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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 7/6 7/5 3/2 7/4 2/1 7/3 12/5 5/2 28/11 8/3 14/5 3/1 42/13 10/3 7/2 4/1 14/3 5/1 28/5 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 0/1 -6/1 1/0 -11/2 -1/1 -2/3 -5/1 -1/2 0/1 -4/1 0/1 1/0 -7/2 1/0 -10/3 1/0 -13/4 -2/1 -1/1 -3/1 -1/1 0/1 -8/3 0/1 1/0 -21/8 0/1 -13/5 0/1 1/1 -18/7 1/0 -5/2 0/1 1/0 -17/7 1/1 2/1 -29/12 0/1 1/1 -12/5 1/1 1/0 -7/3 1/0 -2/1 1/0 -7/4 1/0 -12/7 -3/1 1/0 -17/10 -4/1 -3/1 -5/3 -2/1 1/0 -28/17 -3/1 -1/1 -23/14 -2/1 1/0 -18/11 1/0 -13/8 -3/1 -2/1 -21/13 -2/1 -8/5 -2/1 1/0 -11/7 -3/1 -2/1 -14/9 -2/1 -3/2 -2/1 -1/1 -16/11 -1/1 1/0 -29/20 -2/1 -1/1 -42/29 -1/1 -13/9 -1/1 0/1 -10/7 1/0 -7/5 1/0 -4/3 -2/1 1/0 -9/7 -5/2 -2/1 -14/11 -2/1 -5/4 -2/1 -3/2 -11/9 -4/3 -1/1 -28/23 -1/1 -17/14 -1/1 -2/3 -6/5 1/0 -7/6 -2/1 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 -2/3 7/6 -2/3 6/5 -1/2 11/9 -1/1 -4/5 5/4 -3/4 -2/3 4/3 -2/3 -1/2 7/5 -1/2 10/7 -1/2 13/9 -1/1 0/1 3/2 -1/1 -2/3 8/5 -2/3 -1/2 21/13 -2/3 13/8 -2/3 -3/5 18/11 -1/2 5/3 -2/3 -1/2 17/10 -3/5 -4/7 29/17 -2/3 -3/5 12/7 -3/5 -1/2 7/4 -1/2 2/1 -1/2 7/3 -1/2 12/5 -1/2 -1/3 17/7 -2/5 -1/3 5/2 -1/2 0/1 28/11 -1/1 -1/3 23/9 -1/2 0/1 18/7 -1/2 13/5 -1/3 0/1 21/8 0/1 8/3 -1/2 0/1 11/4 -1/3 0/1 14/5 0/1 3/1 -1/1 0/1 16/5 -1/1 -1/2 29/9 -1/1 0/1 42/13 -1/1 13/4 -1/1 -2/3 10/3 -1/2 7/2 -1/2 4/1 -1/2 0/1 9/2 -1/4 0/1 14/3 0/1 5/1 0/1 1/0 11/2 -2/1 -1/1 28/5 -1/1 17/3 -1/1 -4/5 6/1 -1/2 7/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,112,8,69) (-7/1,1/0) -> (21/13,13/8) Hyperbolic Matrix(13,84,2,13) (-7/1,-6/1) -> (6/1,7/1) Hyperbolic Matrix(29,168,-24,-139) (-6/1,-11/2) -> (-17/14,-6/5) Hyperbolic Matrix(27,140,16,83) (-11/2,-5/1) -> (5/3,17/10) Hyperbolic Matrix(13,56,-10,-43) (-5/1,-4/1) -> (-4/3,-9/7) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(111,364,68,223) (-10/3,-13/4) -> (13/8,18/11) Hyperbolic Matrix(97,308,-40,-127) (-13/4,-3/1) -> (-17/7,-29/12) Hyperbolic Matrix(41,112,-26,-71) (-3/1,-8/3) -> (-8/5,-11/7) Hyperbolic Matrix(127,336,48,127) (-8/3,-21/8) -> (21/8,8/3) Hyperbolic Matrix(43,112,38,99) (-21/8,-13/5) -> (1/1,7/6) Hyperbolic Matrix(141,364,98,253) (-13/5,-18/7) -> (10/7,13/9) Hyperbolic Matrix(197,504,-120,-307) (-18/7,-5/2) -> (-23/14,-18/11) Hyperbolic Matrix(57,140,46,113) (-5/2,-17/7) -> (11/9,5/4) Hyperbolic Matrix(337,812,-232,-559) (-29/12,-12/5) -> (-16/11,-29/20) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(181,308,-124,-211) (-12/7,-17/10) -> (-3/2,-16/11) Hyperbolic Matrix(83,140,16,27) (-17/10,-5/3) -> (5/1,11/2) Hyperbolic Matrix(475,784,186,307) (-5/3,-28/17) -> (28/11,23/9) Hyperbolic