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 -3/4 -1/2 -17/48 -4/15 -1/4 -1/6 -2/13 0/1 1/8 1/7 1/6 2/11 3/16 1/5 2/9 1/4 2/7 5/16 1/3 3/8 2/5 7/16 1/2 9/16 5/8 2/3 11/16 3/4 13/16 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/1 -5/6 1/1 3/1 -9/11 5/1 -13/16 1/0 -4/5 0/1 1/0 -3/4 1/0 -11/15 -1/1 -8/11 0/1 1/0 -5/7 1/1 -17/24 0/1 -12/17 0/1 1/1 -7/10 -1/1 1/1 -9/13 1/1 -11/16 1/0 -2/3 0/1 1/0 -5/8 0/1 -8/13 0/1 1/1 -19/31 1/1 -11/18 1/1 3/1 -3/5 1/1 -4/7 -1/1 0/1 -9/16 0/1 -5/9 1/3 -6/11 0/1 1/2 -1/2 -1/1 1/1 -4/9 3/1 1/0 -7/16 1/0 -3/7 -1/1 -8/19 0/1 1/1 -13/31 1/1 -5/12 1/0 -2/5 0/1 1/0 -3/8 1/0 -4/11 -2/1 1/0 -5/14 -5/3 -1/1 -11/31 -1/1 -17/48 -1/1 -6/17 -1/1 -2/3 -1/3 -1/1 -5/16 0/1 -4/13 0/1 1/1 -3/10 -1/1 1/1 -5/17 -1/1 -12/41 -1/1 1/0 -7/24 -1/1 -2/7 -1/1 0/1 -3/11 1/1 -7/26 -3/1 -1/1 -4/15 -2/1 -1/1 -1/4 0/1 -4/17 2/3 1/1 -3/13 1/1 -5/22 -1/1 1/1 -2/9 1/1 1/0 -3/14 -3/1 -1/1 -1/5 -1/1 -3/16 0/1 -2/11 0/1 1/2 -3/17 1/1 -4/23 1/1 1/0 -1/6 -1/1 1/1 -2/13 0/1 1/1 -1/7 -1/1 -1/8 0/1 0/1 0/1 1/0 1/8 0/1 1/7 1/1 1/6 -1/1 1/1 2/11 -1/2 0/1 3/16 0/1 1/5 1/1 2/9 -1/1 1/0 3/13 -1/1 1/4 0/1 2/7 0/1 1/1 3/10 -1/1 1/1 4/13 -1/1 0/1 5/16 0/1 1/3 1/1 6/17 2/3 1/1 5/14 1/1 5/3 4/11 2/1 1/0 3/8 1/0 2/5 0/1 1/0 3/7 1/1 7/16 1/0 4/9 -3/1 1/0 5/11 -1/1 1/2 -1/1 1/1 5/9 -1/3 9/16 0/1 4/7 0/1 1/1 7/12 0/1 17/29 1/1 10/17 1/1 2/1 3/5 -1/1 11/18 -3/1 -1/1 30/49 -6/5 -1/1 49/80 -1/1 19/31 -1/1 8/13 -1/1 0/1 5/8 0/1 2/3 0/1 1/0 11/16 1/0 9/13 -1/1 16/23 -1/1 1/0 7/10 -1/1 1/1 12/17 -1/1 0/1 17/24 0/1 5/7 -1/1 13/18 1/3 1/1 8/11 0/1 1/0 19/26 1/3 1/1 30/41 1/2 1/1 11/15 1/1 3/4 1/0 13/17 -1/1 10/13 -1/1 0/1 17/22 -1/1 1/1 7/9 1/1 4/5 0/1 1/0 13/16 1/0 9/11 -5/1 14/17 -3/1 -2/1 5/6 -3/1 -1/1 6/7 -2/1 -1/1 7/8 -1/1 8/9 -1/1 1/0 1/1 -1/1 1/0 0/1 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(33,28,-112,-95) (-1/1,-5/6) -> (-3/10,-5/17) Hyperbolic Matrix(95,78,-352,-289) (-5/6,-9/11) -> (-3/11,-7/26) Hyperbolic Matrix(287,234,352,287) (-9/11,-13/16) -> (13/16,9/11) Hyperbolic Matrix(129,104,160,129) (-13/16,-4/5) -> (4/5,13/16) Hyperbolic Matrix(33,26,-80,-63) (-4/5,-3/4) -> (-5/12,-2/5) Hyperbolic Matrix(127,94,-304,-225) (-3/4,-11/15) -> (-13/31,-5/12) Hyperbolic Matrix(63,46,-352,-257) (-11/15,-8/11) -> (-2/11,-3/17) Hyperbolic Matrix(97,70,-176,-127) (-8/11,-5/7) -> (-5/9,-6/11) Hyperbolic Matrix(31,22,224,159) (-5/7,-17/24) -> (1/8,1/7) Hyperbolic Matrix(577,408,816,577) (-17/24,-12/17) -> (12/17,17/24) Hyperbolic Matrix(159,112,-592,-417) (-12/17,-7/10) -> (-7/26,-4/15) Hyperbolic Matrix(95,66,-416,-289) (-7/10,-9/13) -> (-3/13,-5/22) Hyperbolic