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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -4/1 -8/3 -5/2 -2/1 -5/3 -5/4 0/1 1/1 5/4 10/7 3/2 30/19 5/3 2/1 20/9 5/2 8/3 30/11 20/7 3/1 10/3 7/2 11/3 4/1 9/2 5/1 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/2 -6/1 1/1 -5/1 1/0 -14/3 -1/1 -9/2 -1/1 0/1 -4/1 0/1 -11/3 1/0 -7/2 -2/1 -1/1 -10/3 -1/1 -3/1 -1/2 -14/5 -1/1 -11/4 -2/1 -1/1 -30/11 -1/1 -19/7 -7/8 -8/3 -2/3 -5/2 -1/2 -12/5 -2/5 -19/8 -3/7 -2/5 -7/3 -3/8 -9/4 -1/3 0/1 -2/1 -1/3 -11/6 -1/3 -2/7 -20/11 -2/7 -9/5 -1/4 -7/4 -1/3 -2/7 -5/3 -1/4 -13/8 -5/21 -4/17 -21/13 -13/56 -8/5 -2/9 -19/12 -2/9 -1/5 -30/19 -1/5 -11/7 -1/4 -14/9 -3/13 -17/11 -5/22 -20/13 -2/9 -3/2 -2/9 -1/5 -10/7 -1/5 -7/5 -5/26 -11/8 -1/5 -2/11 -4/3 -2/11 -9/7 -1/6 -5/4 -1/6 -11/9 -1/6 -6/5 -3/19 -13/11 -9/58 -20/17 -2/13 -7/6 -2/13 -5/33 -1/1 -1/8 0/1 0/1 1/1 1/10 7/6 5/43 2/17 6/5 3/25 5/4 1/8 14/11 5/39 9/7 1/8 4/3 2/15 11/8 2/15 1/7 7/5 5/36 10/7 1/7 3/2 1/7 2/13 14/9 3/19 11/7 1/6 30/19 1/7 19/12 1/7 2/13 8/5 2/13 5/3 1/6 12/7 4/23 19/11 11/62 7/4 2/11 1/5 9/5 1/6 2/1 1/5 11/5 1/4 20/9 0/1 9/4 0/1 1/5 7/3 3/14 5/2 1/4 13/5 5/18 21/8 2/7 1/3 8/3 2/7 19/7 7/22 30/11 1/3 11/4 1/3 2/5 14/5 1/3 17/6 0/1 1/1 20/7 0/1 3/1 1/4 10/3 1/3 7/2 1/3 2/5 11/3 1/2 4/1 0/1 9/2 0/1 1/3 5/1 1/2 11/2 2/3 1/1 6/1 1/1 13/2 -1/3 0/1 20/3 0/1 7/1 1/4 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,160,8,61) (-7/1,1/0) -> (13/5,21/8) Hyperbolic Matrix(19,120,-16,-101) (-7/1,-6/1) -> (-6/5,-13/11) Hyperbolic Matrix(19,100,-4,-21) (-6/1,-5/1) -> (-5/1,-14/3) Parabolic Matrix(61,280,22,101) (-14/3,-9/2) -> (11/4,14/5) Hyperbolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,22,79) (-11/3,-7/2) -> (7/4,9/5) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(99,280,-64,-181) (-3/1,-14/5) -> (-14/9,-17/11) Hyperbolic Matrix(79,220,14,39) (-14/5,-11/4) -> (11/2,6/1) Hyperbolic Matrix(241,660,88,241) (-11/4,-30/11) -> (30/11,11/4) Hyperbolic Matrix(419,1140,154,419) (-30/11,-19/7) -> (19/7,30/11) Hyperbolic Matrix(141,380,82,221) (-19/7,-8/3) -> (12/7,19/11) Hyperbolic Matrix(39,100,-16,-41) (-8/3,-5/2) -> (-5/2,-12/5) Parabolic Matrix(159,380,100,239) (-12/5,-19/8) -> (19/12,8/5) Hyperbolic Matrix(59,140,8,19) (-19/8,-7/3) -> (7/1,1/0) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(219,400,98,179) (-11/6,-20/11) -> (20/9,9/4) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(79,140,22,39) (-9/5,-7/4) -> (7/2,11/3) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(99,160,86,139) (-13/8,-21/13) -> (1/1,7/6) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(721,1140,456,721) (-19/12,-30/19) -> (30/19,19/12) Hyperbolic