Matrix(477,784,188,309) (-28/17,-23/14) -> (5/2,28/11) Hyperbolic Matrix(223,364,68,111) (-18/11,-13/8) -> (13/4,10/3) Hyperbolic Matrix(69,112,8,13) (-13/8,-21/13) -> (7/1,1/0) Hyperbolic Matrix(209,336,130,209) (-21/13,-8/5) -> (8/5,21/13) Hyperbolic Matrix(125,196,44,69) (-11/7,-14/9) -> (14/5,3/1) Hyperbolic Matrix(127,196,46,71) (-14/9,-3/2) -> (11/4,14/5) Hyperbolic Matrix(1217,1764,376,545) (-29/20,-42/29) -> (42/13,13/4) Hyperbolic Matrix(1219,1764,378,547) (-42/29,-13/9) -> (29/9,42/13) Hyperbolic Matrix(253,364,98,141) (-13/9,-10/7) -> (18/7,13/5) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(153,196,32,41) (-9/7,-14/11) -> (14/3,5/1) Hyperbolic Matrix(155,196,34,43) (-14/11,-5/4) -> (9/2,14/3) Hyperbolic Matrix(113,140,46,57) (-5/4,-11/9) -> (17/7,5/2) Hyperbolic Matrix(643,784,114,139) (-11/9,-28/23) -> (28/5,17/3) Hyperbolic Matrix(645,784,116,141) (-28/23,-17/14) -> (11/2,28/5) Hyperbolic Matrix(71,84,60,71) (-6/5,-7/6) -> (7/6,6/5) Hyperbolic Matrix(99,112,38,43) (-7/6,-1/1) -> (13/5,21/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(211,-308,124,-181) (13/9,3/2) -> (17/10,29/17) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(475,-812,148,-253) (29/17,12/7) -> (16/5,29/9) Hyperbolic Matrix(127,-308,40,-97) (12/5,17/7) -> (3/1,16/5) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,112,8,69) -> Matrix(5,2,-8,-3) Matrix(13,84,2,13) -> Matrix(1,0,-2,1) Matrix(29,168,-24,-139) -> Matrix(1,0,0,1) Matrix(27,140,16,83) -> Matrix(5,2,-8,-3) Matrix(13,56,-10,-43) -> Matrix(1,-2,0,1) Matrix(15,56,4,15) -> Matrix(1,0,-2,1) Matrix(41,140,12,41) -> Matrix(1,2,-2,-3) Matrix(111,364,68,223) -> Matrix(1,4,-2,-7) Matrix(97,308,-40,-127) -> Matrix(1,2,0,1) Matrix(41,112,-26,-71) -> Matrix(1,-2,0,1) Matrix(127,336,48,127) -> Matrix(1,0,-2,1) Matrix(43,112,38,99) -> Matrix(3,-2,-4,3) Matrix(141,364,98,253) -> Matrix(1,0,-2,1) Matrix(197,504,-120,-307) -> Matrix(1,-2,0,1) Matrix(57,140,46,113) -> Matrix(3,-2,-4,3) Matrix(337,812,-232,-559) -> Matrix(1,-2,0,1) Matrix(71,168,30,71) -> Matrix(1,-2,-2,5) Matrix(13,28,6,13) -> Matrix(1,0,-2,1) Matrix(15,28,8,15) -> Matrix(1,4,-2,-7) Matrix(97,168,56,97) -> Matrix(1,6,-2,-11) Matrix(181,308,-124,-211) -> Matrix(1,2,0,1) Matrix(83,140,16,27) -> Matrix(1,2,0,1) Matrix(475,784,186,307) -> Matrix(1,2,-2,-3) Matrix(477,784,188,309) -> Matrix(1,2,-2,-3) Matrix(223,364,68,111) -> Matrix(1,4,-2,-7) Matrix(69,112,8,13) -> Matrix(1,2,0,1) Matrix(209,336,130,209) -> Matrix(1,4,-2,-7) Matrix(125,196,44,69) -> Matrix(1,2,0,1) Matrix(127,196,46,71) -> Matrix(1,2,-4,-7) Matrix(1217,1764,376,545) -> Matrix(3,4,-4,-5) Matrix(1219,1764,378,547) -> Matrix(1,0,0,1) Matrix(253,364,98,141) -> Matrix(1,0,-2,1) Matrix(99,140,70,99) -> Matrix(1,2,-2,-3) Matrix(41,56,30,41) -> Matrix(1,4,-2,-7) Matrix(153,196,32,41) -> Matrix(1,2,2,5) Matrix(155,196,34,43) -> Matrix(1,2,-6,-11) Matrix(113,140,46,57) -> Matrix(1,2,-4,-7) Matrix(643,784,114,139) -> Matrix(7,8,-8,-9) Matrix(645,784,116,141) -> Matrix(5,4,-4,-3) Matrix(71,84,60,71) -> Matrix(1,4,-2,-7) Matrix(99,112,38,43) -> Matrix(1,2,-4,-7) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(139,-168,24,-29) -> Matrix(1,0,0,1) Matrix(43,-56,10,-13) -> Matrix(3,2,-8,-5) Matrix(211,-308,124,-181) -> Matrix(5,2,-8,-3) Matrix(71,-112,26,-41) -> Matrix(3,2,-8,-5) Matrix(307,-504,120,-197) -> Matrix(3,2,-8,-5) Matrix(475,-812,148,-253) -> Matrix(3,2,-8,-5) Matrix(127,-308,40,-97) -> Matrix(5,2,-8,-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): 10 Degree of the the map X: 10 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,13,4,3,12,7)(5,18,8,10,9,16,15)(11,14,37,21,20,24,32)(17,40,39,28,27,26,41)(19,38,43,30,25,36,33)(22,42,44,23,34,46,35); (1,4,16,40,24,38,46,48,42,33,32,17,5,2)(3,10,30,34,41,45,39,44,25,8,7,20,31,11)(6,21,43,47,36,14,13,35,26,18,29,9,28,22)(12,23)(15,19)(27,37); (1,2,8,26,37,36,44,48,46,30,21,27,9,3)(4,14,31,20,6,5,19,42,28,45,41,35,38,15)(7,23,39,16,29,18,17,34,12,11,33,47,43,24)(10,25)(13,22)(32,40)) ----------------------------------------------------------------------- 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): 144 Minimal number of generators: 25 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: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 7/3 12/5 28/11 8/3 14/5 3/1 42/13 10/3 7/2 4/1 14/3 5/1 28/5 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 1/1 -1/1 -2/3 7/6 -2/3 6/5 -1/2 11/9 -1/1 -4/5 5/4 -3/4 -2/3 4/3 -2/3 -1/2 7/5 -1/2 10/7 -1/2 13/9 -1/1 0/1 3/2 -1/1 -2/3 8/5 -2/3 -1/2 21/13 -2/3 13/8 -2/3 -3/5 18/11 -1/2 5/3 -2/3 -1/2 17/10 -3/5 -4/7 29/17 -2/3 -3/5 12/7 -3/5 -1/2 7/4 -1/2 2/1 -1/2 7/3 -1/2 12/5 -1/2 -1/3 17/7 -2/5 -1/3 5/2 -1/2 0/1 28/11 -1/1 -1/3 23/9 -1/2 0/1 18/7 -1/2 13/5 -1/3 0/1 21/8 0/1 8/3 -1/2 0/1 11/4 -1/3 0/1 14/5 0/1 3/1 -1/1 0/1 16/5 -1/1 -1/2 29/9 -1/1 0/1 42/13 -1/1 13/4 -1/1 -2/3 10/3 -1/2 7/2 -1/2 4/1 -1/2 0/1 9/2 -1/4 0/1 14/3 0/1 5/1 0/1 1/0 11/2 -2/1 -1/1 28/5 -1/1 17/3 -1/1 -4/5 6/1 -1/2 7/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(99,-112,61,-69) (1/1,7/6) -> (21/13,13/8) Hyperbolic Matrix(71,-84,11,-13) (7/6,6/5) -> (6/1,7/1) Hyperbolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(113,-140,67,-83) (11/9,5/4) -> (5/3,17/10) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(41,-56,11,-15) (4/3,7/5) -> (7/2,4/1) Hyperbolic