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 -6/1 -9/2 -4/1 -10/3 -3/1 -8/3 -16/7 -2/1 -9/5 -3/2 -4/3 -6/5 0/1 1/1 6/5 4/3 3/2 36/23 12/7 9/5 2/1 24/11 12/5 5/2 8/3 3/1 36/11 10/3 24/7 7/2 11/3 4/1 9/2 24/5 5/1 11/2 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 1/1 1/0 -6/1 1/0 -5/1 -1/1 1/0 -14/3 -3/4 -23/5 -2/3 -1/2 -9/2 -1/2 -4/1 -1/2 1/0 -15/4 -1/2 -11/3 -1/2 0/1 -18/5 -1/2 -7/2 -1/1 -1/2 -10/3 -1/2 -3/1 -1/2 -14/5 -1/2 -25/9 -5/12 -2/5 -36/13 -2/5 -11/4 -2/5 -3/8 -8/3 -1/3 -13/5 -1/2 0/1 -5/2 -1/2 -1/3 -12/5 -1/3 -7/3 -1/3 -3/10 -23/10 -3/10 -2/7 -16/7 -2/7 -25/11 -2/7 -1/4 -9/4 -1/4 -11/5 -2/7 -1/4 -2/1 -1/4 -13/7 -1/4 0/1 -24/13 -1/4 -11/6 -1/4 -2/9 -9/5 -1/4 -25/14 -1/4 -2/9 -16/9 -2/9 -7/4 -3/14 -1/5 -12/7 -1/5 -5/3 -1/5 -1/6 -18/11 -1/6 -13/8 -1/6 0/1 -8/5 -1/5 -11/7 -3/16 -2/11 -3/2 -1/6 -13/9 -1/8 0/1 -36/25 0/1 -23/16 -1/4 0/1 -10/7 -1/6 -17/12 -1/5 -1/6 -24/17 -1/6 -7/5 -1/6 -1/7 -18/13 -1/6 -11/8 -1/6 0/1 -4/3 -1/6 -1/8 -13/10 -1/6 0/1 -9/7 -1/6 -23/18 -1/6 -2/13 -14/11 -3/20 -19/15 -7/48 -1/7 -24/19 -1/7 -5/4 -1/7 -1/8 -11/9 -2/15 -1/8 -6/5 -1/8 -7/6 -1/8 -1/9 -8/7 0/1 -1/1 -1/8 0/1 0/1 0/1 1/1 0/1 1/8 7/6 1/9 1/8 6/5 1/8 5/4 1/8 1/7 14/11 3/20 23/18 2/13 1/6 9/7 1/6 4/3 1/8 1/6 15/11 1/6 11/8 0/1 1/6 18/13 1/6 7/5 1/7 1/6 10/7 1/6 3/2 1/6 14/9 1/6 25/16 5/28 2/11 36/23 2/11 11/7 2/11 3/16 8/5 1/5 13/8 0/1 1/6 5/3 1/6 1/5 12/7 1/5 7/4 1/5 3/14 23/13 3/14 2/9 16/9 2/9 25/14 2/9 1/4 9/5 1/4 11/6 2/9 1/4 2/1 1/4 13/6 0/1 1/4 24/11 1/4 11/5 1/4 2/7 9/4 1/4 25/11 1/4 2/7 16/7 2/7 7/3 3/10 1/3 12/5 1/3 5/2 1/3 1/2 18/7 1/2 13/5 0/1 1/2 8/3 1/3 11/4 3/8 2/5 3/1 1/2 13/4 0/1 1/0 36/11 0/1 23/7 0/1 1/4 10/3 1/2 17/5 1/3 1/2 24/7 1/2 7/2 1/2 1/1 18/5 1/2 11/3 0/1 1/2 4/1 1/2 1/0 13/3 0/1 1/2 9/2 1/2 23/5 1/2 2/3 14/3 3/4 19/4 7/8 1/1 24/5 1/1 5/1 1/1 1/0 11/2 2/1 1/0 6/1 1/0 7/1 -1/1 1/0 8/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(23,168,-10,-73) (-7/1,1/0) -> (-7/3,-23/10) Hyperbolic Matrix(25,168,18,121) (-7/1,-6/1) -> (18/13,7/5) Hyperbolic Matrix(23,120,-14,-73) (-6/1,-5/1) -> (-5/3,-18/11) Hyperbolic Matrix(71,336,-56,-265) (-5/1,-14/3) -> (-14/11,-19/15) Hyperbolic Matrix(119,552,36,167) (-14/3,-23/5) -> (23/7,10/3) Hyperbolic Matrix(95,432,42,191) (-23/5,-9/2) -> (9/4,25/11) Hyperbolic Matrix(23,96,-6,-25) (-9/2,-4/1) -> (-4/1,-15/4) Parabolic Matrix(71,264,32,119) (-15/4,-11/3) -> (11/5,9/4) Hyperbolic Matrix(119,432,46,167) (-11/3,-18/5) -> (18/7,13/5) Hyperbolic Matrix(47,168,40,143) (-18/5,-7/2) -> (7/6,6/5) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(241,672,52,145) (-14/5,-25/9) -> (23/5,14/3) Hyperbolic Matrix(623,1728,190,527) (-25/9,-36/13) -> (36/11,23/7) Hyperbolic Matrix(313,864,96,265) (-36/13,-11/4) -> (13/4,36/11) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(47,120,-38,-97) (-13/5,-5/2) -> (-5/4,-11/9) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(335,768,188,431) (-23/10,-16/7) -> (16/9,25/14) Hyperbolic Matrix(337,768,190,433) (-16/7,-25/11) -> (23/13,16/9) Hyperbolic Matrix(191,432,42,95) (-25/11,-9/4) -> (9/2,23/5) Hyperbolic