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 -3/1 -2/1 -1/1 -2/3 -8/15 0/1 1/2 4/7 8/13 2/3 8/11 3/4 4/5 1/1 5/4 4/3 3/2 8/5 7/4 16/9 2/1 9/4 16/7 5/2 8/3 11/4 3/1 13/4 7/2 15/4 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -15/4 -1/1 1/0 -11/3 -1/2 -7/2 -1/1 0/1 -10/3 1/0 -3/1 1/0 -14/5 -1/2 -11/4 -1/2 0/1 -8/3 0/1 -5/2 0/1 1/1 -12/5 1/0 -19/8 -2/1 1/0 -7/3 -1/2 -16/7 0/1 -9/4 0/1 1/2 -11/5 1/2 -2/1 1/0 -9/5 -1/2 -16/9 0/1 -7/4 0/1 1/0 -5/3 -1/2 -18/11 -1/2 -31/19 -3/10 -13/8 -1/4 0/1 -8/5 0/1 -3/2 0/1 1/1 -13/9 1/0 -10/7 1/2 -27/19 1/2 -17/12 3/4 1/1 -24/17 1/1 -7/5 3/2 -18/13 1/0 -11/8 2/1 1/0 -4/3 1/0 -5/4 -1/1 1/0 -1/1 1/0 -6/7 1/2 -5/6 1/1 2/1 -4/5 1/0 -3/4 -1/1 1/0 -8/11 -1/1 -13/18 -1/1 0/1 -5/7 1/0 -7/10 -1/1 0/1 -16/23 0/1 -9/13 1/2 -2/3 1/0 -9/14 -3/1 -2/1 -16/25 -2/1 -7/11 -3/2 -5/8 -3/2 -1/1 -8/13 -1/1 -11/18 -1/1 -4/5 -14/23 -1/2 -3/5 -1/2 -7/12 0/1 1/0 -4/7 1/0 -9/16 -2/1 1/0 -5/9 1/0 -6/11 -3/2 -7/13 -7/6 -8/15 -1/1 -1/2 -1/1 0/1 0/1 0/1 1/2 0/1 1/1 7/13 7/6 6/11 3/2 5/9 1/0 9/16 2/1 1/0 4/7 1/0 3/5 1/2 11/18 4/5 1/1 8/13 1/1 5/8 1/1 3/2 2/3 1/0 9/13 -1/2 7/10 0/1 1/1 5/7 1/0 13/18 0/1 1/1 8/11 1/1 3/4 1/1 1/0 4/5 1/0 5/6 -2/1 -1/1 1/1 1/0 6/5 1/2 5/4 1/1 1/0 4/3 1/0 11/8 -2/1 1/0 18/13 1/0 7/5 -3/2 31/22 -10/9 -1/1 24/17 -1/1 17/12 -1/1 -3/4 10/7 -1/2 13/9 1/0 3/2 -1/1 0/1 8/5 0/1 13/8 0/1 1/4 5/3 1/2 12/7 1/0 7/4 0/1 1/0 16/9 0/1 25/14 0/1 1/5 9/5 1/2 11/6 2/3 1/1 13/7 1/0 2/1 1/0 11/5 -1/2 9/4 -1/2 0/1 16/7 0/1 7/3 1/2 19/8 2/1 1/0 12/5 1/0 5/2 -1/1 0/1 8/3 0/1 11/4 0/1 1/2 14/5 1/2 3/1 1/0 13/4 0/1 1/2 23/7 3/4 10/3 1/0 17/5 -1/2 24/7 0/1 7/2 0/1 1/1 11/3 1/2 26/7 1/2 15/4 1/1 1/0 4/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(31,120,8,31) (-4/1,-15/4) -> (15/4,4/1) Hyperbolic Matrix(159,592,-112,-417) (-15/4,-11/3) -> (-27/19,-17/12) Hyperbolic Matrix(31,112,44,159) (-11/3,-7/2) -> (7/10,5/7) Hyperbolic Matrix(31,104,-48,-161) (-7/2,-10/3) -> (-2/3,-9/14) Hyperbolic Matrix(33,104,-20,-63) (-10/3,-3/1) -> (-5/3,-18/11) Hyperbolic Matrix(31,88,56,159) (-3/1,-14/5) -> (6/11,5/9) Hyperbolic Matrix(127,352,92,255) (-14/5,-11/4) -> (11/8,18/13) 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(33,80,40,97) (-5/2,-12/5) -> (4/5,5/6) Hyperbolic Matrix(191,456,80,191) (-12/5,-19/8) -> (19/8,12/5) Hyperbolic Matrix(287,680,-176,-417) (-19/8,-7/3) -> (-31/19,-13/8) Hyperbolic Matrix(97,224,-152,-351) (-7/3,-16/7) -> (-16/25,-7/11) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(65,144,116,257) (-9/4,-11/5) -> (5/9,9/16) Hyperbolic