Matrix(287,198,416,287) (-9/13,-11/16) -> (11/16,9/13) 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(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(479,294,624,383) (-8/13,-19/31) -> (13/17,10/13) Hyperbolic Matrix(353,216,-992,-607) (-19/31,-11/18) -> (-5/14,-11/31) Hyperbolic Matrix(95,58,208,127) (-11/18,-3/5) -> (5/11,1/2) Hyperbolic Matrix(31,18,-112,-65) (-3/5,-4/7) -> (-2/7,-3/11) Hyperbolic Matrix(127,72,224,127) (-4/7,-9/16) -> (9/16,4/7) Hyperbolic Matrix(161,90,288,161) (-9/16,-5/9) -> (5/9,9/16) Hyperbolic Matrix(63,34,176,95) (-6/11,-1/2) -> (5/14,4/11) Hyperbolic Matrix(31,14,-144,-65) (-1/2,-4/9) -> (-2/9,-3/14) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(97,42,224,97) (-7/16,-3/7) -> (3/7,7/16) Hyperbolic Matrix(33,14,-224,-95) (-3/7,-8/19) -> (-2/13,-1/7) Hyperbolic Matrix(609,256,992,417) (-8/19,-13/31) -> (19/31,8/13) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(65,24,176,65) (-3/8,-4/11) -> (4/11,3/8) Hyperbolic Matrix(255,92,352,127) (-4/11,-5/14) -> (13/18,8/11) Hyperbolic Matrix(2431,862,3968,1407) (-11/31,-17/48) -> (49/80,19/31) Hyperbolic Matrix(2273,804,3712,1313) (-17/48,-6/17) -> (30/49,49/80) Hyperbolic Matrix(319,112,544,191) (-6/17,-1/3) -> (17/29,10/17) Hyperbolic Matrix(31,10,96,31) (-1/3,-5/16) -> (5/16,1/3) Hyperbolic Matrix(129,40,416,129) (-5/16,-4/13) -> (4/13,5/16) Hyperbolic Matrix(33,10,-208,-63) (-4/13,-3/10) -> (-1/6,-2/13) Hyperbolic Matrix(961,282,1312,385) (-5/17,-12/41) -> (30/41,11/15) Hyperbolic Matrix(479,140,544,159) (-12/41,-7/24) -> (7/8,8/9) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(31,8,-128,-33) (-4/15,-1/4) -> (-1/4,-4/17) Parabolic Matrix(95,22,272,63) (-4/17,-3/13) -> (1/3,6/17) Hyperbolic Matrix(97,22,-560,-127) (-5/22,-2/9) -> (-4/23,-1/6) Hyperbolic Matrix(97,20,160,33) (-3/14,-1/5) -> (3/5,11/18) Hyperbolic Matrix(31,6,160,31) (-1/5,-3/16) -> (3/16,1/5) Hyperbolic Matrix(65,12,352,65) (-3/16,-2/11) -> (2/11,3/16) Hyperbolic Matrix(159,28,176,31) (-3/17,-4/23) -> (8/9,1/1) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(1,0,16,1) (-1/8,0/1) -> (0/1,1/8) Parabolic Matrix(161,-24,208,-31) (1/7,1/6) -> (17/22,7/9) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(65,-14,144,-31) (1/5,2/9) -> (4/9,5/11) Hyperbolic Matrix(289,-66,416,-95) (2/9,3/13) -> (9/13,16/23) Hyperbolic Matrix(159,-38,272,-65) (3/13,1/4) -> (7/12,17/29) Hyperbolic Matrix(65,-18,112,-31) (1/4,2/7) -> (4/7,7/12) Hyperbolic Matrix(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(607,-216,992,-353) (6/17,5/14) -> (11/18,30/49) Hyperbolic