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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -11/30 -3/10 -3/20 0/1 1/6 1/5 1/4 1/3 3/8 2/5 1/2 3/5 2/3 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/2 -5/6 2/3 1/1 -4/5 1/1 -3/4 1/2 1/1 -2/3 1/0 -9/14 0/1 1/3 -7/11 1/2 -5/8 1/2 1/1 -3/5 1/1 -1/2 0/1 1/1 -2/5 1/1 -3/8 1/1 1/0 -7/19 1/0 -11/30 1/0 -4/11 1/0 -1/3 1/2 -3/10 1/1 -2/7 3/2 -1/4 1/1 1/0 -1/5 1/1 -1/6 1/1 2/1 -2/13 7/4 -3/20 2/1 -1/7 5/2 0/1 1/0 1/6 -2/1 -1/1 1/5 -1/1 1/4 -1/1 1/0 1/3 -1/2 5/14 0/1 1/1 4/11 1/0 3/8 -1/1 1/0 2/5 -1/1 1/2 -1/1 0/1 3/5 -1/1 5/8 -1/1 -1/2 12/19 -1/2 19/30 -1/2 7/11 -1/2 2/3 1/0 7/10 -1/1 5/7 -3/4 3/4 -1/1 -1/2 4/5 -1/1 5/6 -1/1 -2/3 11/13 -7/10 17/20 -2/3 6/7 -5/8 1/1 -1/2 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(51,44,-80,-69) (-1/1,-5/6) -> (-9/14,-7/11) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(11,8,-40,-29) (-3/4,-2/3) -> (-2/7,-1/4) Hyperbolic Matrix(31,20,-200,-129) (-2/3,-9/14) -> (-1/6,-2/13) Hyperbolic Matrix(89,56,-240,-151) (-7/11,-5/8) -> (-3/8,-7/19) 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(9,4,20,9) (-1/2,-2/5) -> (2/5,1/2) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(571,210,900,331) (-7/19,-11/30) -> (19/30,7/11) Hyperbolic Matrix(569,208,900,329) (-11/30,-4/11) -> (12/19,19/30) Hyperbolic Matrix(11,4,-80,-29) (-4/11,-1/3) -> (-1/7,0/1) Hyperbolic Matrix(71,22,100,31) (-1/3,-3/10) -> (7/10,5/7) Hyperbolic Matrix(69,20,100,29) (-3/10,-2/7) -> (2/3,7/10) Hyperbolic Matrix(9,2,40,9) (-1/4,-1/5) -> (1/5,1/4) Hyperbolic Matrix(11,2,60,11) (-1/5,-1/6) -> (1/6,1/5) Hyperbolic Matrix(341,52,400,61) (-2/13,-3/20) -> (17/20,6/7) Hyperbolic Matrix(339,50,400,59) (-3/20,-1/7) -> (11/13,17/20) Hyperbolic Matrix(29,-4,80,-11) (0/1,1/6) -> (5/14,4/11) Hyperbolic Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic Matrix(169,-60,200,-71) (1/3,5/14) -> (5/6,11/13) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(69,-44,80,-51) (7/11,2/3) -> (6/7,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(51,44,-80,-69) -> Matrix(3,-2,8,-5) Matrix(49,40,60,49) -> Matrix(5,-4,-6,5) Matrix(31,24,40,31) -> Matrix(3,-2,-4,3) Matrix(11,8,-40,-29) -> Matrix(3,-2,2,-1) Matrix(31,20,-200,-129) -> Matrix(7,-2,4,-1) Matrix(89,56,-240,-151) -> Matrix(3,-2,2,-1) Matrix(49,30,80,49) -> Matrix(3,-2,-4,3) Matrix(11,6,20,11) -> Matrix(1,0,-2,1) Matrix(9,4,20,9) -> Matrix(1,0,-2,1) Matrix(31,12,80,31) -> Matrix(1,-2,0,1) Matrix(571,210,900,331) -> Matrix(1,-4,-2,9) Matrix(569,208,900,329) -> Matrix(1,4,-2,-7) Matrix(11,4,-80,-29) -> Matrix(1,2,0,1) Matrix(71,22,100,31) -> Matrix(5,-4,-6,5) Matrix(69,20,100,29) -> Matrix(3,-4,-2,3) Matrix(9,2,40,9) -> Matrix(1,-2,0,1) Matrix(11,2,60,11) -> Matrix(3,-4,-2,3) Matrix(341,52,400,61) -> Matrix(13,-24,-20,37) Matrix(339,50,400,59) -> Matrix(11,-24,-16,35) Matrix(29,-4,80,-11) -> Matrix(1,2,0,1) Matrix(29,-8,40,-11) -> Matrix(1,2,-2,-3) Matrix(169,-60,200,-71) -> Matrix(3,-2,-4,3) Matrix(151,-56,240,-89) -> Matrix(1,2,-2,-3) Matrix(69,-44,80,-51) -> 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): 8 Degree of the the map X: 8 Degree of the the map Y: 24 Permutation triple for Y: ((1,4,13,21,20,23,16,14,5,2)(3,10,15,8,7,17,22,24,19,11)(6,12)(9,18); (1,2,8,18,17,23,20,19,9,3)(4,12,21,24,22,16,6,5,15,10)(7,11)(13,14); (2,6,4,3,7)(5,13,10,9,8)(11,20,12,16,17)(14,22,18,19,21)) ----------------------------------------------------------------------- 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/6 1/5 1/4 1/3 3/8 2/5 1/2 19/30 7/10 