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): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -3/1 -3/2 0/1 1/1 9/7 3/2 27/17 9/5 2/1 9/4 5/2 18/7 8/3 3/1 27/8 7/2 18/5 4/1 9/2 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 1/0 -5/1 -1/2 0/1 -9/2 0/1 -4/1 1/0 -7/2 -1/1 0/1 -3/1 0/1 -11/4 0/1 1/1 -19/7 0/1 1/1 -27/10 1/1 -8/3 1/0 -5/2 -1/1 1/0 -12/5 -1/2 1/0 -19/8 -1/1 0/1 -7/3 -1/1 0/1 -9/4 0/1 -2/1 1/0 -9/5 -1/1 -7/4 -1/1 0/1 -12/7 -1/2 1/0 -5/3 0/1 1/0 -18/11 1/0 -13/8 -3/1 1/0 -8/5 1/0 -3/2 -1/1 -10/7 -1/2 -27/19 0/1 -17/12 -1/1 0/1 -7/5 -1/1 0/1 -18/13 -1/2 1/0 -11/8 -1/1 0/1 -4/3 1/0 -9/7 -1/1 -5/4 -1/1 -1/2 -6/5 -1/2 1/0 -7/6 -1/1 0/1 -1/1 -1/1 0/1 0/1 -1/2 1/0 1/1 -1/1 0/1 6/5 -1/2 1/0 5/4 -1/1 1/0 9/7 -1/1 4/3 -1/2 7/5 -1/1 0/1 3/2 -1/1 11/7 -1/1 -2/3 19/12 -1/1 -2/3 27/17 -2/3 8/5 -1/2 5/3 -1/2 0/1 12/7 -1/2 1/0 19/11 -1/1 0/1 7/4 -1/1 0/1 9/5 -1/1 2/1 -1/2 9/4 0/1 7/3 -1/1 0/1 12/5 -1/2 1/0 5/2 -1/1 -1/2 18/7 -1/2 13/5 -1/2 -2/5 8/3 -1/2 3/1 0/1 10/3 1/0 27/8 -1/1 17/5 -1/1 0/1 7/2 -1/1 0/1 18/5 -1/2 1/0 11/3 -1/1 0/1 4/1 -1/2 9/2 0/1 5/1 0/1 1/0 6/1 -1/2 1/0 7/1 -1/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,126,-8,-53) (-6/1,1/0) -> (-12/5,-19/8) Hyperbolic Matrix(17,90,10,53) (-6/1,-5/1) -> (5/3,12/7) Hyperbolic Matrix(19,90,4,19) (-5/1,-9/2) -> (9/2,5/1) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(17,54,-6,-19) (-7/2,-3/1) -> (-3/1,-11/4) Parabolic Matrix(125,342,72,197) (-11/4,-19/7) -> (19/11,7/4) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(37,90,30,73) (-5/2,-12/5) -> (6/5,5/4) Hyperbolic Matrix(145,342,92,217) (-19/8,-7/3) -> (11/7,19/12) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(73,126,-62,-107) (-7/4,-12/7) -> (-6/5,-7/6) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(341,486,214,305) (-10/7,-27/19) -> (27/17,8/5) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(89,126,12,17) (-17/12,-7/5) -> (7/1,1/0) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(109,126,32,37) (-7/6,-1/1) -> (17/5,7/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,126,-8,-53) -> Matrix(1,0,0,1) Matrix(17,90,10,53) -> Matrix(1,0,0,1) Matrix(19,90,4,19) -> Matrix(1,0,2,1) Matrix(17,72,4,17) -> Matrix(1,0,-2,1) Matrix(19,72,-14,-53) -> Matrix(1,0,0,1) Matrix(17,54,-6,-19) -> Matrix(1,0,2,1) Matrix(125,342,72,197) -> Matrix(1,0,-2,1) Matrix(359,972,106,287) -> Matrix(1,0,-2,1) Matrix(181,486,54,145) -> Matrix(1,-2,0,1) Matrix(55,144,-34,-89) -> Matrix(1,-2,0,1) Matrix(37,90,30,73) -> Matrix(1,0,0,1) Matrix(145,342,92,217) -> Matrix(1,2,-2,-3) Matrix(55,126,24,55) -> Matrix(1,0,0,1) Matrix(17,36,8,17) -> Matrix(1,0,-2,1) Matrix(19,36,10,19) -> Matrix(1,2,-2,-3) Matrix(71,126,40,71) -> Matrix(1,0,0,1) Matrix(73,126,-62,-107) -> Matrix(1,0,0,1) Matrix(53,90,10,17) -> Matrix(1,0,0,1) Matrix(197,324,76,125) -> Matrix(1,-2,-2,5) Matrix(199,324,78,127) -> Matrix(1,4,