Matrix(63,-26,80,-33) (2/5,3/7) -> (7/9,4/5) Hyperbolic Matrix(127,-70,176,-97) (1/2,5/9) -> (5/7,13/18) Hyperbolic Matrix(223,-132,272,-161) (10/17,3/5) -> (9/11,14/17) Hyperbolic Matrix(737,-514,1008,-703) (16/23,7/10) -> (19/26,30/41) Hyperbolic Matrix(225,-158,272,-191) (7/10,12/17) -> (14/17,5/6) Hyperbolic Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(33,28,-112,-95) -> Matrix(1,-2,0,1) Matrix(95,78,-352,-289) -> Matrix(1,-4,0,1) Matrix(287,234,352,287) -> Matrix(1,-10,0,1) Matrix(129,104,160,129) -> Matrix(1,0,0,1) Matrix(33,26,-80,-63) -> Matrix(1,0,0,1) Matrix(127,94,-304,-225) -> Matrix(1,2,0,1) Matrix(63,46,-352,-257) -> Matrix(1,0,2,1) Matrix(97,70,-176,-127) -> Matrix(1,0,2,1) Matrix(31,22,224,159) -> Matrix(1,0,0,1) Matrix(577,408,816,577) -> Matrix(1,0,-2,1) Matrix(159,112,-592,-417) -> Matrix(1,-2,0,1) Matrix(95,66,-416,-289) -> Matrix(1,0,0,1) Matrix(287,198,416,287) -> Matrix(1,-2,0,1) Matrix(65,44,96,65) -> Matrix(1,0,0,1) Matrix(31,20,48,31) -> Matrix(1,0,0,1) Matrix(129,80,208,129) -> Matrix(1,0,-2,1) Matrix(479,294,624,383) -> Matrix(1,0,-2,1) Matrix(353,216,-992,-607) -> Matrix(3,-4,-2,3) Matrix(95,58,208,127) -> Matrix(1,-2,0,1) Matrix(31,18,-112,-65) -> Matrix(1,0,0,1) Matrix(127,72,224,127) -> Matrix(1,0,2,1) Matrix(161,90,288,161) -> Matrix(1,0,-6,1) Matrix(63,34,176,95) -> Matrix(3,-2,2,-1) Matrix(31,14,-144,-65) -> Matrix(1,-2,0,1) Matrix(127,56,288,127) -> Matrix(1,-6,0,1) Matrix(97,42,224,97) -> Matrix(1,2,0,1) Matrix(33,14,-224,-95) -> Matrix(1,0,0,1) Matrix(609,256,992,417) -> Matrix(1,0,-2,1) Matrix(31,12,80,31) -> Matrix(1,0,0,1) Matrix(65,24,176,65) -> Matrix(1,4,0,1) Matrix(255,92,352,127) -> Matrix(1,2,0,1) Matrix(2431,862,3968,1407) -> Matrix(9,10,-10,-11) Matrix(2273,804,3712,1313) -> Matrix(9,8,-8,-7) Matrix(319,112,544,191) -> Matrix(1,0,2,1) Matrix(31,10,96,31) -> Matrix(1,0,2,1) Matrix(129,40,416,129) -> Matrix(1,0,-2,1) Matrix(33,10,-208,-63) -> Matrix(1,0,0,1) Matrix(961,282,1312,385) -> Matrix(1,0,2,1) Matrix(479,140,544,159) -> Matrix(1,0,0,1) Matrix(193,56,224,65) -> Matrix(3,2,-2,-1) Matrix(31,8,-128,-33) -> Matrix(1,0,2,1) Matrix(95,22,272,63) -> Matrix(1,0,0,1) Matrix(97,22,-560,-127) -> Matrix(1,0,0,1) Matrix(97,20,160,33) -> Matrix(1,0,0,1) Matrix(31,6,160,31) -> Matrix(1,0,2,1) Matrix(65,12,352,65) -> Matrix(1,0,-4,1) Matrix(159,28,176,31) -> Matrix(1,-2,0,1) Matrix(159,22,224,31) -> Matrix(1,0,0,1) Matrix(1,0,16,1) -> Matrix(1,0,0,1) Matrix(161,-24,208,-31) -> Matrix(1,0,0,1) Matrix(257,-46,352,-63) -> Matrix(1,0,2,1) Matrix(65,-14,