Matrix(419,660,266,419) (-30/19,-11/7) -> (11/7,30/19) Hyperbolic Matrix(179,280,140,219) (-11/7,-14/9) -> (14/11,9/7) Hyperbolic Matrix(259,400,90,139) (-17/11,-20/13) -> (20/7,3/1) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(59,80,14,19) (-11/8,-4/3) -> (4/1,9/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(79,100,-64,-81) (-9/7,-5/4) -> (-5/4,-11/9) Parabolic Matrix(181,220,116,141) (-11/9,-6/5) -> (14/9,11/7) Hyperbolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(121,140,70,81) (-7/6,-1/1) -> (19/11,7/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,160,8,61) -> Matrix(1,-2,4,-7) Matrix(19,120,-16,-101) -> Matrix(5,-2,-32,13) Matrix(19,100,-4,-21) -> Matrix(1,-2,0,1) Matrix(61,280,22,101) -> Matrix(1,2,2,5) Matrix(19,80,14,59) -> Matrix(3,2,22,15) Matrix(21,80,16,61) -> Matrix(1,-2,8,-15) Matrix(39,140,22,79) -> Matrix(1,0,6,1) Matrix(41,140,12,41) -> Matrix(3,4,8,11) Matrix(19,60,6,19) -> Matrix(3,2,10,7) Matrix(99,280,-64,-181) -> Matrix(1,-2,-4,9) Matrix(79,220,14,39) -> Matrix(1,0,2,1) Matrix(241,660,88,241) -> Matrix(3,4,8,11) Matrix(419,1140,154,419) -> Matrix(15,14,46,43) Matrix(141,380,82,221) -> Matrix(13,10,74,57) Matrix(39,100,-16,-41) -> Matrix(7,4,-16,-9) Matrix(159,380,100,239) -> Matrix(9,4,56,25) Matrix(59,140,8,19) -> Matrix(5,2,12,5) Matrix(61,140,44,101) -> Matrix(7,2,52,15) Matrix(19,40,-10,-21) -> Matrix(5,2,-18,-7) Matrix(219,400,98,179) -> Matrix(7,2,38,11) Matrix(221,400,100,181) -> Matrix(7,2,24,7) Matrix(79,140,22,39) -> Matrix(1,0,6,1) Matrix(59,100,-36,-61) -> Matrix(23,6,-96,-25) Matrix(99,160,86,139) -> Matrix(43,10,374,87) Matrix(199,320,74,119) -> Matrix(35,8,118,27) Matrix(201,320,76,121) -> Matrix(1,0,8,1) Matrix(721,1140,456,721) -> Matrix(19,4,128,27) Matrix(419,660,266,419) -> Matrix(9,2,58,13) Matrix(179,280,140,219) -> Matrix(7,2,52,15) Matrix(259,400,90,139) -> Matrix(9,2,58,13) Matrix(261,400,92,141) -> Matrix(9,2,4,1) Matrix(41,60,28,41) -> Matrix(19,4,128,27) Matrix(99,140,70,99) -> Matrix(51,10,362,71) Matrix(101,140,44,61) -> Matrix(11,2,60,11) Matrix(59,80,14,19) -> Matrix(11,2,38,7) Matrix(61,80,16,21) -> Matrix(11,2,16,3) Matrix(79,100,-64,-81) -> Matrix(47,8,-288,-49) Matrix(181,220,116,141) -> Matrix(37,6,228,37) Matrix(339,400,50,59) -> Matrix(13,2,110,17) Matrix(341,400,52,61) -> Matrix(13,2,-72,-11) Matrix(121,140,70,81) -> Matrix(53,8,298,45) Matrix(1,0,2,1) -> Matrix(1,0,18,1) Matrix(101,-120,16,-19) -> Matrix(17,-2,-8,1) Matrix(81,-100,64,-79) -> Matrix(65,-8,512,-63) Matrix(181,-280,64,-99) -> Matrix(13,-2,20,-3) Matrix(61,-100,36,-59) -> Matrix(37,-6,216,-35) Matrix(21,-40,10,-19) -> Matrix(11,-2,50,-9) Matrix(41,-100,16,-39) -> Matrix(17,-4,64,-15) Matrix(21,-100,4,-19) -> 