/10 0 4 4/13 0 8 5/16 -1/1 4 2 1/3 (-1/1,-1/2) 0 8 3/8 -1/2 1 2 2/5 0 8 5/12 -1/1 1 2 3/7 -1/2 1 8 1/2 0 4 5/9 -1/2 1 8 9/16 -1/2 3 2 4/7 0 8 7/12 0/1 2 2 3/5 (-1/1,0/1) 0 8 5/8 -1/1 1 2 2/3 0 8 11/16 -1/1 4 2 9/13 -3/4 1 8 7/10 0 4 12/17 0 8 17/24 -2/3 2 2 5/7 (-2/3,-1/2) 0 8 3/4 -1/2 1 2 7/9 (-1/2,0/1) 0 8 4/5 0 8 13/16 0/1 3 2 9/11 (-1/1,0/1) 0 8 5/6 0 4 6/7 0 8 7/8 -1/1 1 2 1/1 -1/2 1 8 1/0 0/1 1 2 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(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(17,14,96,79) (-5/6,-4/5) -> (1/6,2/11) Glide Reflection Matrix(33,26,80,63) (-4/5,-3/4) -> (2/5,5/12) Glide Reflection Matrix(47,34,112,81) (-3/4,-5/7) -> (5/12,3/7) Glide Reflection Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(49,34,160,111) (-7/10,-2/3) -> (3/10,4/13) Glide Reflection Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,-80,-49) (-5/8,-3/5) -> (-5/8,-3/5) Reflection Matrix(31,18,112,65) (-3/5,-4/7) -> (3/11,2/7) Glide Reflection 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(97,42,-224,-97) (-7/16,-3/7) -> (-7/16,-3/7) Reflection Matrix(63,26,80,33) (-3/7,-2/5) -> (7/9,4/5) Glide Reflection Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(17,6,-48,-17) (-3/8,-1/3) -> (-3/8,-1/3) Reflection Matrix(111,34,160,49) (-1/3,-3/10) -> (9/13,7/10) Glide Reflection Matrix(239,70,-816,-239) (-5/17,-7/24) -> (-5/17,-7/24) Reflection Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(65,18,112,31) (-2/7,-1/4) -> (4/7,7/12) Glide Reflection Matrix(47,10,80,17) (-1/4,-1/5) -> (7/12,3/5) Glide Reflection Matrix(79,14,96,17) (-1/5,-1/6) -> (9/11,5/6) Glide Reflection Matrix(15,2,-112,-15) (-1/7,-1/8) -> (-1/7,-1/8) Reflection Matrix(113,12,160,17) (-1/8,0/1) -> (12/17,17/24) Hyperbolic Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(207,-38,256,-47) (2/11,3/16) -> (4/5,13/16) Glide 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(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(175,-54,256,-79) (4/13,5/16) -> (2/3,11/16) Glide Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(287,-198,416,-287) (11/16,9/13) -> (11/16,9/13) Reflection Matrix(239,-170,336,-239) (17/24,5/7) -> (17/24,5/7) Reflection Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(287,-234,352,-287) (13/16,9/11) -> (13/16,9/11) Reflection Matrix(15,-14,16,-15) (7/8,1/1) -> (7/8,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(33,28,-112,-95) -> Matrix(1,0,-2,1) 0/1 Matrix(17,14,96,79) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(33,26,80,63) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(47,34,112,81) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(17,12,-112,-79) -> Matrix(1,0,0,1) Matrix(49,34,160,111) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(31,20,48,31) -> Matrix(1,0,0,1) Matrix(49,30,-80,-49) -> Matrix(3,2,-4,-3) (-5/8,-3/5) -> (-1/1,-1/2) Matrix(31,18,112,65) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(15,8,-32,-17) -> Matrix(1,0,0,1) Matrix(145,64,256,113) -> Matrix(3,2,-8,-5) -1/2 Matrix(97,42,-224,-97) -> Matrix(-1,0,4,1) (-7/16,-3/7) -> (-1/2,0/1) Matrix(63,26,80,33) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(31,12,80,31) -> Matrix(1,0,0,1) Matrix(17,6,-48,-17) -> Matrix(-1,0,4,1) (-3/8,-1/3) -> (-1/2,0/1) Matrix(111,34,160,49) -> Matrix(7,2,-10,-3) Matrix(239,70,-816,-239) -> Matrix(-1,0,6,1) (-5/17,-7/24) -> (-1/3,0/1) Matrix(193,56,224,65) -> Matrix(1,0,2,1) 0/1 Matrix(65,18,112,31) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(47,10,80,17) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(79,14,96,17) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(15,2,-112,-15) -> Matrix(-1,0,4,1) (-1/7,-1/8) -> (-1/2,0/1) Matrix(113,12,160,17) -> Matrix(1,2,-2,-3) -1/1 Matrix(79,-12,112,-17) -> Matrix(1,2,-2,-3) -1/1 Matrix(207,-38,256,-47) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(31,-6,160,-31) -> Matrix(1,0,0,-1) (3/16,1/5) -> (0/1,1/0) Matrix(17,-4,64,-15) -> Matrix(1,2,-2,-3) -1/1 Matrix(95,-28,112,-33) -> Matrix(1,0,0,1) Matrix(175,-54,256,-79) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(31,-10,96,-31) -> Matrix(3,2,-4,-3) (5/16,1/3) -> (-1/1,-1/2) Matrix(17,-6,48,-17) -> Matrix(3,2,-4,-3) (1/3,3/8) -> (-1/1,-1/2) Matrix(17,-8,32,-15) -> Matrix(1,0,0,1) Matrix(161,-90,288,-161) -> Matrix(3,2,-4,-3) (5/9,9/16) -> (-1/1,-1/2) Matrix(49,-30,80,-49) -> Matrix(-1,0,2,1) (3/5,5/8) -> (-1/1,0/1) Matrix(287,-198,416,-287) -> Matrix(7,6,-8,-7) (11/16,9/13) -> (-1/1,-3/4) Matrix(239,-170,336,-239) -> Matrix(7,4,-12,-7) (17/24,5/7) -> (-2/3,-1/2) Matrix(49,-36,64,-47) -> Matrix(3,2,-8,-5) -1/2 Matrix(287,-234,352,-287) -> Matrix(-1,0,2,1) (13/16,9/11) -> (-1/1,0/1) Matrix(15,-14,16,-15) -> Matrix(3,2,-4,-3) (7/8,1/1) -> (-1/1,-1/2) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.