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: 12 Genus: 19 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 3/2 2/1 3/1 33/10 11/3 22/5 33/7 11/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -7/33 -6/1 -2/11 -5/1 -5/33 -9/2 -3/22 -22/5 -2/15 -13/3 -13/99 -4/1 -4/33 -11/3 -1/9 -7/2 -7/66 -17/5 -17/165 -10/3 -10/99 -3/1 -1/11 -11/4 -1/12 -8/3 -8/99 -5/2 -5/66 -12/5 -4/55 -19/8 -19/264 -7/3 -7/99 -2/1 -2/33 -7/4 -7/132 -19/11 -19/363 -12/7 -4/77 -17/10 -17/330 -22/13 -2/39 -5/3 -5/99 -8/5 -8/165 -11/7 -1/21 -3/2 -1/22 -13/9 -13/297 -23/16 -23/528 -33/23 -1/23 -10/7 -10/231 -17/12 -17/396 -7/5 -7/165 -18/13 -6/143 -11/8 -1/24 -4/3 -4/99 -13/10 -13/330 -22/17 -2/51 -9/7 -3/77 -14/11 -14/363 -33/26 -1/26 -19/15 -19/495 -5/4 -5/132 -11/9 -1/27 -6/5 -2/55 -7/6 -7/198 -1/1 -1/33 0/1 0/1 1/1 1/33 7/6 7/198 6/5 2/55 5/4 5/132 9/7 3/77 22/17 2/51 13/10 13/330 4/3 4/99 11/8 1/24 7/5 7/165 17/12 17/396 10/7 10/231 3/2 1/22 11/7 1/21 8/5 8/165 5/3 5/99 12/7 4/77 19/11 19/363 7/4 7/132 2/1 2/33 7/3 7/99 19/8 19/264 12/5 4/55 17/7 17/231 22/9 2/27 5/2 5/66 8/3 8/99 11/4 1/12 3/1 1/11 13/4 13/132 23/7 23/231 33/10 1/10 10/3 10/99 17/5 17/165 7/2 7/66 18/5 6/55 11/3 1/9 4/1 4/33 13/3 13/99 22/5 2/15 9/2 3/22 14/3 14/99 33/7 1/7 19/4 19/132 5/1 5/33 11/2 1/6 6/1 2/11 7/1 7/33 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,66,2,19) (-7/1,1/0) -> (17/5,7/2) Hyperbolic Matrix(31,198,18,115) (-7/1,-6/1) -> (12/7,19/11) Hyperbolic Matrix(25,132,-18,-95) (-6/1,-5/1) -> (-7/5,-18/13) Hyperbolic Matrix(29,132,-20,-91) (-5/1,-9/2) -> (-3/2,-13/9) Hyperbolic Matrix(89,396,20,89) (-9/2,-22/5) -> (22/5,9/2) Hyperbolic Matrix(151,660,62,271) (-22/5,-13/3) -> (17/7,22/9) Hyperbolic Matrix(47,198,14,59) (-13/3,-4/1) -> (10/3,17/5) Hyperbolic Matrix(35,132,22,83) (-4/1,-11/3) -> (11/7,8/5) Hyperbolic Matrix(37,132,-30,-107) (-11/3,-7/2) -> (-5/4,-11/9) Hyperbolic Matrix(19,66,2,7) (-7/2,-17/5) -> (7/1,1/0) Hyperbolic Matrix(59,198,14,47) (-17/5,-10/3) -> (4/1,13/3) Hyperbolic Matrix(41,132,-32,-103) (-10/3,-3/1) -> (-9/7,-14/11) Hyperbolic Matrix(23,66,8,23) (-3/1,-11/4) -> (11/4,3/1) Hyperbolic Matrix(49,132,36,97) (-11/4,-8/3) -> (4/3,11/8) Hyperbolic Matrix(25,66,14,37) (-8/3,-5/2) -> (7/4,2/1) Hyperbolic Matrix(109,264,-64,-155) (-5/2,-12/5) -> (-12/7,-17/10) Hyperbolic Matrix(83,198,70,167) (-12/5,-19/8) -> (7/6,6/5) Hyperbolic Matrix(167,396,-132,-313) (-19/8,-7/3) -> (-19/15,-5/4) Hyperbolic Matrix(29,66,18,41) (-7/3,-2/1) -> (8/5,5/3) Hyperbolic Matrix(37,66,14,25) (-2/1,-7/4) -> (5/2,8/3) Hyperbolic Matrix(305,528,-212,-367) (-7/4,-19/11) -> (-13/9,-23/16) Hyperbolic Matrix(115,198,18,31) (-19/11,-12/7) -> (6/1,7/1) Hyperbolic Matrix(389,660,300,509) (-17/10,-22/13) -> (22/17,13/10) Hyperbolic Matrix(235,396,54,91) (-22/13,-5/3) -> (13/3,22/5) Hyperbolic Matrix(41,66,18,29) (-5/3,-8/5) -> (2/1,7/3) Hyperbolic Matrix(83,132,22,35) (-8/5,-11/7) -> (11/3,4/1) Hyperbolic Matrix(43,66,28,43) (-11/7,-3/2) -> (3/2,11/7) Hyperbolic Matrix(965,1386,204,293) (-23/16,-33/23) -> (33/7,19/4) Hyperbolic Matrix(553,792,118,169) (-33/23,-10/7) -> (14/3,33/7) Hyperbolic