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): 24 Degree of the the map X: 24 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,23,24,7)(3,12,36,13,4)(5,18,10,9,19)(8,28,16,15,29)(11,34,22,21,35)(14,37,26,25,38)(17,42,30,33,39)(31,41,40,43,32); (1,4,16,41,34,46,37,17,5,2)(3,10,27,8,7,26,43,48,33,11)(6,22)(9,32)(12,35,42,47,31,25,24,15,20,19)(13,14)(18,39,38,44,21,40,28,36,45,23)(29,30); (1,2,8,30,35,44,38,31,9,3)(4,14,39,48,43,21,6,5,20,15)(7,25)(10,23,22,41,47,42,37,13,28,27)(11,12)(16,40)(17,18)(19,32,26,46,34,33,29,24,45,36)) ----------------------------------------------------------------------- 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: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -2/1 -5/3 0/1 1/1 5/4 10/7 3/2 5/3 2/1 20/9 7/3 5/2 3/1 10/3 11/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 1/0 -9/2 -1/1 0/1 -4/1 0/1 -11/3 1/0 -7/2 -2/1 -1/1 -10/3 -1/1 -3/1 -1/2 -5/2 -1/2 -7/3 -3/8 -9/4 -1/3 0/1 -2/1 -1/3 -5/3 -1/4 -8/5 -2/9 -19/12 -2/9 -1/5 -30/19 -1/5 -11/7 -1/4 -3/2 -2/9 -1/5 -10/7 -1/5 -7/5 -5/26 -11/8 -1/5 -2/11 -4/3 -2/11 -9/7 -1/6 -5/4 -1/6 -1/1 -1/8 0/1 0/1 1/1 1/10 5/4 1/8 9/7 1/8 4/3 2/15 11/8 2/15 1/7 7/5 5/36 10/7 1/7 3/2 1/7 2/13 5/3 1/6 7/4 2/11 1/5 9/5 1/6 2/1 1/5 11/5 1/4 20/9 0/1 9/4 0/1 1/5 7/3 3/14 5/2 1/4 3/1 1/4 10/3 1/3 7/2 1/3 2/5 11/3 1/2 4/1 0/1 5/1 1/2 6/1 1/1 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,22,79) (-11/3,-7/2) -> (7/4,9/5) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(51,110,-32,-69) (-9/4,-2/1) -> (-8/5,-19/12) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(411,650,184,291) (-19/12,-30/19) -> (20/9,9/4) Hyperbolic Matrix(349,550,158,249) (-30/19,-11/7) -> (11/5,20/9) Hyperbolic Matrix(71,110,20,31) (-11/7,-3/2) -> (7/2,11/3) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(51,70,8,11) (-11/8,-4/3) -> (6/1,1/0) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,50,-2,-11) -> Matrix(1,-1,0,1) Matrix(19,80,14,59) -> Matrix(3,2,22,15) Matrix(21,80,16,61) -> Matrix(1,-2,8,-15) Matrix(39,140,22,79) -> Matrix(1,0,6,1) Matrix(41,140,12,41) -> Matrix(3,4,8,11) Matrix(19,60,6,19) -> Matrix(3,2,10,7) Matrix(11,30,4,11) -> Matrix(1,1,2,3) Matrix(29,70,12,29) -> Matrix(7,3,30,13) Matrix(61,140,44,101) -> Matrix(7,2,52,15) Matrix(51,110,-32,-69) -> Matrix(1,1,-6,-5) Matrix(29,50,-18,-31) -> Matrix(11,3,-48,-13) Matrix(411,650,184,291) -> Matrix(5,1,34,7) Matrix(349,550,158,249) -> Matrix(5,1,24,5) Matrix(71,110,20,31) -> Matrix(13,3,30,7) Matrix(41,60,28,41) -> Matrix(19,4,128,27) Matrix(99,140,70,99) -> Matrix(51,10,362,71) Matrix(101,140,44,61) -> Matrix(11,2,60,11) Matrix(51,70,8,11) -> Matrix(5,1,-6,-1) Matrix(61,80,16,21) -> Matrix(11,2,16,3) Matrix(71,90,56,71) -> Matrix(41,7,322,55) Matrix(9,10,8,9) -> Matrix(7,1,62,9) Matrix(1,0,2,1) -> Matrix(1,0,18,1) Matrix(31,-50,18,-29) -> Matrix(19,-3,108,-17) Matrix(21,-40,10,-19) -> Matrix(11,-2,50,-9) Matrix(11,-50,2,-9) -> Matrix(3,-1,4,-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): 24 ----------------------------------------------------------------------- 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 9 1 1/1 1/10 1 10 5/4 1/8 4 2 9/7 1/8 1 10 4/3 2/15 1 5 11/8 (2/15,1/7) 0 10 7/5 5/36 1 10 10/7 1/7 7 1 3/2 (1/7,2/13) 0 10 5/3 1/6 3 2 7/4 (2/11,1/5) 0 10 9/5 1/6 1 10 2/1 1/5 1 5 11/5 1/4 1 10 20/9 0/1 1 1 9/4 (0/1,1/5) 0 10 7/3 3/14 1 10 5/2 1/4 2 2 3/1 1/4 1 10 10/3 1/3 3 1 7/2 (1/3,2/5) 0 10 11/3 1/2 1 10 4/1 0/1 1 5 5/1 1/2 1 2 6/1 1/1 1 5 1/0 (0/1,1/1) 0 10 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(61,-80,16,-21) (9/7,4/3) -> (11/3,4/1) Glide Reflection Matrix(51,-70,8,-11) (4/3,11/8) -> (6/1,1/0) Glide Reflection Matrix(101,-140,44,-61) (11/8,7/5) -> (9/4,7/3) Glide Reflection Matrix(99,-140,70,-99) (7/5,10/7) -> (7/5,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(79,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(199,-440,90,-199) (11/5,20/9) -> (11/5,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic 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(1,0,2,-1) -> Matrix(1,0,20,-1) (0/1,1/1) -> (0/1,1/10) Matrix(9,-10,8,-9) -> Matrix(9,-1,80,-9) (1/1,5/4) -> (1/10,1/8) Matrix(71,-90,56,-71) -> Matrix(55,-7,432,-55) (5/4,9/7) -> (1/8,7/54) Matrix(61,-80,16,-21) -> Matrix(15,-2,22,-3) Matrix(51,-70,8,-11) -> Matrix(7,-1,-8,1) Matrix(101,-140,44,-61) -> Matrix(15,-2,82,-11) Matrix(99,-140,70,-99) -> Matrix(71,-10,504,-71) (7/5,10/7) -> (5/36,1/7) Matrix(41,-60,28,-41) -> Matrix(27,-4,182,-27) (10/7,3/2) -> (1/7,2/13) Matrix(31,-50,18,-29) -> Matrix(19,-3,108,-17) 1/6 Matrix(79,-140,22,-39) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(21,-40,10,-19) -> Matrix(11,-2,50,-9) 1/5 Matrix(199,-440,90,-199) -> Matrix(1,0,8,-1) (11/5,20/9) -> (0/1,1/4) Matrix(161,-360,72,-161) -> Matrix(1,0,10,-1) (20/9,9/4) -> (0/1,1/5) Matrix(29,-70,12,-29) -> Matrix(13,-3,56,-13) (7/3,5/2) -> (3/14,1/4) Matrix(11,-30,4,-11) -> Matrix(3,-1,8,-3) (5/2,3/1) -> (1/4,1/2) Matrix(19,-60,6,-19) -> Matrix(7,-2,24,-7) (3/1,10/3) -> (1/4,1/3) Matrix(41,-140,12,-41) -> Matrix(11,-4,30,-11) (10/3,7/2) -> (1/3,2/5) Matrix(11,-50,2,-9) -> Matrix(3,-1,4,-1) 1/2 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.