-2,-7) Matrix(35,54,-24,-37) -> Matrix(1,2,-2,-3) Matrix(341,486,214,305) -> Matrix(5,2,-8,-3) Matrix(685,972,432,613) -> Matrix(1,2,-2,-3) Matrix(89,126,12,17) -> Matrix(1,0,0,1) Matrix(233,324,64,89) -> Matrix(1,0,0,1) Matrix(235,324,66,91) -> Matrix(1,0,0,1) Matrix(55,72,42,55) -> Matrix(1,2,-2,-3) Matrix(71,90,56,71) -> Matrix(3,2,-2,-1) Matrix(73,90,30,37) -> Matrix(1,0,0,1) Matrix(109,126,32,37) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(107,-126,62,-73) -> Matrix(1,0,0,1) Matrix(53,-72,14,-19) -> Matrix(1,0,0,1) Matrix(37,-54,24,-35) -> Matrix(1,2,-2,-3) Matrix(89,-144,34,-55) -> Matrix(3,2,-8,-5) Matrix(53,-126,8,-19) -> Matrix(1,0,0,1) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) 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): 6 Degree of the the map X: 6 Degree of the the map Y: 36 Permutation triple for Y: ((2,6,23,13,4,3,12,24,7)(5,18,16,15,10,9,27,8,19)(11,31,36,26,21,20,35,34,14)(17,22,29)(25,28,33); (1,4,16,33,36,31,17,5,2)(3,10,11)(6,21,32,14,13,28,27,9,22)(7,26,8)(12,29,20,19,30,15,34,25,24); (1,2,8,28,34,35,29,9,3)(4,14,15)(5,20,6)(7,25,16,18,17,12,11,32,21)(10,30,19,26,33,13,23,22,31)) ----------------------------------------------------------------------- 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): 108 Minimal number of generators: 19 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: 16 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 6/5 3/2 2/1 9/4 18/7 8/3 3/1 27/8 18/5 4/1 9/2 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/2 1/0 1/1 -1/1 0/1 6/5 -1/2 1/0 5/4 -1/1 1/0 9/7 -1/1 4/3 -1/2 7/5 -1/1 0/1 3/2 -1/1 11/7 -1/1 -2/3 19/12 -1/1 -2/3 27/17 -2/3 8/5 -1/2 5/3 -1/2 0/1 12/7 -1/2 1/0 19/11 -1/1 0/1 7/4 -1/1 0/1 9/5 -1/1 2/1 -1/2 9/4 0/1 7/3 -1/1 0/1 12/5 -1/2 1/0 5/2 -1/1 -1/2 18/7 -1/2 13/5 -1/2 -2/5 8/3 -1/2 3/1 0/1 10/3 1/0 27/8 -1/1 17/5 -1/1 0/1 7/2 -1/1 0/1 18/5 -1/2 1/0 11/3 -1/1 0/1 4/1 -1/2 9/2 0/1 5/1 0/1 1/0 6/1 -1/2 1/0 7/1 -1/1 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(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(73,-90,43,-53) (6/5,5/4) -> (5/3,12/7) Hyperbolic Matrix(71,-90,15,-19) (5/4,9/7) -> (9/2,5/1) Hyperbolic Matrix(55,-72,13,-17) (9/7,4/3) -> (4/1,9/2) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(217,-342,125,-197) (11/7,19/12) -> (19/11,7/4) Hyperbolic Matrix(613,-972,181,-287) (19/12,27/17) -> (27/8,17/5) Hyperbolic Matrix(305,-486,91,-145) (27/17,8/5) -> (10/3,27/8) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(71,-126,31,-55) (7/4,9/5) -> (9/4,7/3) Hyperbolic Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(37,-90,7,-17) (12/5,5/2) -> (5/1,6/1) Hyperbolic Matrix(127,-324,49,-125) (5/2,18/7) -> (18/7,13/5) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(37,-126,5,-17) (17/5,7/2) -> (7/1,1/0) Hyperbolic