Matrix(63,136,44,95) (-11/5,-2/1) -> (10/7,13/9) Hyperbolic Matrix(31,56,-36,-65) (-2/1,-9/5) -> (-1/1,-6/7) Hyperbolic Matrix(161,288,-232,-415) (-9/5,-16/9) -> (-16/23,-9/13) Hyperbolic Matrix(127,224,72,127) (-16/9,-7/4) -> (7/4,16/9) Hyperbolic Matrix(33,56,-56,-95) (-7/4,-5/3) -> (-3/5,-7/12) Hyperbolic Matrix(191,312,352,575) (-18/11,-31/19) -> (7/13,6/11) 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(127,184,176,255) (-3/2,-13/9) -> (5/7,13/18) Hyperbolic Matrix(95,136,44,63) (-13/9,-10/7) -> (2/1,11/5) Hyperbolic Matrix(129,184,68,97) (-10/7,-27/19) -> (13/7,2/1) Hyperbolic Matrix(577,816,408,577) (-17/12,-24/17) -> (24/17,17/12) Hyperbolic Matrix(159,224,-296,-417) (-24/17,-7/5) -> (-7/13,-8/15) Hyperbolic Matrix(127,176,184,255) (-7/5,-18/13) -> (2/3,9/13) Hyperbolic Matrix(255,352,92,127) (-18/13,-11/8) -> (11/4,14/5) Hyperbolic Matrix(65,88,48,65) (-11/8,-4/3) -> (4/3,11/8) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(33,40,-52,-63) (-5/4,-1/1) -> (-7/11,-5/8) Hyperbolic Matrix(161,136,-264,-223) (-6/7,-5/6) -> (-11/18,-14/23) Hyperbolic Matrix(97,80,40,33) (-5/6,-4/5) -> (12/5,5/2) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(65,48,88,65) (-3/4,-8/11) -> (8/11,3/4) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(255,184,176,127) (-13/18,-5/7) -> (13/9,3/2) Hyperbolic Matrix(159,112,44,31) (-5/7,-7/10) -> (7/2,11/3) Hyperbolic Matrix(735,512,412,287) (-7/10,-16/23) -> (16/9,25/14) Hyperbolic Matrix(255,176,184,127) (-9/13,-2/3) -> (18/13,7/5) Hyperbolic Matrix(511,328,148,95) (-9/14,-16/25) -> (24/7,7/2) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(383,232,104,63) (-14/23,-3/5) -> (11/3,26/7) 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(257,144,116,65) (-9/16,-5/9) -> (11/5,9/4) Hyperbolic Matrix(159,88,56,31) (-5/9,-6/11) -> (14/5,3/1) Hyperbolic Matrix(383,208,116,63) (-6/11,-7/13) -> (23/7,10/3) Hyperbolic Matrix(513,272,364,193) (-8/15,-1/2) -> (31/22,24/17) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(417,-224,296,-159) (1/2,7/13) -> (7/5,31/22) Hyperbolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(289,-176,156,-95) (3/5,11/18) -> (11/6,13/7) Hyperbolic Matrix(63,-40,52,-33) (5/8,2/3) -> (6/5,5/4) Hyperbolic Matrix(415,-288,232,-161) (9/13,7/10) -> (25/14,9/5) Hyperbolic Matrix(65,-56,36,-31) (5/6,1/1) -> (9/5,11/6) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(417,-592,112,-159) (17/12,10/7) -> (26/7,15/4) Hyperbolic Matrix(63,-104,20,-33) (13/8,5/3) -> (3/1,13/4) Hyperbolic