Matrix(139,198,106,151) (-10/7,-17/12) -> (13/10,4/3) Hyperbolic Matrix(47,66,42,59) (-17/12,-7/5) -> (1/1,7/6) Hyperbolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(151,198,106,139) (-4/3,-13/10) -> (17/12,10/7) Hyperbolic Matrix(305,396,124,161) (-13/10,-22/17) -> (22/9,5/2) Hyperbolic Matrix(307,396,238,307) (-22/17,-9/7) -> (9/7,22/17) Hyperbolic Matrix(623,792,188,239) (-14/11,-33/26) -> (33/10,10/3) Hyperbolic Matrix(1093,1386,332,421) (-33/26,-19/15) -> (23/7,33/10) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(167,198,70,83) (-6/5,-7/6) -> (19/8,12/5) Hyperbolic Matrix(59,66,42,47) (-7/6,-1/1) -> (7/5,17/12) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(103,-132,32,-41) (5/4,9/7) -> (3/1,13/4) Hyperbolic Matrix(95,-132,18,-25) (11/8,7/5) -> (5/1,11/2) Hyperbolic Matrix(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(155,-264,64,-109) (5/3,12/7) -> (12/5,17/7) Hyperbolic Matrix(229,-396,48,-83) (19/11,7/4) -> (19/4,5/1) Hyperbolic Matrix(223,-528,68,-161) (7/3,19/8) -> (13/4,23/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,66,2,19) -> Matrix(7,2,66,19) Matrix(31,198,18,115) -> Matrix(31,6,594,115) Matrix(25,132,-18,-95) -> Matrix(25,4,-594,-95) Matrix(29,132,-20,-91) -> Matrix(29,4,-660,-91) Matrix(89,396,20,89) -> Matrix(89,12,660,89) Matrix(151,660,62,271) -> Matrix(151,20,2046,271) Matrix(47,198,14,59) -> Matrix(47,6,462,59) Matrix(35,132,22,83) -> Matrix(35,4,726,83) Matrix(37,132,-30,-107) -> Matrix(37,4,-990,-107) Matrix(19,66,2,7) -> Matrix(19,2,66,7) Matrix(59,198,14,47) -> Matrix(59,6,462,47) Matrix(41,132,-32,-103) -> Matrix(41,4,-1056,-103) Matrix(23,66,8,23) -> Matrix(23,2,264,23) Matrix(49,132,36,97) -> Matrix(49,4,1188,97) Matrix(25,66,14,37) -> Matrix(25,2,462,37) Matrix(109,264,-64,-155) -> Matrix(109,8,-2112,-155) Matrix(83,198,70,167) -> Matrix(83,6,2310,167) Matrix(167,396,-132,-313) -> Matrix(167,12,-4356,-313) Matrix(29,66,18,41) -> Matrix(29,2,594,41) Matrix(37,66,14,25) -> Matrix(37,2,462,25) Matrix(305,528,-212,-367) -> Matrix(305,16,-6996,-367) Matrix(115,198,18,31) -> Matrix(115,6,594,31) Matrix(389,660,300,509) -> Matrix(389,20,9900,509) Matrix(235,396,54,91) -> Matrix(235,12,1782,91) Matrix(41,66,18,29) -> Matrix(41,2,594,29) Matrix(83,132,22,35) -> Matrix(83,4,726,35) Matrix(43,66,28,43) -> Matrix(43,2,924,43) Matrix(965,1386,204,293) -> Matrix(965,42,6732,293) Matrix(553,792,118,169) -> Matrix(553,24,3894,169) Matrix(139,198,106,151) -> Matrix(139,6,3498,151) Matrix(47,66,42,59) -> Matrix(47,2,1386,59) Matrix(191,264,34,47) -> Matrix(191,8,1122,47) Matrix(97,132,36,49) -> Matrix(97,4,1188,49) Matrix(151,198,106,139) -> Matrix(151,6,3498,139) Matrix(305,396,124,161) -> Matrix(305,12,4092,161) Matrix(307,396,238,307) -> Matrix(307,12,7854,307) Matrix(623,792,188,239) -> Matrix(623,24,6204,239) Matrix(1093,1386,332,421) -> Matrix(1093,42,10956,421) Matrix(217,264,60,73) -> Matrix(217,8,1980,73) Matrix(167,198,70,83) -> Matrix(167,6,2310,83) Matrix(59,66,42,47) -> Matrix(59,2,1386,47) Matrix(1,0,2,1) -> Matrix(1,0,66,1) Matrix(107,-132,30,-37) -> Matrix(107,-4,990,-37) Matrix(103,-132,32,-41) -> Matrix(103,-4,1056,-41) Matrix(95,-132,18,-25) -> Matrix(95,-4,594,-25) Matrix(91,-132,20,-29) -> Matrix(91,-4,660,-29) Matrix(155,-264,64,-109) -> Matrix(155,-8,2112,-109) Matrix(229,-396,48,-83) -> Matrix(229,-12,1584,-83) Matrix(223,-528,68,-161) -> Matrix(223,-16,2244,-161) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 