Matrix(97,216,22,49) (-9/4,-11/5) -> (13/3,9/2) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(311,576,142,263) (-13/7,-24/13) -> (24/11,11/5) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(145,264,106,193) (-11/6,-9/5) -> (15/11,11/8) Hyperbolic Matrix(241,432,188,337) (-9/5,-25/14) -> (23/18,9/7) Hyperbolic Matrix(121,216,14,25) (-25/14,-16/9) -> (8/1,1/0) Hyperbolic Matrix(95,168,-82,-145) (-16/9,-7/4) -> (-7/6,-8/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(265,432,192,313) (-18/11,-13/8) -> (11/8,18/13) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(599,864,382,551) (-13/9,-36/25) -> (36/23,11/7) Hyperbolic Matrix(1201,1728,768,1105) (-36/25,-23/16) -> (25/16,36/23) Hyperbolic Matrix(385,552,302,433) (-23/16,-10/7) -> (14/11,23/18) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(121,168,18,25) (-7/5,-18/13) -> (6/1,7/1) Hyperbolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(71,96,-54,-73) (-11/8,-4/3) -> (-4/3,-13/10) Parabolic Matrix(167,216,92,119) (-13/10,-9/7) -> (9/5,11/6) Hyperbolic Matrix(337,432,188,241) (-9/7,-23/18) -> (25/14,9/5) Hyperbolic Matrix(527,672,338,431) (-23/18,-14/11) -> (14/9,25/16) Hyperbolic Matrix(455,576,94,119) (-19/15,-24/19) -> (24/5,5/1) Hyperbolic Matrix(457,576,96,121) (-24/19,-5/4) -> (19/4,24/5) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(191,216,84,95) (-8/7,-1/1) -> (25/11,16/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(145,-168,82,-95) (1/1,7/6) -> (7/4,23/13) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(73,-96,54,-71) (9/7,4/3) -> (4/3,15/11) Parabolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(73,-168,10,-23) (16/7,7/3) -> (7/1,8/1) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(23,168,-10,-73) -> Matrix(3,-2,-10,7) Matrix(25,168,18,121) -> Matrix(1,0,6,1) Matrix(23,120,-14,-73) -> Matrix(1,2,-6,-11) Matrix(71,336,-56,-265) -> Matrix(7,6,-48,-41) Matrix(119,552,36,167) -> Matrix(3,2,10,7) Matrix(95,432,42,191) -> Matrix(7,4,26,15) Matrix(23,96,-6,-25) -> Matrix(1,0,0,1) Matrix(71,264,32,119) -> Matrix(3,2,10,7) Matrix(119,432,46,167) -> Matrix(1,0,4,1) Matrix(47,168,40,143) -> Matrix(1,0,10,1) Matrix(71,240,-50,-169) -> Matrix(1,0,-4,1) Matrix(23,72,-8,-25) -> Matrix(3,2,-8,-5) Matrix(241,672,52,145) -> Matrix(19,8,26,11) Matrix(623,1728,190,527) -> Matrix(5,2,32,13) Matrix(313,864,96,265) -> Matrix(5,2,-8,-3) Matrix(71,192,44,119) -> Matrix(5,2,22,9) Matrix(73,192,46,121) -> Matrix(7,2,38,11) Matrix(47,120,-38,-97) -> Matrix(5,2,-38,-15) Matrix(49,120,20,49) -> Matrix(5,2,12,5) Matrix(71,168,30,71) -> Matrix(19,6,60,19) Matrix(335,768,188,431) -> Matrix(27,8,118,35) Matrix(337,768,190,433) -> Matrix(29,8,134,37) Matrix(191,432,42,95) -> Matrix(15,4,26,7) Matrix(97,216,22,49) -> Matrix(7,2,10,3) Matrix(23,48,-12,-25) -> Matrix(7,2,-32,-9) Matrix(311,576,142,263) -> Matrix(7,2,24,7) Matrix(313,576,144,265) -> Matrix(9,2,40,9) Matrix(145,264,106,193) -> Matrix(9,2,58,13) Matrix(241,432,188,337) -> Matrix(17,4,106,25) Matrix(121,216,14