Matrix(191,-440,56,-129) (16/7,7/3) -> (17/5,24/7) Hyperbolic Matrix(223,-528,68,-161) (7/3,19/8) -> (13/4,23/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(31,120,8,31) -> Matrix(1,2,0,1) Matrix(159,592,-112,-417) -> Matrix(3,2,4,3) Matrix(31,112,44,159) -> Matrix(1,0,2,1) Matrix(31,104,-48,-161) -> Matrix(1,-2,0,1) Matrix(33,104,-20,-63) -> Matrix(1,0,-2,1) Matrix(31,88,56,159) -> Matrix(1,2,0,1) Matrix(127,352,92,255) -> Matrix(3,2,-2,-1) Matrix(65,176,24,65) -> Matrix(1,0,4,1) Matrix(31,80,12,31) -> Matrix(1,0,-2,1) Matrix(33,80,40,97) -> Matrix(1,-2,0,1) Matrix(191,456,80,191) -> Matrix(1,4,0,1) Matrix(287,680,-176,-417) -> Matrix(1,2,-4,-7) Matrix(97,224,-152,-351) -> Matrix(7,2,-4,-1) Matrix(127,288,56,127) -> Matrix(1,0,-4,1) Matrix(65,144,116,257) -> Matrix(3,-2,2,-1) Matrix(63,136,44,95) -> Matrix(1,0,-2,1) Matrix(31,56,-36,-65) -> Matrix(1,0,2,1) Matrix(161,288,-232,-415) -> Matrix(1,0,4,1) Matrix(127,224,72,127) -> Matrix(1,0,0,1) Matrix(33,56,-56,-95) -> Matrix(1,0,0,1) Matrix(191,312,352,575) -> Matrix(11,4,8,3) Matrix(129,208,80,129) -> Matrix(1,0,8,1) Matrix(31,48,20,31) -> Matrix(1,0,-2,1) Matrix(127,184,176,255) -> Matrix(1,0,0,1) Matrix(95,136,44,63) -> Matrix(1,0,-2,1) Matrix(129,184,68,97) -> Matrix(3,-2,2,-1) Matrix(577,816,408,577) -> Matrix(7,-6,-8,7) Matrix(159,224,-296,-417) -> Matrix(9,-10,-8,9) Matrix(127,176,184,255) -> Matrix(1,-2,0,1) Matrix(255,352,92,127) -> Matrix(1,-2,2,-3) Matrix(65,88,48,65) -> Matrix(1,-4,0,1) Matrix(31,40,24,31) -> Matrix(1,2,0,1) Matrix(33,40,-52,-63) -> Matrix(3,2,-2,-1) Matrix(161,136,-264,-223) -> Matrix(3,-2,-4,3) Matrix(97,80,40,33) -> Matrix(1,-2,0,1) Matrix(31,24,40,31) -> Matrix(1,2,0,1) Matrix(65,48,88,65) -> Matrix(1,2,0,1) Matrix(287,208,396,287) -> Matrix(1,0,2,1) Matrix(255,184,176,127) -> Matrix(1,0,0,1) Matrix(159,112,44,31) -> Matrix(1,0,2,1) Matrix(735,512,412,287) -> Matrix(1,0,6,1) Matrix(255,176,184,127) -> Matrix(1,-2,0,1) Matrix(511,328,148,95) -> Matrix(1,2,2,5) Matrix(129,80,208,129) -> Matrix(5,6,4,5) Matrix(287,176,468,287) -> Matrix(9,8,10,9) Matrix(383,232,104,63) -> Matrix(3,2,4,3) Matrix(193,112,112,65) -> Matrix(1,0,0,1) Matrix(127,72,224,127) -> Matrix(1,4,0,1) Matrix(257,144,116,65) -> Matrix(1,2,-2,-3) Matrix(159,88,56,31) -> Matrix(1,2,0,1) Matrix(383,208,116,63) -> Matrix(3,4,2,3) Matrix(513,272,364,193) -> Matrix(11,10,-10,-9) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(417,-224,296,-159) -> Matrix(9,-10,-8,9) Matrix(95,-56,56,-33) -> Matrix(