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 -11/28 -5/14 -13/42 -3/14 -5/28 0/1 1/7 1/6 1/5 1/4 2/7 3/10 1/3 3/8 2/5 5/12 3/7 1/2 4/7 2/3 5/7 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/11 -6/7 1/10 -5/6 1/10 -9/11 2/19 3/28 -4/5 2/19 1/9 -3/4 1/9 1/8 -5/7 1/8 -7/10 1/8 -9/13 2/15 1/7 -2/3 0/1 1/8 -5/8 1/8 1/7 -13/21 1/8 -8/13 2/15 1/7 -11/18 1/8 -3/5 2/15 1/7 -10/17 2/15 3/22 -17/29 4/29 1/7 -7/12 3/22 1/7 -4/7 1/7 -1/2 1/6 -3/7 1/5 -5/12 1/5 3/14 -7/17 3/14 2/9 -2/5 1/5 2/9 -11/28 2/9 -9/23 2/9 3/13 -7/18 1/4 -5/13 1/5 2/9 -8/21 1/4 -3/8 1/5 1/4 -4/11 2/9 1/4 -5/14 1/4 -1/3 0/1 1/4 -5/16 1/5 1/4 -9/29 0/1 1/5 -13/42 1/5 -4/13 1/5 2/9 -3/10 1/4 -2/7 1/4 -1/4 1/4 1/3 -2/9 4/13 1/3 -3/14 1/3 -1/5 1/3 2/5 -2/11 3/8 2/5 -5/28 2/5 -3/17 2/5 5/12 -1/6 1/2 -1/7 1/2 0/1 0/1 1/1 1/7 1/0 1/6 1/0 2/11 -2/1 -3/2 1/5 -2/1 -1/1 1/4 -1/1 -1/2 2/7 -1/2 3/10 -1/2 4/13 -2/5 -1/3 1/3 -1/2 0/1 3/8 -1/2 -1/3 8/21 -1/2 5/13 -2/5 -1/3 7/18 -1/2 2/5 -2/5 -1/3 7/17 -2/5 -3/8 12/29 -4/11 -1/3 5/12 -3/8 -1/3 3/7 -1/3 1/2 -1/4 4/7 -1/5 7/12 -1/5 -3/16 10/17 -3/16 -2/11 3/5 -1/5 -2/11 17/28 -2/11 14/23 -2/11 -3/17 11/18 -1/6 8/13 -1/5 -2/11 13/21 -1/6 5/8 -1/5 -1/6 7/11 -2/11 -1/6 9/14 -1/6 2/3 -1/6 0/1 11/16 -1/5 -1/6 20/29 -1/5 0/1 29/42 -1/5 9/13 -1/5 -2/11 7/10 -1/6 5/7 -1/6 3/4 -1/6 -1/7 7/9 -4/27 -1/7 11/14 -1/7 4/5 -1/7 -2/15 9/11 -3/22 -2/15 23/28 -2/15 14/17 -2/15 -5/38 5/6 -1/8 6/7 -1/8 1/1 -1/9 0/1 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(43,38,112,99) (-1/1,-6/7) -> (8/21,5/13) Hyperbolic Matrix(71,60,84,71) (-6/7,-5/6) -> (5/6,6/7) Hyperbolic Matrix(29,24,-168,-139) (-5/6,-9/11) -> (-3/17,-1/6) Hyperbolic Matrix(57,46,140,113) (-9/11,-4/5) -> (2/5,7/17) Hyperbolic Matrix(13,10,-56,-43) (-4/5,-3/4) -> (-1/4,-2/9) Hyperbolic Matrix(41,30,56,41) (-3/4,-5/7) -> (5/7,3/4) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(141,98,364,253) (-7/10,-9/13) -> (5/13,7/18) Hyperbolic Matrix(181,124,-308,-211) (-9/13,-2/3) -> (-10/17,-17/29) Hyperbolic Matrix(41,26,-112,-71) (-2/3,-5/8) -> (-3/8,-4/11) Hyperbolic Matrix(209,130,336,209) (-5/8,-13/21) -> (13/21,5/8) Hyperbolic Matrix(13,8,112,69) (-13/21,-8/13) -> (0/1,1/7) Hyperbolic Matrix(111,68,364,223) (-8/13,-11/18) -> (3/10,4/13) Hyperbolic Matrix(197,120,-504,-307) (-11/18,-3/5) -> (-9/23,-7/18) Hyperbolic Matrix(27,16,140,83) (-3/5,-10/17) -> (2/11,1/5) Hyperbolic Matrix(253,148,-812,-475) (-17/29,-7/12) -> (-5/16,-9/29) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(13,6,28,13) (-1/2,-3/7) -> (3/7,1/2) Hyperbolic Matrix(71,30,168,71) (-3/7,-5/12) -> (5/12,3/7) Hyperbolic Matrix(97,40,-308,-127) (-5/12,-7/17) -> (-1/3,-5/16) Hyperbolic Matrix(113,46,140,57) (-7/17,-2/5) -> (4/5,9/11) Hyperbolic