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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -9/4 -2/1 -3/2 -1/1 -3/4 -2/3 0/1 1/2 6/11 3/5 2/3 3/4 1/1 6/5 5/4 3/2 12/7 7/4 2/1 9/4 5/2 11/4 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 1/0 -11/4 -1/1 1/0 -8/3 -1/1 -5/2 -1/2 -12/5 0/1 -7/3 1/0 -9/4 -1/1 0/1 -2/1 0/1 -7/4 1/2 1/1 -12/7 1/1 -5/3 1/0 -3/2 1/0 -7/5 1/0 -18/13 -1/1 -11/8 -1/1 1/0 -4/3 -1/1 -5/4 -1/2 0/1 -6/5 0/1 -1/1 1/0 -3/4 -1/1 0/1 -5/7 1/0 -12/17 -1/1 -7/10 -1/2 -2/3 -1/1 -5/8 -2/3 -1/2 -3/5 -1/2 -7/12 -1/2 -1/3 -4/7 0/1 -9/16 -1/1 0/1 -5/9 1/0 -6/11 -1/1 -1/2 -1/2 0/1 0/1 1/2 1/2 6/11 1/1 5/9 1/0 4/7 0/1 3/5 1/2 5/8 1/2 2/3 2/3 1/1 3/4 0/1 1/1 4/5 1/1 1/1 1/0 6/5 0/1 5/4 0/1 1/2 4/3 1/1 3/2 1/0 8/5 -1/1 13/8 -1/1 1/0 18/11 -1/1 5/3 1/0 12/7 -1/1 7/4 -1/1 -1/2 9/5 -1/2 2/1 0/1 9/4 0/1 1/1 16/7 0/1 7/3 1/0 12/5 0/1 5/2 1/2 18/7 1/1 13/5 1/0 8/3 1/1 11/4 1/1 1/0 3/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(23,66,8,23) (-3/1,-11/4) -> (11/4,3/1) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(25,66,-36,-95) (-8/3,-5/2) -> (-7/10,-2/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(23,54,20,47) (-12/5,-7/3) -> (1/1,6/5) Hyperbolic Matrix(47,108,-84,-193) (-7/3,-9/4) -> (-9/16,-5/9) Hyperbolic Matrix(25,54,-44,-95) (-9/4,-2/1) -> (-4/7,-9/16) Hyperbolic Matrix(23,42,-40,-73) (-2/1,-7/4) -> (-7/12,-4/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(95,132,172,239) (-7/5,-18/13) -> (6/11,5/9) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(23,30,36,47) (-4/3,-5/4) -> (5/8,2/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(23,18,-32,-25) (-1/1,-3/4) -> (-3/4,-5/7) Parabolic Matrix(265,186,104,73) (-12/17,-7/10) -> (5/2,18/7) Hyperbolic Matrix(47,30,36,23) (-2/3,-5/8) -> (5/4,4/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(143,84,80,47) (-3/5,-7/12) -> (7/4,9/5) Hyperbolic Matrix(217,120,132,73) (-5/9,-6/11) -> (18/11,5/3) Hyperbolic Matrix(23,12,44,23) (-6/11,-1/2) -> (1/2,6/11) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(193,-108,84,-47) (5/9,4/7) -> (16/7,7/3) Hyperbolic Matrix(73,-42,40,-23) (4/7,3/5) -> (9/5,2/1) Hyperbolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(73,-60,28,-23) (4/5,1/1) -> (13/5,8/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(145,-246,56,-95) (5/3,12/7) -> (18/7,13/5) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(23,66,8,23) -> Matrix(1,2,0,1) Matrix(71,192,44,119) -> Matrix(1,0,0,1) Matrix(25,66,-36,-95) -> Matrix(1,0,0,1) Matrix(49,120,20,49) -> Matrix(1,0,4,1) Matrix(23,54,20,47) -> Matrix(1,0,0,1) Matrix(47,108,-84,-193) -> Matrix(1,0,0,1) Matrix(25,54,-44,-95) -> Matrix(1,0,0,1) Matrix(23,42,-40,-73) -> Matrix(1,0,-4,1) Matrix(97,168,56,97) -> Matrix(3,-2,-4,3) Matrix(71,120,-100,-169) -> Matrix(1,-2,0,1) Matrix(23,36,-16,-25) -> Matrix(1,-2,0,1) Matrix(95,132,172,239) -> Matrix(1,2,0,1) Matrix(313,432,192,265) -> Matrix(1,0,0,1) Matrix(97,132,36,49) -> Matrix(1,2,0,1) Matrix(23,30,36,47) -> Matrix(3,2,4,3) Matrix(49,60,40,49) -> Matrix(1,0,4,1) Matrix(47,54,20,23) -> Matrix(1,0,0,1) Matrix(23,18,-32,-25) -> Matrix(1,0,0,1) Matrix(265,186,104,73) -> Matrix(3,2,4,3) Matrix(47,30,36,23) -> Matrix(3,2,4,3) Matrix(49,30,80,49) -> Matrix(7,4,12,7) Matrix(143,84,80,47) -> Matrix(5,2,-8,-3) Matrix(217,120,132,73) -> Matrix(1,0,0,1) Matrix(23,12,44,23) -> Matrix(3,2,4,3) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(193,-108,84,-47) -> Matrix(1,0,0,1) Matrix(73,-42,40,-23) -> Matrix(1,0,-4,1) Matrix(25,-18,32,-23) -> Matrix(1,0,0,1) Matrix(73,-60,28,-23) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,-2,0,1) Matrix(145,-246,56,-95) -> Matrix(1,2,0,1) Matrix(73,-162,32,-71) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 