12 Genus: 19 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 48 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,11,19,5,18,39,31,47,44,20,40,16,15,24,13,4,3,12,21,10,9,32,25,42,48,35,34,27,8,26,14,7)(17,37,36,38,28,30,29,22,23,41,33)(43,45,46); (1,4,16,41,20,19,28,27,44,29,32,45,22,6,12,37,48,42,23,14,13,36,46,34,38,31,9,30,15,39,17,5,2)(3,10,35,40,43,18,25,8,7,24,11)(21,26,33); (1,2,8,28,34,10,33,16,35,37,39,43,36,12,11,22,44,47,38,13,7,23,45,40,41,42,18,17,26,25,29,9,3)(4,14,21,6,5,20,27,46,32,31,15)(19,24,30)) ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 4 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 3/1 11/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/33 4/3 4/99 3/2 1/22 11/7 1/21 8/5 8/165 5/3 5/99 7/4 7/132 2/1 2/33 5/2 5/66 8/3 8/99 11/4 1/12 3/1 1/11 7/2 7/66 11/3 1/9 4/1 4/33 5/1 5/33 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(26,-33,15,-19) (1/1,4/3) -> (5/3,7/4) Hyperbolic Matrix(23,-33,7,-10) (4/3,3/2) -> (3/1,7/2) Hyperbolic Matrix(43,-66,15,-23) (3/2,11/7) -> (11/4,3/1) Hyperbolic Matrix(104,-165,29,-46) (11/7,8/5) -> (7/2,11/3) Hyperbolic Matrix(41,-66,23,-37) (8/5,5/3) -> (7/4,2/1) Hyperbolic Matrix(14,-33,3,-7) (2/1,5/2) -> (4/1,5/1) Hyperbolic Matrix(13,-33,2,-5) (5/2,8/3) -> (5/1,1/0) Hyperbolic Matrix(49,-132,13,-35) (8/3,11/4) -> (11/3,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,33,1) Matrix(26,-33,15,-19) -> Matrix(26,-1,495,-19) Matrix(23,-33,7,-10) -> Matrix(23,-1,231,-10) Matrix(43,-66,15,-23) -> Matrix(43,-2,495,-23) Matrix(104,-165,29,-46) -> Matrix(104,-5,957,-46) Matrix(41,-66,23,-37) -> Matrix(41,-2,759,-37) Matrix(14,-33,3,-7) -> Matrix(14,-1,99,-7) Matrix(13,-33,2,-5) -> Matrix(13,-1,66,-5) Matrix(49,-132,13,-35) -> Matrix(49,-4,429,-35) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 48 Minimal number of generators: 9 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: 4 Genus: 3 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 33 1 2/1 2/33 1 33 5/2 5/66 1 33 8/3 8/99 1 33 11/4 1/12 11 3 3/1 1/11 3 11 11/3 1/9 11 3 4/1 4/33 1 33 5/1 5/33 1 33 1/0 1/0 1 33 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(14,-33,3,-7) (2/1,5/2) -> (4/1,5/1) Hyperbolic Matrix(13,-33,2,-5) (5/2,8/3) -> (5/1,1/0) Hyperbolic Matrix(49,-132,13,-35) (8/3,11/4) -> (11/3,4/1) Hyperbolic Matrix(23,-66,8,-23) (11/4,3/1) -> (11/4,3/1) Reflection Matrix(10,-33,3,-10) (3/1,11/3) -> (3/1,11/3) 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(1,0,33,-1) (0/1,2/1) -> (0/1,2/33) Matrix(14,-33,3,-7) -> Matrix(14,-1,99,-7) Matrix(13,-33,2,-5) -> Matrix(13,-1,66,-5) Matrix(49,-132,13,-35) -> Matrix(49,-4,429,-35) Matrix(23,-66,8,-23) -> Matrix(23,-2,264,-23) (11/4,3/1) -> (1/12,1/11) Matrix(10,-33,3,-10) -> Matrix(10,-1,99,-10) (3/1,11/3) -> (1/11,1/9) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.