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): 180 Minimal number of generators: 31 Number of equivalence classes of cusps: 6 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 29/11 29/8 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -5/29 -9/2 -9/58 -4/1 -4/29 -11/3 -11/87 -7/2 -7/58 -3/1 -3/29 -11/4 -11/116 -8/3 -8/87 -5/2 -5/58 -7/3 -7/87 -9/4 -9/116 -2/1 -2/29 -9/5 -9/145 -7/4 -7/116 -12/7 -12/203 -5/3 -5/87 -13/8 -13/232 -21/13 -21/377 -29/18 -1/18 -8/5 -8/145 -3/2 -3/58 -7/5 -7/145 -18/13 -18/377 -29/21 -1/21 -11/8 -11/232 -4/3 -4/87 -9/7 -9/203 -5/4 -5/116 -6/5 -6/145 -1/1 -1/29 0/1 0/1 1/1 1/29 5/4 5/116 9/7 9/203 4/3 4/87 11/8 11/232 7/5 7/145 3/2 3/58 11/7 11/203 8/5 8/145 5/3 5/87 7/4 7/116 9/5 9/145 2/1 2/29 9/4 9/116 7/3 7/87 12/5 12/145 5/2 5/58 13/5 13/145 21/8 21/232 29/11 1/11 8/3 8/87 3/1 3/29 7/2 7/58 18/5 18/145 29/8 1/8 11/3 11/87 4/1 4/29 9/2 9/58 5/1 5/29 6/1 6/29 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,58,-4,-21) (-5/1,1/0) -> (-3/1,-11/4) Hyperbolic Matrix(25,116,14,65) (-5/1,-9/2) -> (7/4,9/5) Hyperbolic Matrix(13,58,2,9) (-9/2,-4/1) -> (6/1,1/0) Hyperbolic Matrix(15,58,8,31) (-4/1,-11/3) -> (9/5,2/1) Hyperbolic Matrix(81,290,-50,-179) (-11/3,-7/2) -> (-13/8,-21/13) Hyperbolic Matrix(17,58,12,41) (-7/2,-3/1) -> (7/5,3/2) Hyperbolic Matrix(43,116,10,27) (-11/4,-8/3) -> (4/1,9/2) Hyperbolic Matrix(45,116,-26,-67) (-8/3,-5/2) -> (-7/4,-12/7) Hyperbolic Matrix(49,116,-30,-71) (-5/2,-7/3) -> (-5/3,-13/8) Hyperbolic Matrix(51,116,40,91) (-7/3,-9/4) -> (5/4,9/7) Hyperbolic Matrix(27,58,20,43) (-9/4,-2/1) -> (4/3,11/8) Hyperbolic Matrix(31,58,8,15) (-2/1,-9/5) -> (11/3,4/1) Hyperbolic Matrix(65,116,14,25) (-9/5,-7/4) -> (9/2,5/1) Hyperbolic Matrix(103,174,-74,-125) (-12/7,-5/3) -> (-7/5,-18/13) Hyperbolic Matrix(575,928,158,255) (-21/13,-29/18) -> (29/8,11/3) Hyperbolic Matrix(469,754,130,209) (-29/18,-8/5) -> (18/5,29/8) Hyperbolic Matrix(37,58,-30,-47) (-8/5,-3/2) -> (-5/4,-6/5) Hyperbolic Matrix(41,58,12,17) (-3/2,-7/5) -> (3/1,7/2) Hyperbolic Matrix(545,754,206,285) (-18/13,-29/21) -> (29/11,8/3) Hyperbolic Matrix(673,928,256,353) (-29/21,-11/8) -> (21/8,29/11) Hyperbolic Matrix(43,58,20,27) (-11/8,-4/3) -> (2/1,9/4) Hyperbolic Matrix(89,116,56,73) (-4/3,-9/7) -> (11/7,8/5) Hyperbolic Matrix(91,116,40,51) (-9/7,-5/4) -> (9/4,7/3) Hyperbolic Matrix(49,58,38,45) (-6/5,-1/1) -> (9/7,4/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(47,-58,30,-37) (1/1,5/4) -> (3/2,11/7) Hyperbolic Matrix(209,-290,80,-111) (11/8,7/5) -> (13/5,21/8) Hyperbolic Matrix(71,-116,30,-49) (8/5,5/3) -> (7/3,12/5) Hyperbolic Matrix(67,-116,26,-45) (5/3,7/4) -> (5/2,13/5) Hyperbolic Matrix(71,-174,20,-49) (12/5,5/2) -> (7/2,18/5) Hyperbolic Matrix(21,-58,4,-11) (8/3,3/1) -> (5/1,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,58,-4,-21) -> Matrix(11,2,-116,-21) Matrix(25,116,14,65) -> Matrix(25,4,406,65) Matrix(13,58,2,9) -> Matrix(13,2,58,9) Matrix(15,58,8,31) -> Matrix(15,2,232,31) Matrix(81,290,-50,-179) -> Matrix(81,10,-1450,-179) Matrix(17,58,12,41) -> Matrix(17,2,348,41) Matrix(43,116,10,27) -> Matrix(43,4,290,27) Matrix(45,116,-26,-67) -> Matrix(45,4,-754,-67) Matrix(49,116,-30,-71) -> Matrix(49,4,-870,-71) Matrix(51,116,40,91) -> Matrix(51,4,1160,91) Matrix(27,58,20,43) -> Matrix