,25) -> Matrix(9,2,4,1) Matrix(95,168,-82,-145) -> Matrix(9,2,-86,-19) Matrix(97,168,56,97) -> Matrix(29,6,140,29) Matrix(71,120,42,71) -> Matrix(11,2,60,11) Matrix(265,432,192,313) -> Matrix(1,0,12,1) Matrix(119,192,44,71) -> Matrix(9,2,22,5) Matrix(121,192,46,73) -> Matrix(11,2,38,7) Matrix(47,72,-32,-49) -> Matrix(11,2,-72,-13) Matrix(599,864,382,551) -> Matrix(19,2,104,11) Matrix(1201,1728,768,1105) -> Matrix(3,2,16,11) Matrix(385,552,302,433) -> Matrix(9,2,58,13) Matrix(407,576,118,167) -> Matrix(11,2,16,3) Matrix(409,576,120,169) -> Matrix(13,2,32,5) Matrix(121,168,18,25) -> Matrix(1,0,6,1) Matrix(191,264,34,47) -> Matrix(13,2,6,1) Matrix(71,96,-54,-73) -> Matrix(1,0,0,1) Matrix(167,216,92,119) -> Matrix(13,2,58,9) Matrix(337,432,188,241) -> Matrix(25,4,106,17) Matrix(527,672,338,431) -> Matrix(53,8,298,45) Matrix(455,576,94,119) -> Matrix(55,8,48,7) Matrix(457,576,96,121) -> Matrix(57,8,64,9) Matrix(217,264,60,73) -> Matrix(15,2,22,3) Matrix(143,168,40,47) -> Matrix(1,0,10,1) Matrix(191,216,84,95) -> Matrix(15,2,52,7) Matrix(1,0,2,1) -> Matrix(1,0,16,1) Matrix(145,-168,82,-95) -> Matrix(19,-2,86,-9) Matrix(97,-120,38,-47) -> Matrix(15,-2,38,-5) Matrix(265,-336,56,-71) -> Matrix(41,-6,48,-7) Matrix(73,-96,54,-71) -> Matrix(1,0,0,1) Matrix(169,-240,50,-71) -> Matrix(1,0,-4,1) Matrix(49,-72,32,-47) -> Matrix(13,-2,72,-11) Matrix(73,-120,14,-23) -> Matrix(11,-2,6,-1) Matrix(25,-48,12,-23) -> Matrix(9,-2,32,-7) Matrix(73,-168,10,-23) -> Matrix(7,-2,-10,3) Matrix(25,-72,8,-23) -> Matrix(5,-2,8,-3) Matrix(25,-96,6,-23) -> 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): 20 Degree of the the map X: 20 Degree of the the map Y: 64 Permutation triple for Y: ((2,6,24,56,25,7)(3,12,42,43,13,4)(5,18,19)(8,30,31)(9,36,10)(11,22,21)(14,28,27)(15,47,16)(17,37)(20,29,59,35,48,54)(32,46)(33,34)(38,45,52,51,60,39)(40,49)(41,55,50)(44,58,57); (1,4,16,49,57,59,62,60,50,17,5,2)(3,10,39,11)(6,22,32,31,51,61,35,9,34,14,13,23)(7,28,29,8)(12,26,25,58,46,15,38,53,20,19,33,41)(18,52,44,43)(21,54,64,45,27,37,36,56,63,42,30,40)(24,55,48,47); (1,2,8,32,58,52,64,54,55,33,9,3)(4,14,45,15)(5,20,21,6)(7,26,12,11,40,16,48,61,51,18,17,27)(10,37,50,24,23,13,44,49,30,29,53,38)(19,43,63,56,47,46,22,39,62,59,28,34)(25,36,35,57)(31,42,41,60)) ----------------------------------------------------------------------- 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): 96 Minimal number of generators: 17 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: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 3/2 8/5 2/1 12/5 8/3 3/1 24/7 4/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/8 4/3 1/8 1/6 7/5 1/7 1/6 10/7 1/6 3/2 1/6 11/7 2/11 3/16 8/5 1/5 13/8 0/1 1/6 5/3 1/6 1/5 12/7 1/5 7/4 1/5 3/14 9/5 1/4 2/1 1/4 9/4 1/4 7/3 3/10 1/3 12/5 1/3 5/2 1/3 1/2 13/5 0/1 1/2 8/3 1/3 11/4 3/8 2/5 3/1 1/2 10/3 1/2 17/5 1/3 1/2 24/7 1/2 7/2 1/2 1/1 18/5 1/2 11/3 0/1 1/2 4/1 1/2 1/0 5/1 1/1 1/0 11/2 2/1 1/0 6/1 1/0 7/1 -1/1 1/0 1/0 0/1 1/0 