144,-31) -> Matrix(1,-2,0,1) Matrix(289,-66,416,-95) -> Matrix(1,0,0,1) Matrix(159,-38,272,-65) -> Matrix(1,0,2,1) Matrix(65,-18,112,-31) -> Matrix(1,0,0,1) Matrix(95,-28,112,-33) -> Matrix(1,-2,0,1) Matrix(321,-98,416,-127) -> Matrix(1,0,0,1) Matrix(607,-216,992,-353) -> Matrix(3,-4,-2,3) Matrix(63,-26,80,-33) -> Matrix(1,0,0,1) Matrix(127,-70,176,-97) -> Matrix(1,0,2,1) Matrix(223,-132,272,-161) -> Matrix(1,-4,0,1) Matrix(737,-514,1008,-703) -> Matrix(1,0,2,1) Matrix(225,-158,272,-191) -> Matrix(1,-2,0,1) Matrix(97,-72,128,-95) -> Matrix(1,-2,0,1) 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): 14 Degree of the the map X: 14 Degree of the the map Y: 64 Permutation triple for Y: ((1,4,16,45,46,17,5,2)(3,10,35,58,26,8,7,11)(6,14,13,9,30,60,28,22)(12,15,29,53,20,19,24,38)(18,23,40,39,43,62,61,50)(21,33,32,44,49,48,27,41)(25,57,31,42,54,47,56,36)(34,59,52,51,63,64,55,37); (1,2,8,28,53,62,32,57,64,63,56,48,61,29,9,3)(4,14,41,51,50,58,35,43,55,21,6,5,20,54,42,15)(7,19,52,60,30,37,12,11,36,18,17,49,44,16,39,25)(10,33,38,24,27,26,59,45,31,13,40,23,22,47,46,34); (2,6,23,36,63,41,24,7)(3,12,33,55,57,39,13,4)(5,18,51,19)(8,27,56,22)(9,31,32,10)(11,25)(14,21)(15,37,43,16)(17,47,20,28,52,26,50,48)(29,42,45,44,62,35,34,30)(46,59)(53,61)) ----------------------------------------------------------------------- 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 (0/1,1/0) 0 16 1/8 0/1 1 2 1/7 1/1 1 16 1/6 0 8 2/11 (-1/2,0/1) 0 16 3/16 0/1 3 1 1/5 1/1 1 16 2/9 (-1/1,1/0) 0 16 3/13 -1/1 1 16 1/4 0/1 1 4 2/7 (0/1,1/1) 0 16 3/10 0 8 4/13 (-1/1,0/1) 0 16 5/16 0/1 2 1 1/3 1/1 1 16 6/17 (2/3,1/1) 0 16 5/14 0 8 4/11 (2/1,1/0) 0 16 3/8 1/0 2 2 2/5 (0/1,1/0) 0 16 3/7 1/1 1 16 7/16 1/0 4 1 4/9 (-3/1,1/0) 0 16 5/11 -1/1 1 16 1/2 0 8 5/9 -1/3 1 16 9/16 0/1 4 1 4/7 (0/1,1/1) 0 16 7/12 0/1 1 4 17/29 1/1 1 16 10/17 (1/1,2/1) 0 16 3/5 -1/1 1 16 11/18 0 8 30/49 (-6/5,-1/1) 0 16 49/80 -1/1 9 1 19/31 -1/1 1 16 8/13 (-1/1,0/1) 0 16 5/8 0/1 1 2 2/3 (0/1,1/0) 0 16 11/16 1/0 1 1 9/13 -1/1 1 16 16/23 (-1/1,1/0) 0 16 7/10 0 8 12/17 (-1/1,0/1) 0 16 17/24 0/1 1 2 5/7 -1/1 1 16 13/18 0 8 8/11 (0/1,1/0) 0 16 19/26 0 8 30/41 (1/2,1/1) 0 16 11/15 1/1 1 16 3/4 1/0 1 4 13/17 -1/1 1 16 10/13 (-1/1,0/1) 0 16 17/22 0 8 7/9 1/1 1 16 4/5 (0/1,1/0) 0 16 13/16 1/0 5 1 9/11 -5/1 1 16 14/17 (-3/1,-2/1) 0 16 5/6 0 8 6/7 (-2/1,-1/1) 0 16 7/8 -1/1 1 2 8/9 (-1/1,1/0) 0 16 1/1 -1/1 1 16 1/0 0/1 1 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(1,0,16,-1) (0/1,1/8) -> (0/1,1/8) Reflection Matrix(159,-22,224,-31) (1/8,1/7) -> (17/24,5/7) Glide Reflection