Matrix(91,-324,25,-89) (7/2,18/5) -> (18/5,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,1,-2,-1) Matrix(107,-126,62,-73) -> Matrix(1,0,0,1) Matrix(73,-90,43,-53) -> Matrix(1,1,-2,-1) Matrix(71,-90,15,-19) -> Matrix(1,1,0,1) Matrix(55,-72,13,-17) -> Matrix(1,1,-4,-3) Matrix(53,-72,14,-19) -> Matrix(1,0,0,1) Matrix(37,-54,24,-35) -> Matrix(1,2,-2,-3) Matrix(217,-342,125,-197) -> Matrix(1,1,-4,-3) Matrix(613,-972,181,-287) -> Matrix(1,1,-4,-3) Matrix(305,-486,91,-145) -> Matrix(5,3,-2,-1) Matrix(89,-144,34,-55) -> Matrix(3,2,-8,-5) Matrix(71,-126,31,-55) -> Matrix(1,1,-2,-1) Matrix(19,-36,9,-17) -> Matrix(1,1,-4,-3) Matrix(53,-126,8,-19) -> Matrix(1,0,0,1) Matrix(37,-90,7,-17) -> Matrix(1,1,-2,-1) Matrix(127,-324,49,-125) -> Matrix(5,3,-12,-7) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) Matrix(37,-126,5,-17) -> Matrix(1,1,-2,-1) Matrix(91,-324,25,-89) -> Matrix(1,1,-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): 6 ----------------------------------------------------------------------- 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 (-1/1,0/1).(-1/2,1/0) 0 1 2/1 -1/2 1 9 9/4 0/1 1 1 7/3 (-1/1,0/1) 0 9 12/5 (-1/1,0/1).(-1/2,1/0) 0 3 5/2 (-1/1,-1/2) 0 9 18/7 -1/2 3 1 8/3 -1/2 1 9 3/1 0/1 1 3 10/3 1/0 1 9 27/8 -1/1 1 1 17/5 (-1/1,0/1) 0 9 7/2 (-1/1,0/1) 0 9 18/5 (-1/1,0/1).(-1/2,1/0) 0 1 4/1 -1/2 1 9 9/2 0/1 2 1 5/1 (0/1,1/0) 0 9 6/1 (-1/1,0/1).(-1/2,1/0) 0 3 7/1 (-1/1,0/1) 0 9 1/0 (-1/1,0/1) 0 9 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(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(37,-90,7,-17) (12/5,5/2) -> (5/1,6/1) Hyperbolic Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(271,-918,80,-271) (27/8,17/5) -> (27/8,17/5) Reflection Matrix(37,-126,5,-17) (17/5,7/2) -> (7/1,1/0) Hyperbolic Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(19,-72,5,-19) (18/5,4/1) -> (18/5,4/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) 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(1,1,0,-1) (0/1,2/1) -> (-1/2,1/0) Matrix(17,-36,8,-17) -> Matrix(-1,0,4,1) (2/1,9/4) -> (-1/2,0/1) Matrix(55,-126,24,-55) -> Matrix(-1,0,2,1) (9/4,7/3) -> (-1/1,0/1) Matrix(53,-126,8,-19) -> Matrix(1,0,0,1) Matrix(37,-90,7,-17) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(71,-180,28,-71) -> Matrix(3,2,-4,-3) (5/2,18/7) -> (-1/1,-1/2) Matrix(55,-144,21,-55) -> Matrix(3,1,-8,-3) (18/7,8/3) -> (-1/2,-1/4) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) 0/1 Matrix(161,-540,48,-161) -> Matrix(1,2,0,-1) (10/3,27/8) -> (-1/1,1/0) Matrix(271,-918,80,-271) -> Matrix(-1,0,2,1) (27/8,17/5) -> (-1/1,0/1) Matrix(37,-126,5,-17) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(71,-252,20,-71) -> Matrix(-1,0,2,1) (7/2,18/5) -> (-1/1,0/1) Matrix(19,-72,5,-19) -> Matrix(1,1,0,-1) (18/5,4/1) -> (-1/2,1/0) Matrix(17,-72,4,-17) -> Matrix(-1,0,4,1) (4/1,9/2) -> (-1/2,0/1) Matrix(19,-90,4,-19) -> Matrix(1,0,0,-1) (9/2,5/1) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.