17/20 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/6 -2/1 -1/1 1/5 -1/1 1/4 -1/1 1/0 1/3 -1/2 5/14 0/1 1/1 4/11 1/0 3/8 -1/1 1/0 2/5 -1/1 1/2 -1/1 0/1 3/5 -1/1 5/8 -1/1 -1/2 12/19 -1/2 19/30 -1/2 7/11 -1/2 2/3 1/0 7/10 -1/1 5/7 -3/4 3/4 -1/1 -1/2 4/5 -1/1 5/6 -1/1 -2/3 11/13 -7/10 17/20 -2/3 6/7 -5/8 1/1 -1/2 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(29,-4,80,-11) (0/1,1/6) -> (5/14,4/11) Hyperbolic Matrix(49,-9,60,-11) (1/6,1/5) -> (4/5,5/6) Hyperbolic Matrix(31,-7,40,-9) (1/5,1/4) -> (3/4,4/5) Hyperbolic Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic Matrix(169,-60,200,-71) (1/3,5/14) -> (5/6,11/13) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(49,-19,80,-31) (3/8,2/5) -> (3/5,5/8) Hyperbolic Matrix(11,-5,20,-9) (2/5,1/2) -> (1/2,3/5) Parabolic Matrix(571,-361,900,-569) (12/19,19/30) -> (19/30,7/11) Parabolic Matrix(69,-44,80,-51) (7/11,2/3) -> (6/7,1/1) Hyperbolic Matrix(71,-49,100,-69) (2/3,7/10) -> (7/10,5/7) Parabolic Matrix(341,-289,400,-339) (11/13,17/20) -> (17/20,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-2,1) Matrix(29,-4,80,-11) -> Matrix(1,2,0,1) Matrix(49,-9,60,-11) -> Matrix(3,4,-4,-5) Matrix(31,-7,40,-9) -> Matrix(1,2,-2,-3) Matrix(29,-8,40,-11) -> Matrix(1,2,-2,-3) Matrix(169,-60,200,-71) -> Matrix(3,-2,-4,3) Matrix(151,-56,240,-89) -> Matrix(1,2,-2,-3) Matrix(49,-19,80,-31) -> Matrix(1,2,-2,-3) Matrix(11,-5,20,-9) -> Matrix(1,0,0,1) Matrix(571,-361,900,-569) -> Matrix(7,4,-16,-9) Matrix(69,-44,80,-51) -> Matrix(5,2,-8,-3) Matrix(71,-49,100,-69) -> Matrix(3,4,-4,-5) Matrix(341,-289,400,-339) -> Matrix(35,24,-54,-37) 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): 4 ----------------------------------------------------------------------- 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 1/0 1 10 1/6 (-2/1,-1/1) 0 5 1/5 -1/1 1 2 1/4 (-1/1,1/0) 0 5 3/10 -1/1 4 1 1/3 -1/2 1 10 7/20 0/1 6 1 5/14 (0/1,1/1) 0 5 4/11 1/0 1 10 11/30 1/0 4 1 3/8 (-1/1,1/0) 0 5 2/5 -1/1 1 2 1/2 (-1/1,0/1) 0 5 1/0 0/1 2 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(29,-4,80,-11) (0/1,1/6) -> (5/14,4/11) Hyperbolic Matrix(11,-2,60,-11) (1/6,1/5) -> (1/6,1/5) Reflection Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(11,-3,40,-11) (1/4,3/10) -> (1/4,3/10) Reflection Matrix(19,-6,60,-19) (3/10,1/3) -> (3/10,1/3) Reflection Matrix(41,-14,120,-41) (1/3,7/20) -> (1/3,7/20) Reflection Matrix(99,-35,280,-99) (7/20,5/14) -> (7/20,5/14) Reflection Matrix(241,-88,660,-241) (4/11,11/30) -> (4/11,11/30) Reflection Matrix(89,-33,240,-89) (11/30,3/8) -> (11/30,3/8) Reflection Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(9,-4,20,-9) (2/5,1/2) -> (2/5,1/2) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(29,-4,80,-11) -> Matrix(1,2,0,1) 1/0 Matrix(11,-2,60,-11) -> Matrix(3,4,-2,-3) (1/6,1/5) -> (-2/1,-1/1) Matrix(9,-2,40,-9) -> Matrix(1,2,0,-1) (1/5,1/4) -> (-1/1,1/0) Matrix(11,-3,40,-11) -> Matrix(1,2,0,-1) (1/4,3/10) -> (-1/1,1/0) Matrix(19,-6,60,-19) -> Matrix(3,2,-4,-3) (3/10,1/3) -> (-1/1,-1/2) Matrix(41,-14,120,-41) -> Matrix(-1,0,4,1) (1/3,7/20) -> (-1/2,0/1) Matrix(99,-35,280,-99) -> Matrix(1,0,2,-1) (7/20,5/14) -> (0/1,1/1) Matrix(241,-88,660,-241) -> Matrix(-1,2,0,1) (4/11,11/30) -> (1/1,1/0) Matrix(89,-33,240,-89) -> Matrix(1,2,0,-1) (11/30,3/8) -> (-1/1,1/0) Matrix(31,-12,80,-31) -> Matrix(1,2,0,-1) (3/8,2/5) -> (-1/1,1/0) Matrix(9,-4,20,-9) -> Matrix(-1,0,2,1) (2/5,1/2) -> (-1/1,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,2,1) (1/2,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.