Matrix(161,-24,208,-31) (1/7,1/6) -> (17/22,7/9) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(65,-12,352,-65) (2/11,3/16) -> (2/11,3/16) Reflection Matrix(31,-6,160,-31) (3/16,1/5) -> (3/16,1/5) Reflection Matrix(65,-14,144,-31) (1/5,2/9) -> (4/9,5/11) Hyperbolic Matrix(289,-66,416,-95) (2/9,3/13) -> (9/13,16/23) Hyperbolic Matrix(159,-38,272,-65) (3/13,1/4) -> (7/12,17/29) Hyperbolic Matrix(65,-18,112,-31) (1/4,2/7) -> (4/7,7/12) Hyperbolic Matrix(95,-28,112,-33) (2/7,3/10) -> (5/6,6/7) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(129,-40,416,-129) (4/13,5/16) -> (4/13,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(319,-112,544,-191) (1/3,6/17) -> (17/29,10/17) Glide Reflection Matrix(607,-216,992,-353) (6/17,5/14) -> (11/18,30/49) Hyperbolic Matrix(255,-92,352,-127) (5/14,4/11) -> (13/18,8/11) Glide Reflection Matrix(65,-24,176,-65) (4/11,3/8) -> (4/11,3/8) Reflection Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(63,-26,80,-33) (2/5,3/7) -> (7/9,4/5) Hyperbolic Matrix(97,-42,224,-97) (3/7,7/16) -> (3/7,7/16) Reflection Matrix(127,-56,288,-127) (7/16,4/9) -> (7/16,4/9) Reflection Matrix(127,-58,208,-95) (5/11,1/2) -> (3/5,11/18) Glide Reflection Matrix(127,-70,176,-97) (1/2,5/9) -> (5/7,13/18) Hyperbolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(223,-132,272,-161) (10/17,3/5) -> (9/11,14/17) Hyperbolic Matrix(4801,-2940,7840,-4801) (30/49,49/80) -> (30/49,49/80) Reflection Matrix(3039,-1862,4960,-3039) (49/80,19/31) -> (49/80,19/31) Reflection Matrix(479,-294,624,-383) (19/31,8/13) -> (13/17,10/13) Glide Reflection Matrix(129,-80,208,-129) (8/13,5/8) -> (8/13,5/8) Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(65,-44,96,-65) (2/3,11/16) -> (2/3,11/16) Reflection Matrix(287,-198,416,-287) (11/16,9/13) -> (11/16,9/13) Reflection Matrix(737,-514,1008,-703) (16/23,7/10) -> (19/26,30/41) Hyperbolic Matrix(225,-158,272,-191) (7/10,12/17) -> (14/17,5/6) Hyperbolic Matrix(577,-408,816,-577) (12/17,17/24) -> (12/17,17/24) Reflection Matrix(161,-118,176,-129) (30/41,11/15) -> (8/9,1/1) Glide Reflection Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic Matrix(129,-104,160,-129) (4/5,13/16) -> (4/5,13/16) Reflection Matrix(287,-234,352,-287) (13/16,9/11) -> (13/16,9/11) Reflection Matrix(97,-84,112,-97) (6/7,7/8) -> (6/7,7/8) Reflection Matrix(127,-112,144,-127) (7/8,8/9) -> (7/8,8/9) 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(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,16,-1) -> Matrix(1,0,0,-1) (0/1,1/8) -> (0/1,1/0) Matrix(159,-22,224,-31) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(161,-24,208,-31) -> Matrix(1,0,0,1) Matrix(257,-46,352,-63) -> Matrix(1,0,2,1) 