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(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(67,-96,37,-53) (10/7,3/2) -> (9/5,2/1) Hyperbolic Matrix(77,-120,43,-67) (3/2,11/7) -> (7/4,9/5) Hyperbolic Matrix(121,-192,75,-119) (11/7,8/5) -> (8/5,13/8) Parabolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(71,-120,29,-49) (5/3,12/7) -> (12/5,5/2) Hyperbolic Matrix(97,-168,41,-71) (12/7,7/4) -> (7/3,12/5) Hyperbolic Matrix(43,-96,13,-29) (2/1,9/4) -> (3/1,10/3) Hyperbolic Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(73,-192,27,-71) (13/5,8/3) -> (8/3,11/4) Parabolic Matrix(169,-576,49,-167) (17/5,24/7) -> (24/7,7/2) Parabolic Matrix(47,-168,7,-25) (7/2,18/5) -> (6/1,7/1) Hyperbolic Matrix(73,-264,13,-47) (18/5,11/3) -> (11/2,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,8,1) Matrix(19,-24,4,-5) -> Matrix(7,-1,8,-1) Matrix(53,-72,14,-19) -> Matrix(7,-1,8,-1) Matrix(169,-240,50,-71) -> Matrix(1,0,-4,1) Matrix(67,-96,37,-53) -> Matrix(5,-1,26,-5) Matrix(77,-120,43,-67) -> Matrix(17,-3,74,-13) Matrix(121,-192,75,-119) -> Matrix(11,-2,50,-9) Matrix(73,-120,14,-23) -> Matrix(11,-2,6,-1) Matrix(71,-120,29,-49) -> Matrix(11,-2,28,-5) Matrix(97,-168,41,-71) -> Matrix(29,-6,92,-19) Matrix(43,-96,13,-29) -> Matrix(3,-1,10,-3) Matrix(53,-120,19,-43) -> Matrix(11,-3,26,-7) Matrix(19,-48,2,-5) -> Matrix(3,-1,-2,1) Matrix(73,-192,27,-71) -> Matrix(7,-2,18,-5) Matrix(169,-576,49,-167) -> Matrix(5,-2,8,-3) Matrix(47,-168,7,-25) -> Matrix(1,0,-2,1) Matrix(73,-264,13,-47) -> Matrix(3,-2,2,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 10 ----------------------------------------------------------------------- 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 8 1 2/1 1/4 2 12 9/4 1/4 2 8 7/3 (3/10,1/3) 0 24 12/5 1/3 8 2 5/2 (1/3,1/2) 0 24 13/5 (0/1,1/2) 0 24 8/3 1/3 2 3 11/4 (3/8,2/5) 0 24 3/1 1/2 2 8 10/3 1/2 2 12 24/7 1/2 2 1 7/2 (1/2,1/1) 0 24 18/5 1/2 2 4 4/1 (1/2,1/0) 0 6 6/1 1/0 2 4 7/1 (-1/1,1/0) 0 24 1/0 (0/1,1/0) 0 24 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(43,-96,13,-29) (2/1,9/4) -> (3/1,10/3) Hyperbolic Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(73,-192,27,-71) (13/5,8/3) -> (8/3,11/4) Parabolic Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(47,-168,7,-25) (7/2,18/5) -> (6/1,7/1) Hyperbolic Matrix(19,-72,5,-19) (18/5,4/1) -> (18/5,4/1) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,8,-1) (0/1,2/1) -> (0/1,1/4) Matrix(43,-96,13,-29) -> Matrix(3,-1,10,-3) (0/1,1/3).(1/4,1/2) Matrix(53,-120,19,-43) -> Matrix(11,-3,26,-7) Matrix(71,-168,30,-71) -> Matrix(19,-6,60,-19) (7/3,12/5) -> (3/10,1/3) Matrix(49,-120,20,-49) -> Matrix(5,-2,12,-5) (12/5,5/2) -> (1/3,1/2) Matrix(19,-48,2,-5) -> Matrix(3,-1,-2,1) Matrix(73,-192,27,-71) -> Matrix(7,-2,18,-5) 1/3 Matrix(71,-240,21,-71) -> Matrix(1,0,4,-1) (10/3,24/7) -> (0/1,1/2) Matrix(97,-336,28,-97) -> Matrix(3,-2,4,-3) (24/7,7/2) -> (1/2,1/1) Matrix(47,-168,7,-25) -> Matrix(1,0,-2,1) 0/1 Matrix(19,-72,5,-19) -> Matrix(-1,1,0,1) (18/5,4/1) -> (1/2,1/0) Matrix(5,-24,1,-5) -> Matrix(-1,1,0,1) (4/1,6/1) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.