5 Degree of the the map X: 10 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,10,21,20,26,11)(4,13,30,19,14,5)(6,17,12,24,9,18)(7,16,15,28,22,8)(23,27)(25,29)(31,32); (1,5,16,27,26,20,31,15,14,29,17,6)(2,8,22,25,10,24,32,30,13,23,9,3)(4,11,28,12)(7,21,19,18); (1,3,4)(2,6,19,32,20,7)(5,15)(9,10)(11,27,13,12,29,22)(14,21,25)(16,18,23)(24,28,31)) ----------------------------------------------------------------------- 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): 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: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -3/2 -5/4 -6/5 -1/1 -3/4 0/1 1/2 3/5 3/4 1/1 3/2 2/1 5/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 1/0 -5/2 -1/2 -12/5 0/1 -7/3 1/0 -2/1 0/1 -3/2 1/0 -4/3 -1/1 -5/4 -1/2 0/1 -6/5 0/1 -1/1 1/0 -3/4 -1/1 0/1 -5/7 1/0 -2/3 -1/1 -5/8 -2/3 -1/2 -3/5 -1/2 -1/2 -1/2 0/1 0/1 1/2 1/2 3/5 1/2 5/8 1/2 2/3 2/3 1/1 3/4 0/1 1/1 4/5 1/1 1/1 1/0 3/2 1/0 5/3 1/0 12/7 -1/1 7/4 -1/1 -1/2 2/1 0/1 7/3 1/0 12/5 0/1 5/2 1/2 3/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(13,30,16,37) (-7/3,-2/1) -> (4/5,1/1) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(23,30,36,47) (-4/3,-5/4) -> (5/8,2/3) Hyperbolic Matrix(83,102,48,59) (-5/4,-6/5) -> (12/7,7/4) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(23,18,-32,-25) (-1/1,-3/4) -> (-3/4,-5/7) Parabolic Matrix(35,24,16,11) (-5/7,-2/3) -> (2/1,7/3) Hyperbolic Matrix(37,24,20,13) (-2/3,-5/8) -> (7/4,2/1) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(11,6,20,11) (-3/5,-1/2) -> (1/2,3/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(11,30,4,11) -> Matrix(1,1,0,1) Matrix(49,120,20,49) -> Matrix(1,0,4,1) Matrix(61,144,36,85) -> Matrix(1,-1,0,1) Matrix(13,30,16,37) -> Matrix(1,1,0,1) Matrix(11,18,-8,-13) -> Matrix(1,-1,0,1) Matrix(23,30,36,47) -> Matrix(3,2,4,3) Matrix(83,102,48,59) -> Matrix(3,1,-4,-1) Matrix(47,54,20,23) -> Matrix(1,0,0,1) Matrix(23,18,-32,-25) -> Matrix(1,0,0,1) Matrix(35,24,16,11) -> Matrix(1,1,0,1) Matrix(37,24,20,13) -> Matrix(1,1,-4,-3) Matrix(49,30,80,49) -> Matrix(7,4,12,7) Matrix(11,6,20,11) -> Matrix(1,1,0,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(25,-18,32,-23) -> Matrix(1,0,0,1) Matrix(13,-18,8,-11) -> Matrix(1,-1,0,1) 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 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 0/1 2 2 1/2 1/2 2 12 3/5 1/2 3 4 5/8 (1/2,2/3) 0 12 2/3 1/1 2 6 3/4 0 4 4/5 1/1 2 6 1/1 1/0 1 12 3/2 1/0 2 4 5/3 1/0 1 12 12/7 -1/1 4 2 7/4 (-1/1,-1/2) 0 12 2/1 0/1 2 6 7/3 1/0 1 12 12/5 0/1 4 2 5/2 1/2 2 12 3/1 1/0 1 4 1/0 (0/1,1/0) 0 12 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(37,-24,20,-13) (5/8,2/3) -> (7/4,2/1) Glide Reflection Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(37,-30,16,-13) (4/5,1/1) -> (2/1,7/3) Glide Reflection Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(85,-144,36,-61) (5/3,12/7) -> (7/3,12/5) Glide Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,4,-1) -> Matrix(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Matrix(11,-6,20,-11) -> Matrix(-1,1,0,1) (1/2,3/5) -> (1/2,1/0) Matrix(49,-30,80,-49) -> Matrix(7,-4,12,-7) (3/5,5/8) -> (1/2,2/3) Matrix(37,-24,20,-13) -> Matrix(1,-1,-4,3) Matrix(25,-18,32,-23) -> Matrix(1,0,0,1) Matrix(37,-30,16,-13) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(13,-18,8,-11) -> Matrix(1,-1,0,1) 1/0 Matrix(85,-144,36,-61) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(97,-168,56,-97) -> Matrix(3,2,-4,-3) (12/7,7/4) -> (-1/1,-1/2) Matrix(49,-120,20,-49) -> Matrix(1,0,4,-1) (12/5,5/2) -> (0/1,1/2) Matrix(11,-30,4,-11) -> Matrix(-1,1,0,1) (5/2,3/1) -> (1/2,1/0) Matrix(-1,6,0,1) -> Matrix(1,0,0,-1) (3/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.