1,0,0,1) Matrix(289,-176,156,-95) -> Matrix(3,-2,2,-1) Matrix(63,-40,52,-33) -> Matrix(1,-2,2,-3) Matrix(415,-288,232,-161) -> Matrix(1,0,4,1) Matrix(65,-56,36,-31) -> Matrix(1,0,2,1) Matrix(95,-112,28,-33) -> Matrix(1,0,-2,1) Matrix(417,-592,112,-159) -> Matrix(3,2,4,3) Matrix(63,-104,20,-33) -> Matrix(1,0,-2,1) Matrix(191,-440,56,-129) -> Matrix(1,0,-4,1) Matrix(223,-528,68,-161) -> Matrix(1,-2,2,-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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 21 Degree of the the map X: 21 Degree of the the map Y: 64 Permutation triple for Y: ((1,2)(3,10,34,58,50,49,35,11)(4,15,46,32,62,47,16,5)(6,21,53,43,57,42,41,22)(7,27,38,63,56,51,28,8)(9,19,18,31)(12,39,40,13)(14,37)(17,33)(20,36,44,30)(23,25,24,55)(26,54)(29,52)(45,59)(48,60)(61,64); (1,5,19,41,45,15,44,57,61,32,31,53,48,47,20,6)(2,8,30,34,60,51,18,50,64,63,36,35,59,27,9,3)(4,13,42,17,16,23,22,54,62,39,21,52,46,24,43,14)(7,25,10,33,38,12,11,37,56,55,49,29,28,40,58,26); (1,3,12,4)(2,6,23,56,64,57,24,7)(5,17,10,9,32,52,49,18)(8,29,21,20)(11,36,47,54,58,30,15,14)(13,28,60,53,39,38,59,41)(16,48,34,25)(19,27,26,22)(31,51,37,43)(33,42,44,63)(35,55,46,45)(40,62,61,50)) ----------------------------------------------------------------------- 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 0/1 0/1 1 2 1/2 (0/1,1/1) 0 16 7/13 7/6 1 16 6/11 3/2 2 8 5/9 1/0 1 16 9/16 (2/1,1/0) 0 16 4/7 1/0 2 4 3/5 1/2 1 16 11/18 (4/5,1/1) 0 16 8/13 1/1 7 2 5/8 (1/1,3/2) 0 16 2/3 1/0 2 8 9/13 -1/2 1 16 7/10 (0/1,1/1) 0 16 5/7 1/0 1 16 13/18 (0/1,1/1) 0 16 8/11 1/1 1 2 3/4 (1/1,1/0) 0 16 4/5 1/0 5 4 5/6 (-2/1,-1/1) 0 16 1/1 1/0 1 16 6/5 1/2 2 8 5/4 (1/1,1/0) 0 16 4/3 1/0 3 4 11/8 (-2/1,1/0) 0 16 18/13 1/0 2 8 7/5 -3/2 1 16 31/22 (-10/9,-1/1) 0 16 24/17 -1/1 13 2 17/12 (-1/1,-3/4) 0 16 10/7 -1/2 1 8 13/9 1/0 1 16 3/2 (-1/1,0/1) 0 16 8/5 0/1 5 2 13/8 (0/1,1/4) 0 16 5/3 1/2 1 16 12/7 1/0 2 4 7/4 (0/1,1/0) 0 16 16/9 0/1 5 2 25/14 (0/1,1/5) 0 16 9/5 1/2 1 16 11/6 (2/3,1/1) 0 16 13/7 1/0 1 16 2/1 1/0 1 8 11/5 -1/2 1 16 9/4 (-1/2,0/1) 0 16 16/7 0/1 7 2 7/3 1/2 1 16 19/8 (2/1,1/0) 0 16 12/5 1/0 5 4 5/2 (-1/1,0/1) 0 16 8/3 0/1 3 2 11/4 (0/1,1/2) 0 16 14/5 1/2 2 8 3/1 1/0 1 16 13/4 (0/1,1/2) 0 16 23/7 3/4 1 16 10/3 1/0 2 8 17/5 -1/2 1 16 24/7 0/1 7 2 7/2 (0/1,1/1) 0 16 11/3 1/2 1 16 26/7 1/2 1 8 15/4 (1/1,1/0) 0 16 4/1 1/0 1 4 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(417,-224,296,-159) (1/2,7/13) -> (7/5,31/22) Hyperbolic Matrix(383,-208,116,-63) (7/13,6/11) -> (23/7,10/3) Glide