Matrix(99,-140,29,-41) (7/5,10/7) -> (10/3,7/2) Hyperbolic Matrix(253,-364,155,-223) (10/7,13/9) -> (13/8,18/11) Hyperbolic Matrix(211,-308,124,-181) (13/9,3/2) -> (17/10,29/17) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(209,-336,79,-127) (8/5,21/13) -> (21/8,8/3) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(475,-812,148,-253) (29/17,12/7) -> (16/5,29/9) Hyperbolic Matrix(97,-168,41,-71) (12/7,7/4) -> (7/3,12/5) Hyperbolic Matrix(15,-28,7,-13) (7/4,2/1) -> (2/1,7/3) Parabolic Matrix(127,-308,40,-97) (12/5,17/7) -> (3/1,16/5) Hyperbolic Matrix(57,-140,11,-27) (17/7,5/2) -> (5/1,11/2) Hyperbolic Matrix(309,-784,121,-307) (5/2,28/11) -> (28/11,23/9) Parabolic Matrix(141,-364,43,-111) (18/7,13/5) -> (13/4,10/3) Hyperbolic Matrix(43,-112,5,-13) (13/5,21/8) -> (7/1,1/0) Hyperbolic Matrix(71,-196,25,-69) (11/4,14/5) -> (14/5,3/1) Parabolic Matrix(547,-1764,169,-545) (29/9,42/13) -> (42/13,13/4) Parabolic Matrix(43,-196,9,-41) (9/2,14/3) -> (14/3,5/1) Parabolic Matrix(141,-784,25,-139) (11/2,28/5) -> (28/5,17/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,2,-2,-3) Matrix(99,-112,61,-69) -> Matrix(11,8,-18,-13) Matrix(71,-84,11,-13) -> Matrix(3,2,-8,-5) Matrix(139,-168,24,-29) -> Matrix(1,0,0,1) Matrix(113,-140,67,-83) -> Matrix(11,8,-18,-13) Matrix(43,-56,10,-13) -> Matrix(3,2,-8,-5) Matrix(41,-56,11,-15) -> Matrix(3,2,-8,-5) Matrix(99,-140,29,-41) -> Matrix(1,0,0,1) Matrix(253,-364,155,-223) -> Matrix(5,2,-8,-3) Matrix(211,-308,124,-181) -> Matrix(5,2,-8,-3) Matrix(71,-112,26,-41) -> Matrix(3,2,-8,-5) Matrix(209,-336,79,-127) -> Matrix(3,2,-8,-5) Matrix(307,-504,120,-197) -> Matrix(3,2,-8,-5) Matrix(475,-812,148,-253) -> Matrix(3,2,-8,-5) Matrix(97,-168,41,-71) -> Matrix(7,4,-16,-9) Matrix(15,-28,7,-13) -> Matrix(3,2,-8,-5) Matrix(127,-308,40,-97) -> Matrix(5,2,-8,-3) Matrix(57,-140,11,-27) -> Matrix(1,0,2,1) Matrix(309,-784,121,-307) -> Matrix(1,0,0,1) Matrix(141,-364,43,-111) -> Matrix(5,2,-8,-3) Matrix(43,-112,5,-13) -> Matrix(1,0,2,1) Matrix(71,-196,25,-69) -> Matrix(1,0,2,1) Matrix(547,-1764,169,-545) -> Matrix(1,2,-2,-3) Matrix(43,-196,9,-41) -> Matrix(1,0,4,1) Matrix(141,-784,25,-139) -> Matrix(5,6,-6,-7) 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): 5 ----------------------------------------------------------------------- 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 -1/1 2 1 2/1 -1/2 2 7 7/3 -1/2 1 2 12/5 (-1/2,-1/3) 0 7 17/7 (-2/5,-1/3) 0 14 5/2 (-1/2,0/1) 0 14 28/11 (-1/2,0/1) 0 1 18/7 -1/2 2 7 13/5 (-1/3,0/1) 0 14 21/8 0/1 2 2 8/3 (-1/2,0/1) 0 7 14/5 0/1 2 1 3/1 (-1/1,0/1) 