Matrix(477,188,784,309) (-2/5,-11/28) -> (17/28,14/23) Hyperbolic Matrix(475,186,784,307) (-11/28,-9/23) -> (3/5,17/28) Hyperbolic Matrix(253,98,364,141) (-7/18,-5/13) -> (9/13,7/10) Hyperbolic Matrix(99,38,112,43) (-5/13,-8/21) -> (6/7,1/1) Hyperbolic Matrix(127,48,336,127) (-8/21,-3/8) -> (3/8,8/21) Hyperbolic Matrix(127,46,196,71) (-4/11,-5/14) -> (9/14,2/3) Hyperbolic Matrix(125,44,196,69) (-5/14,-1/3) -> (7/11,9/14) Hyperbolic Matrix(1219,378,1764,547) (-9/29,-13/42) -> (29/42,9/13) Hyperbolic Matrix(1217,376,1764,545) (-13/42,-4/13) -> (20/29,29/42) Hyperbolic Matrix(223,68,364,111) (-4/13,-3/10) -> (11/18,8/13) Hyperbolic Matrix(41,12,140,41) (-3/10,-2/7) -> (2/7,3/10) Hyperbolic Matrix(15,4,56,15) (-2/7,-1/4) -> (1/4,2/7) Hyperbolic Matrix(155,34,196,43) (-2/9,-3/14) -> (11/14,4/5) Hyperbolic Matrix(153,32,196,41) (-3/14,-1/5) -> (7/9,11/14) Hyperbolic Matrix(83,16,140,27) (-1/5,-2/11) -> (10/17,3/5) Hyperbolic Matrix(645,116,784,141) (-2/11,-5/28) -> (23/28,14/17) Hyperbolic Matrix(643,114,784,139) (-5/28,-3/17) -> (9/11,23/28) Hyperbolic Matrix(13,2,84,13) (-1/6,-1/7) -> (1/7,1/6) Hyperbolic Matrix(69,8,112,13) (-1/7,0/1) -> (8/13,13/21) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(43,-10,56,-13) (1/5,1/4) -> (3/4,7/9) Hyperbolic Matrix(127,-40,308,-97) (4/13,1/3) -> (7/17,12/29) Hyperbolic Matrix(71,-26,112,-41) (1/3,3/8) -> (5/8,7/11) Hyperbolic Matrix(307,-120,504,-197) (7/18,2/5) -> (14/23,11/18) Hyperbolic Matrix(559,-232,812,-337) (12/29,5/12) -> (11/16,20/29) Hyperbolic Matrix(211,-124,308,-181) (7/12,10/17) -> (2/3,11/16) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-20,1) Matrix(43,38,112,99) -> Matrix(21,-2,-52,5) Matrix(71,60,84,71) -> Matrix(19,-2,-142,15) Matrix(29,24,-168,-139) -> Matrix(39,-4,88,-9) Matrix(57,46,140,113) -> Matrix(1,0,-12,1) Matrix(13,10,-56,-43) -> Matrix(17,-2,60,-7) Matrix(41,30,56,41) -> Matrix(17,-2,-110,13) Matrix(99,70,140,99) -> Matrix(15,-2,-82,11) Matrix(141,98,364,253) -> Matrix(1,0,-10,1) Matrix(181,124,-308,-211) -> Matrix(13,-2,98,-15) Matrix(41,26,-112,-71) -> Matrix(15,-2,68,-9) Matrix(209,130,336,209) -> Matrix(15,-2,-82,11) Matrix(13,8,112,69) -> Matrix(15,-2,8,-1) Matrix(111,68,364,223) -> Matrix(1,0,-10,1) Matrix(197,120,-504,-307) -> Matrix(31,-4,132,-17) Matrix(27,16,140,83) -> Matrix(1,0,-8,1) Matrix(253,148,-812,-475) -> Matrix(29,-4,138,-19) Matrix(97,56,168,97) -> Matrix(43,-6,-222,31) Matrix(15,8,28,15) -> Matrix(13,-2,-58,9) Matrix(13,6,28,13) -> Matrix(11,-2,-38,7) Matrix(71,30,168,71) -> Matrix(29,-6,-82,17) Matrix(97,40,-308,-127) -> Matrix(9,-2,50,-11) Matrix(113,46,140,57) -> Matrix(1,0,-12,1) Matrix(477,188,784,309) -> Matrix(37,-8,-208,45) Matrix(475,186,784,307) -> Matrix(35,-8,-188,43) Matrix(253,98,364,141) -> Matrix(1,0,-10,1) Matrix(99,38,112,43) -> Matrix(9,-2,-76,17) Matrix(127,48,336,127) -> Matrix(9,-2,-22,5) Matrix(127,46,196,71) -> Matrix(9,-2,-58,13) Matrix(125,44,196,69) -> Matrix(7,-2,-38,11) Matrix(1219,378,1764,547) -> Matrix(11,-2,-60,11) Matrix(1217,376,1764,545) -> Matrix(9,-2,-40,9) Matrix(223,68,364,111) -> Matrix(1,0,-10,1) Matrix(41,12,140,41) -> Matrix(9,-2,-22,5) Matrix(15,4,56,15) -> Matrix(7,-2,-10,3) Matrix(155,34,196,43) -> Matrix(19,-6,-136,43) Matrix(153,32,196,41) -> Matrix(17,-6,-116,41) Matrix(83,16,140,27) -> Matrix(1,0,-8,1) Matrix(645,116,784,141) -> Matrix(41,-16,-310,121) Matrix(643,114,784,139) -> Matrix(39,-16,-290,119) Matrix(13,2,84,13) -> Matrix(5,-2,-2,1) Matrix(69,8,112,13) -> Matrix(3,-2,-16,11) Matrix(139,-24,168,-29) -> Matrix(1,4,-8,-31) Matrix(43,-10,56,-13) -> Matrix(3,2,-20,-13) Matrix(127,-40,308,-97) -> Matrix(7,2,-18,-5) Matrix(71,-26,112,-41) -> Matrix(5,2,-28,-11) Matrix(307,-120,504,-197) -> Matrix(9,4,-52,-23) Matrix(559,-232,812,-337) -> Matrix(11,4,-58,-21) Matrix(211,-124,308,-181) -> Matrix(11,2,-50,-9) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 48 Permutation triple for Y: ((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); (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)) ----------------------------------------------------------------------- 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 1/7 1/6 1/5 1/4 2/7 3/10 1/3 3/8 5/12 3/7 1/2 17/28 9/14 29/42 11/14 23/28 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/7 1/0 1/6 1/0 2/11 -2/1 -3/2 1/5 -2/1 -1/1 1/4 -1/1 -1/2 2/7 -1/2 3/10 -1/2 4/13 -2/5 -1/3 1/3 -1/2 0/1 3/8 -1/2 -1/3 8/21 -1/2 5/13 -2/5 -1/3 7/18 -1/2 2/5 -2/5 -1/3 7/17 -2/5 -3/8 12/29 -4/11 -1/3 5/12 -3/8 -1/3 3/7 -1/3 1/2 -1/4 4/7 -1/5 7/12 -1/5 -3/16 10/17 -3/16 -2/11 3/5 -1/5 -2/11 17/28 -2/11 14/23 -2/11 -3/17 11/18 -1/6 8/13 -1/5 -2/11 13/21 -1/6 5/8 -1/5 -1/6 7/11 -2/11 -1/6 9/14 -1/6 2/3 -1/6 0/1 11/16 -1/5 -1/6 20/29 -1/5 0/1 29/42 -1/5 9/13 -1/5 -2/11 7/10 -1/6 5/7 -1/6 3/4 -1/6 -1/7 7/9 -4/27 -1/7 11/14 -1/7 4/5 -1/7 -2/15 9/11 -3/22 -2/15 23/28 -2/15 14/17 -2/15 -5/38 5/6 -1/8 6/7 -1/8 1/1 -1/9 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(43,-5,112,-13) (0/1,1/7) -> (8/21,5/13) Hyperbolic Matrix(71,-11,84,-13) (1/7,1/6) -> (5/6,6/7) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(57,-11,140,-27) (2/11,1/5) -> (2/5,7/17) Hyperbolic Matrix(43,-10,56,-13) (1/5,1/4) -> (3/4,7/9) Hyperbolic Matrix(41,-11,56,-15) (1/4,2/7) -> (5/7,3/4) Hyperbolic Matrix(99,-29,140,-41) (2/7,3/10) -> (7/10,5/7) Hyperbolic Matrix(141,-43,364,-111) (3/10,4/13) -> (5/13,7/18) Hyperbolic Matrix(127,-40,308,-97) (4/13,1/3) -> (7/17,12/29) Hyperbolic