Reflection Matrix(159,-88,56,-31) (6/11,5/9) -> (14/5,3/1) Glide Reflection Matrix(257,-144,116,-65) (5/9,9/16) -> (11/5,9/4) Glide Reflection 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(289,-176,156,-95) (3/5,11/18) -> (11/6,13/7) Hyperbolic Matrix(287,-176,468,-287) (11/18,8/13) -> (11/18,8/13) Reflection Matrix(129,-80,208,-129) (8/13,5/8) -> (8/13,5/8) Reflection Matrix(63,-40,52,-33) (5/8,2/3) -> (6/5,5/4) Hyperbolic Matrix(255,-176,184,-127) (2/3,9/13) -> (18/13,7/5) Glide Reflection Matrix(415,-288,232,-161) (9/13,7/10) -> (25/14,9/5) Hyperbolic Matrix(159,-112,44,-31) (7/10,5/7) -> (7/2,11/3) Glide Reflection Matrix(255,-184,176,-127) (5/7,13/18) -> (13/9,3/2) Glide Reflection Matrix(287,-208,396,-287) (13/18,8/11) -> (13/18,8/11) Reflection Matrix(65,-48,88,-65) (8/11,3/4) -> (8/11,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(97,-80,40,-33) (4/5,5/6) -> (12/5,5/2) Glide Reflection Matrix(65,-56,36,-31) (5/6,1/1) -> (9/5,11/6) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic 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(255,-352,92,-127) (11/8,18/13) -> (11/4,14/5) Glide Reflection Matrix(1055,-1488,748,-1055) (31/22,24/17) -> (31/22,24/17) Reflection Matrix(577,-816,408,-577) (24/17,17/12) -> (24/17,17/12) Reflection Matrix(417,-592,112,-159) (17/12,10/7) -> (26/7,15/4) Hyperbolic Matrix(95,-136,44,-63) (10/7,13/9) -> (2/1,11/5) Glide Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) Reflection Matrix(63,-104,20,-33) (13/8,5/3) -> (3/1,13/4) Hyperbolic Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(127,-224,72,-127) (7/4,16/9) -> (7/4,16/9) Reflection Matrix(449,-800,252,-449) (16/9,25/14) -> (16/9,25/14) Reflection Matrix(193,-360,52,-97) (13/7,2/1) -> (11/3,26/7) Glide Reflection Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(191,-440,56,-129) (16/7,7/3) -> (17/5,24/7) Hyperbolic Matrix(223,-528,68,-161) (7/3,19/8) -> (13/4,23/7) Hyperbolic Matrix(191,-456,80,-191) (19/8,12/5) -> (19/8,12/5) Reflection 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(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(31,-120,8,-31) (15/4,4/1) -> (15/4,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) 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(1,0,4,-1) -> Matrix(1,0,2,-1) (0/1,1/2) -> (0/1,1/1) Matrix(417,-224,296,-159) -> Matrix(9,-10,-8,9) Matrix(383,-208,116,-63) -> Matrix(3,-4,2,-3) *** -> (1/1,2/1) Matrix(159,-88,56,-31) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(257,-144,116,-65) -> Matrix(1,-2,-2,3) Matrix(127,-72,224,-127) -> Matrix(-1,4,0,1) (9/16,4/7) -> (2/1,1/0) Matrix(95,-56,56,-33) -> Matrix(1,0,0,1) Matrix(289,-176,156,-95) -> Matrix(3,-2,2,-1) 