0 14 16/5 (-1/1,-1/2) 0 7 42/13 -1/1 2 1 13/4 (-1/1,-2/3) 0 14 10/3 -1/2 2 7 7/2 -1/2 1 2 4/1 (-1/2,0/1) 0 7 14/3 0/1 4 1 5/1 (0/1,1/0) 0 14 11/2 (-2/1,-1/1) 0 14 28/5 -1/1 6 1 6/1 -1/2 2 7 7/1 0/1 2 2 1/0 (-1/1,0/1) 0 14 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(127,-308,40,-97) (12/5,17/7) -> (3/1,16/5) Hyperbolic Matrix(57,-140,11,-27) (17/7,5/2) -> (5/1,11/2) Hyperbolic Matrix(111,-280,44,-111) (5/2,28/11) -> (5/2,28/11) Reflection Matrix(197,-504,77,-197) (28/11,18/7) -> (28/11,18/7) Reflection Matrix(141,-364,43,-111) (18/7,13/5) -> (13/4,10/3) Hyperbolic Matrix(43,-112,5,-13) (13/5,21/8) -> (7/1,1/0) Hyperbolic Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(41,-112,15,-41) (8/3,14/5) -> (8/3,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(209,-672,65,-209) (16/5,42/13) -> (16/5,42/13) Reflection Matrix(337,-1092,104,-337) (42/13,13/4) -> (42/13,13/4) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(13,-56,3,-13) (4/1,14/3) -> (4/1,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(111,-616,20,-111) (11/2,28/5) -> (11/2,28/5) Reflection Matrix(29,-168,5,-29) (28/5,6/1) -> (28/5,6/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) 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(1,0,1,-1) -> Matrix(3,2,-4,-3) (0/1,2/1) -> (-1/1,-1/2) Matrix(13,-28,6,-13) -> Matrix(-1,0,4,1) (2/1,7/3) -> (-1/2,0/1) Matrix(71,-168,30,-71) -> Matrix(5,2,-12,-5) (7/3,12/5) -> (-1/2,-1/3) Matrix(127,-308,40,-97) -> Matrix(5,2,-8,-3) -1/2 Matrix(57,-140,11,-27) -> Matrix(1,0,2,1) 0/1 Matrix(111,-280,44,-111) -> Matrix(-1,0,4,1) (5/2,28/11) -> (-1/2,0/1) Matrix(197,-504,77,-197) -> Matrix(-1,0,4,1) (28/11,18/7) -> (-1/2,0/1) Matrix(141,-364,43,-111) -> Matrix(5,2,-8,-3) -1/2 Matrix(43,-112,5,-13) -> Matrix(1,0,2,1) 0/1 Matrix(127,-336,48,-127) -> Matrix(-1,0,4,1) (21/8,8/3) -> (-1/2,0/1) Matrix(41,-112,15,-41) -> Matrix(-1,0,4,1) (8/3,14/5) -> (-1/2,0/1) Matrix(29,-84,10,-29) -> Matrix(-1,0,2,1) (14/5,3/1) -> (-1/1,0/1) Matrix(209,-672,65,-209) -> Matrix(3,2,-4,-3) (16/5,42/13) -> (-1/1,-1/2) Matrix(337,-1092,104,-337) -> Matrix(5,4,-6,-5) (42/13,13/4) -> (-1/1,-2/3) Matrix(41,-140,12,-41) -> Matrix(3,2,-4,-3) (10/3,7/2) -> (-1/1,-1/2) Matrix(15,-56,4,-15) -> Matrix(-1,0,4,1) (7/2,4/1) -> (-1/2,0/1) Matrix(13,-56,3,-13) -> Matrix(-1,0,4,1) (4/1,14/3) -> (-1/2,0/1) Matrix(29,-140,6,-29) -> Matrix(1,0,0,-1) (14/3,5/1) -> (0/1,1/0) Matrix(111,-616,20,-111) -> Matrix(3,4,-2,-3) (11/2,28/5) -> (-2/1,-1/1) Matrix(29,-168,5,-29) -> Matrix(3,2,-4,-3) (28/5,6/1) -> (-1/1,-1/2) Matrix(13,-84,2,-13) -> Matrix(-1,0,4,1) (6/1,7/1) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.