Matrix(71,-26,112,-41) (1/3,3/8) -> (5/8,7/11) Hyperbolic Matrix(209,-79,336,-127) (3/8,8/21) -> (13/21,5/8) Hyperbolic Matrix(307,-120,504,-197) (7/18,2/5) -> (14/23,11/18) Hyperbolic Matrix(559,-232,812,-337) (12/29,5/12) -> (11/16,20/29) Hyperbolic Matrix(97,-41,168,-71) (5/12,3/7) -> (4/7,7/12) Hyperbolic Matrix(15,-7,28,-13) (3/7,1/2) -> (1/2,4/7) Parabolic Matrix(211,-124,308,-181) (7/12,10/17) -> (2/3,11/16) Hyperbolic Matrix(113,-67,140,-83) (10/17,3/5) -> (4/5,9/11) Hyperbolic Matrix(477,-289,784,-475) (3/5,17/28) -> (17/28,14/23) Parabolic Matrix(253,-155,364,-223) (11/18,8/13) -> (9/13,7/10) Hyperbolic Matrix(99,-61,112,-69) (8/13,13/21) -> (6/7,1/1) Hyperbolic Matrix(127,-81,196,-125) (7/11,9/14) -> (9/14,2/3) Parabolic Matrix(1219,-841,1764,-1217) (20/29,29/42) -> (29/42,9/13) Parabolic Matrix(155,-121,196,-153) (7/9,11/14) -> (11/14,4/5) Parabolic Matrix(645,-529,784,-643) (9/11,23/28) -> (23/28,14/17) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-10,1) Matrix(43,-5,112,-13) -> Matrix(1,-2,-2,5) Matrix(71,-11,84,-13) -> Matrix(1,2,-8,-15) Matrix(139,-24,168,-29) -> Matrix(1,4,-8,-31) Matrix(57,-11,140,-27) -> Matrix(1,0,-2,1) Matrix(43,-10,56,-13) -> Matrix(3,2,-20,-13) Matrix(41,-11,56,-15) -> Matrix(3,2,-20,-13) Matrix(99,-29,140,-41) -> Matrix(5,2,-28,-11) Matrix(141,-43,364,-111) -> Matrix(1,0,0,1) Matrix(127,-40,308,-97) -> Matrix(7,2,-18,-5) Matrix(71,-26,112,-41) -> Matrix(5,2,-28,-11) Matrix(209,-79,336,-127) -> Matrix(5,2,-28,-11) Matrix(307,-120,504,-197) -> Matrix(9,4,-52,-23) Matrix(559,-232,812,-337) -> Matrix(11,4,-58,-21) Matrix(97,-41,168,-71) -> Matrix(17,6,-88,-31) Matrix(15,-7,28,-13) -> Matrix(7,2,-32,-9) Matrix(211,-124,308,-181) -> Matrix(11,2,-50,-9) Matrix(113,-67,140,-83) -> Matrix(1,0,-2,1) Matrix(477,-289,784,-475) -> Matrix(43,8,-242,-45) Matrix(253,-155,364,-223) -> Matrix(1,0,0,1) Matrix(99,-61,112,-69) -> Matrix(11,2,-94,-17) Matrix(127,-81,196,-125) -> Matrix(11,2,-72,-13) Matrix(1219,-841,1764,-1217) -> Matrix(9,2,-50,-11) Matrix(155,-121,196,-153) -> Matrix(41,6,-294,-43) Matrix(645,-529,784,-643) -> Matrix(119,16,-900,-121) 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): 8 ----------------------------------------------------------------------- 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 (0/1,1/1) 0 14 1/7 1/0 2 2 1/6 1/0 2 7 5/28 -2/1 4 1 2/11 (-2/1,-3/2) 0 14 1/5 (-2/1,-1/1) 0 14 3/14 -1/1 6 1 1/4 (-1/1,-1/2) 0 7 2/7 -1/2 2 2 3/10 -1/2 2 7 4/13 (-2/5,-1/3) 0 14 1/3 (-1/2,0/1) 0 14 5/14 -1/2 2 1 3/8 (-1/2,-1/3) 0 7 8/21 -1/2 2 2 5/13 (-2/5,-1/3) 0 14 7/18 -1/2 2 7 11/28 -2/5 2 1 2/5 (-2/5,-1/3) 0 14 7/17 (-2/5,-3/8) 0 14 12/29 (-4/11,-1/3) 0 14 29/70 -1/3 2 1 5/12 (-3/8,-1/3) 0 7 3/7 -1/3 4 2 1/2 -1/4 2 7 1/0 