1/1 Matrix(287,-176,468,-287) -> Matrix(9,-8,10,-9) (11/18,8/13) -> (4/5,1/1) Matrix(129,-80,208,-129) -> Matrix(5,-6,4,-5) (8/13,5/8) -> (1/1,3/2) Matrix(63,-40,52,-33) -> Matrix(1,-2,2,-3) 1/1 Matrix(255,-176,184,-127) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(415,-288,232,-161) -> Matrix(1,0,4,1) 0/1 Matrix(159,-112,44,-31) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(255,-184,176,-127) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(287,-208,396,-287) -> Matrix(1,0,2,-1) (13/18,8/11) -> (0/1,1/1) Matrix(65,-48,88,-65) -> Matrix(-1,2,0,1) (8/11,3/4) -> (1/1,1/0) Matrix(31,-24,40,-31) -> Matrix(-1,2,0,1) (3/4,4/5) -> (1/1,1/0) Matrix(97,-80,40,-33) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(65,-56,36,-31) -> Matrix(1,0,2,1) 0/1 Matrix(95,-112,28,-33) -> Matrix(1,0,-2,1) 0/1 Matrix(31,-40,24,-31) -> Matrix(-1,2,0,1) (5/4,4/3) -> (1/1,1/0) Matrix(65,-88,48,-65) -> Matrix(1,4,0,-1) (4/3,11/8) -> (-2/1,1/0) Matrix(255,-352,92,-127) -> Matrix(1,2,2,3) Matrix(1055,-1488,748,-1055) -> Matrix(19,20,-18,-19) (31/22,24/17) -> (-10/9,-1/1) Matrix(577,-816,408,-577) -> Matrix(7,6,-8,-7) (24/17,17/12) -> (-1/1,-3/4) Matrix(417,-592,112,-159) -> Matrix(3,2,4,3) Matrix(95,-136,44,-63) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(31,-48,20,-31) -> Matrix(-1,0,2,1) (3/2,8/5) -> (-1/1,0/1) Matrix(129,-208,80,-129) -> Matrix(1,0,8,-1) (8/5,13/8) -> (0/1,1/4) Matrix(63,-104,20,-33) -> Matrix(1,0,-2,1) 0/1 Matrix(97,-168,56,-97) -> Matrix(1,0,0,-1) (12/7,7/4) -> (0/1,1/0) Matrix(127,-224,72,-127) -> Matrix(1,0,0,-1) (7/4,16/9) -> (0/1,1/0) Matrix(449,-800,252,-449) -> Matrix(1,0,10,-1) (16/9,25/14) -> (0/1,1/5) Matrix(193,-360,52,-97) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(127,-288,56,-127) -> Matrix(-1,0,4,1) (9/4,16/7) -> (-1/2,0/1) Matrix(191,-440,56,-129) -> Matrix(1,0,-4,1) 0/1 Matrix(223,-528,68,-161) -> Matrix(1,-2,2,-3) 1/1 Matrix(191,-456,80,-191) -> Matrix(-1,4,0,1) (19/8,12/5) -> (2/1,1/0) Matrix(31,-80,12,-31) -> Matrix(-1,0,2,1) (5/2,8/3) -> (-1/1,0/1) Matrix(65,-176,24,-65) -> Matrix(1,0,4,-1) (8/3,11/4) -> (0/1,1/2) Matrix(97,-336,28,-97) -> Matrix(1,0,2,-1) (24/7,7/2) -> (0/1,1/1) Matrix(31,-120,8,-31) -> Matrix(-1,2,0,1) (15/4,4/1) -> (1/1,1/0) Matrix(-1,8,0,1) -> Matrix(1,0,0,-1) (4/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.