0/1 Matrix(65,-12,352,-65) -> Matrix(-1,0,4,1) (2/11,3/16) -> (-1/2,0/1) Matrix(31,-6,160,-31) -> Matrix(1,0,2,-1) (3/16,1/5) -> (0/1,1/1) Matrix(65,-14,144,-31) -> Matrix(1,-2,0,1) 1/0 Matrix(289,-66,416,-95) -> Matrix(1,0,0,1) Matrix(159,-38,272,-65) -> Matrix(1,0,2,1) 0/1 Matrix(65,-18,112,-31) -> Matrix(1,0,0,1) Matrix(95,-28,112,-33) -> Matrix(1,-2,0,1) 1/0 Matrix(321,-98,416,-127) -> Matrix(1,0,0,1) Matrix(129,-40,416,-129) -> Matrix(-1,0,2,1) (4/13,5/16) -> (-1/1,0/1) Matrix(31,-10,96,-31) -> Matrix(1,0,2,-1) (5/16,1/3) -> (0/1,1/1) Matrix(319,-112,544,-191) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(607,-216,992,-353) -> Matrix(3,-4,-2,3) Matrix(255,-92,352,-127) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(65,-24,176,-65) -> Matrix(-1,4,0,1) (4/11,3/8) -> (2/1,1/0) Matrix(31,-12,80,-31) -> Matrix(1,0,0,-1) (3/8,2/5) -> (0/1,1/0) Matrix(63,-26,80,-33) -> Matrix(1,0,0,1) Matrix(97,-42,224,-97) -> Matrix(-1,2,0,1) (3/7,7/16) -> (1/1,1/0) Matrix(127,-56,288,-127) -> Matrix(1,6,0,-1) (7/16,4/9) -> (-3/1,1/0) Matrix(127,-58,208,-95) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(127,-70,176,-97) -> Matrix(1,0,2,1) 0/1 Matrix(161,-90,288,-161) -> Matrix(-1,0,6,1) (5/9,9/16) -> (-1/3,0/1) Matrix(127,-72,224,-127) -> Matrix(1,0,2,-1) (9/16,4/7) -> (0/1,1/1) Matrix(223,-132,272,-161) -> Matrix(1,-4,0,1) 1/0 Matrix(4801,-2940,7840,-4801) -> Matrix(11,12,-10,-11) (30/49,49/80) -> (-6/5,-1/1) Matrix(3039,-1862,4960,-3039) -> Matrix(7,6,-8,-7) (49/80,19/31) -> (-1/1,-3/4) Matrix(479,-294,624,-383) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(129,-80,208,-129) -> Matrix(-1,0,2,1) (8/13,5/8) -> (-1/1,0/1) Matrix(31,-20,48,-31) -> Matrix(1,0,0,-1) (5/8,2/3) -> (0/1,1/0) Matrix(65,-44,96,-65) -> Matrix(1,0,0,-1) (2/3,11/16) -> (0/1,1/0) Matrix(287,-198,416,-287) -> Matrix(1,2,0,-1) (11/16,9/13) -> (-1/1,1/0) Matrix(737,-514,1008,-703) -> Matrix(1,0,2,1) 0/1 Matrix(225,-158,272,-191) -> Matrix(1,-2,0,1) 1/0 Matrix(577,-408,816,-577) -> Matrix(-1,0,2,1) (12/17,17/24) -> (-1/1,0/1) Matrix(161,-118,176,-129) -> Matrix(3,-2,-2,1) Matrix(97,-72,128,-95) -> Matrix(1,-2,0,1) 1/0 Matrix(129,-104,160,-129) -> Matrix(1,0,0,-1) (4/5,13/16) -> (0/1,1/0) Matrix(287,-234,352,-287) -> Matrix(1,10,0,-1) (13/16,9/11) -> (-5/1,1/0) Matrix(97,-84,112,-97) -> Matrix(3,4,-2,-3) (6/7,7/8) -> (-2/1,-1/1) Matrix(127,-112,144,-127) -> Matrix(1,2,0,-1) (7/8,8/9) -> (-1/1,1/0) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.