0/1 10 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(43,-5,112,-13) (0/1,1/7) -> (8/21,5/13) Hyperbolic Matrix(13,-2,84,-13) (1/7,1/6) -> (1/7,1/6) Reflection Matrix(29,-5,168,-29) (1/6,5/28) -> (1/6,5/28) Reflection Matrix(111,-20,616,-111) (5/28,2/11) -> (5/28,2/11) Reflection Matrix(57,-11,140,-27) (2/11,1/5) -> (2/5,7/17) Hyperbolic Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(13,-3,56,-13) (3/14,1/4) -> (3/14,1/4) Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(41,-12,140,-41) (2/7,3/10) -> (2/7,3/10) Reflection Matrix(141,-43,364,-111) (3/10,4/13) -> (5/13,7/18) Hyperbolic Matrix(127,-40,308,-97) (4/13,1/3) -> (7/17,12/29) Hyperbolic Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(41,-15,112,-41) (5/14,3/8) -> (5/14,3/8) Reflection Matrix(127,-48,336,-127) (3/8,8/21) -> (3/8,8/21) Reflection Matrix(197,-77,504,-197) (7/18,11/28) -> (7/18,11/28) Reflection Matrix(111,-44,280,-111) (11/28,2/5) -> (11/28,2/5) Reflection Matrix(1681,-696,4060,-1681) (12/29,29/70) -> (12/29,29/70) Reflection Matrix(349,-145,840,-349) (29/70,5/12) -> (29/70,5/12) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(13,-6,28,-13) (3/7,1/2) -> (3/7,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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) -> (0/1,1/1) Matrix(43,-5,112,-13) -> Matrix(1,-2,-2,5) Matrix(13,-2,84,-13) -> Matrix(1,2,0,-1) (1/7,1/6) -> (-1/1,1/0) Matrix(29,-5,168,-29) -> Matrix(1,4,0,-1) (1/6,5/28) -> (-2/1,1/0) Matrix(111,-20,616,-111) -> Matrix(7,12,-4,-7) (5/28,2/11) -> (-2/1,-3/2) Matrix(57,-11,140,-27) -> Matrix(1,0,-2,1) 0/1 Matrix(29,-6,140,-29) -> Matrix(3,4,-2,-3) (1/5,3/14) -> (-2/1,-1/1) Matrix(13,-3,56,-13) -> Matrix(3,2,-4,-3) (3/14,1/4) -> (-1/1,-1/2) Matrix(15,-4,56,-15) -> Matrix(3,2,-4,-3) (1/4,2/7) -> (-1/1,-1/2) Matrix(41,-12,140,-41) -> Matrix(5,2,-12,-5) (2/7,3/10) -> (-1/2,-1/3) Matrix(141,-43,364,-111) -> Matrix(1,0,0,1) Matrix(127,-40,308,-97) -> Matrix(7,2,-18,-5) -1/3 Matrix(29,-10,84,-29) -> Matrix(-1,0,4,1) (1/3,5/14) -> (-1/2,0/1) Matrix(41,-15,112,-41) -> Matrix(5,2,-12,-5) (5/14,3/8) -> (-1/2,-1/3) Matrix(127,-48,336,-127) -> Matrix(5,2,-12,-5) (3/8,8/21) -> (-1/2,-1/3) Matrix(197,-77,504,-197) -> Matrix(9,4,-20,-9) (7/18,11/28) -> (-1/2,-2/5) Matrix(111,-44,280,-111) -> Matrix(11,4,-30,-11) (11/28,2/5) -> (-2/5,-1/3) Matrix(1681,-696,4060,-1681) -> Matrix(23,8,-66,-23) (12/29,29/70) -> (-4/11,-1/3) Matrix(349,-145,840,-349) -> Matrix(17,6,-48,-17) (29/70,5/12) -> (-3/8,-1/3) Matrix(71,-30,168,-71) -> Matrix(17,6,-48,-17) (5/12,3/7) -> (-3/8,-1/3) Matrix(13,-6,28,-13) -> Matrix(7,2,-24,-7) (3/7,1/2) -> (-1/3,-1/4) Matrix(-1,1,0,1) -> Matrix(-1,0,8,1) (1/2,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.