(27,2,580,43) Matrix(31,58,8,15) -> Matrix(31,2,232,15) Matrix(65,116,14,25) -> Matrix(65,4,406,25) Matrix(103,174,-74,-125) -> Matrix(103,6,-2146,-125) Matrix(575,928,158,255) -> Matrix(575,32,4582,255) Matrix(469,754,130,209) -> Matrix(469,26,3770,209) Matrix(37,58,-30,-47) -> Matrix(37,2,-870,-47) Matrix(41,58,12,17) -> Matrix(41,2,348,17) Matrix(545,754,206,285) -> Matrix(545,26,5974,285) Matrix(673,928,256,353) -> Matrix(673,32,7424,353) Matrix(43,58,20,27) -> Matrix(43,2,580,27) Matrix(89,116,56,73) -> Matrix(89,4,1624,73) Matrix(91,116,40,51) -> Matrix(91,4,1160,51) Matrix(49,58,38,45) -> Matrix(49,2,1102,45) Matrix(1,0,2,1) -> Matrix(1,0,58,1) Matrix(47,-58,30,-37) -> Matrix(47,-2,870,-37) Matrix(209,-290,80,-111) -> Matrix(209,-10,2320,-111) Matrix(71,-116,30,-49) -> Matrix(71,-4,870,-49) Matrix(67,-116,26,-45) -> Matrix(67,-4,754,-45) Matrix(71,-174,20,-49) -> Matrix(71,-6,580,-49) Matrix(21,-58,4,-11) -> Matrix(21,-2,116,-11) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 180 Minimal number of generators: 31 Number of equivalence classes of cusps: 6 Genus: 13 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 30 Degree of the the map Y: 30 Permutation triple for Y: ((2,6,22,10,9,24,21,29,28,11,26,8,25,13,4,3,12,19,5,18,14,27,30,20,17,16,15,23,7); (1,4,16,18,12,14,13,28,15,8,7,24,26,25,30,27,22,9,11,3,10,19,21,6,20,23,17,5,2); (1,2,8,24,22,21,7,20,25,15,4,14,16,23,28,29,19,18,17,6,5,10,27,12,11,13,26,9,3)) ----------------------------------------------------------------------- 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, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+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): 30 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/29 5/4 5/116 4/3 4/87 3/2 3/58 5/3 5/87 2/1 2/29 5/2 5/58 3/1 3/29 4/1 4/29 5/1 5/29 1/0 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(24,-29,5,-6) (1/1,5/4) -> (4/1,5/1) Hyperbolic Matrix(23,-29,4,-5) (5/4,4/3) -> (5/1,1/0) Hyperbolic Matrix(21,-29,8,-11) (4/3,3/2) -> (5/2,3/1) Hyperbolic Matrix(18,-29,5,-8) (3/2,5/3) -> (3/1,4/1) Hyperbolic Matrix(17,-29,10,-17) (5/3,2/1) -> (5/3,2/1) Elliptic Matrix(12,-29,5,-12) (2/1,5/2) -> (2/1,5/2) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,29,1) Matrix(24,-29,5,-6) -> Matrix(24,-1,145,-6) Matrix(23,-29,4,-5) -> Matrix(23,-1,116,-5) Matrix(21,-29,8,-11) -> Matrix(21,-1,232,-11) Matrix(18,-29,5,-8) -> Matrix(18,-1,145,-8) Matrix(17,-29,10,-17) -> Matrix(17,-1,290,-17) Matrix(12,-29,5,-12) -> Matrix(12,-1,145,-12) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 30 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 2 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 29 1 2/1 2/29 1 29 5/2 5/58 1 29 3/1 3/29 1 29 4/1 4/29 1 29 5/1 5/29 1 29 1/0 1/0 1 29 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(12,-29,5,-12) (2/1,5/2) -> (2/1,5/2) Elliptic Matrix(11,-29,3,-8) (5/2,3/1) -> (3/1,4/1) Glide Reflection Matrix(6,-29,1,-5) (4/1,5/1) -> (5/1,1/0) Glide 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(1,0,29,-1) (0/1,2/1) -> (0/1,2/29) Matrix(12,-29,5,-12) -> Matrix(12,-1,145,-12) (0/1,1/12).(1/13,1/11) Matrix(11,-29,3,-8) -> Matrix(11,-1,87,-8) Matrix(6,-29,1,-5) -> Matrix(6,-1,29,-5) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.