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: 12 Genus: 11 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/2 8/13 1/1 13/8 2/1 5/2 11/3 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/42 -4/9 5/189 -3/7 4/147 -2/5 1/35 -5/13 8/273 -8/21 13/441 -3/8 5/168 -1/3 2/63 -2/7 5/147 -5/18 13/378 -3/11 8/231 -4/15 11/315 -1/4 1/28 -1/5 4/105 -2/11 3/77 -1/6 5/126 0/1 1/21 1/5 2/35 1/4 5/84 1/3 4/63 3/8 11/168 2/5 1/15 1/2 1/14 3/5 8/105 8/13 1/13 13/21 34/441 5/8 13/168 7/11 6/77 2/3 5/63 7/10 17/210 12/17 29/357 5/7 4/49 3/4 1/12 1/1 2/21 5/4 3/28 4/3 1/9 7/5 4/35 17/12 29/252 10/7 17/147 3/2 5/42 8/5 13/105 13/8 1/8 18/11 29/231 5/3 8/63 7/4 11/84 9/5 2/15 11/6 17/126 2/1 1/7 7/3 10/63 5/2 1/6 13/5 6/35 8/3 11/63 3/1 4/21 7/2 3/14 18/5 23/105 11/3 2/9 15/4 19/84 19/5 8/35 4/1 5/21 9/2 11/42 23/5 4/15 14/3 17/63 5/1 2/7 6/1 1/3 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(47,22,32,15) (-1/2,-4/9) -> (10/7,3/2) Hyperbolic Matrix(55,24,-204,-89) (-4/9,-3/7) -> (-3/11,-4/15) Hyperbolic Matrix(49,20,22,9) (-3/7,-2/5) -> (2/1,7/3) Hyperbolic Matrix(31,12,-168,-65) (-2/5,-5/13) -> (-1/5,-2/11) Hyperbolic Matrix(339,130,206,79) (-5/13,-8/21) -> (18/11,5/3) Hyperbolic Matrix(437,166,308,117) (-8/21,-3/8) -> (17/12,10/7) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(45,14,16,5) (-1/3,-2/7) -> (8/3,3/1) Hyperbolic Matrix(265,74,376,105) (-2/7,-5/18) -> (7/10,12/17) Hyperbolic Matrix(289,80,466,129) (-5/18,-3/11) -> (13/21,5/8) Hyperbolic Matrix(177,46,50,13) (-4/15,-1/4) -> (7/2,18/5) Hyperbolic Matrix(29,6,24,5) (-1/4,-1/5) -> (1/1,5/4) Hyperbolic Matrix(133,24,72,13) (-2/11,-1/6) -> (11/6,2/1) Hyperbolic Matrix(19,2,28,3) (-1/6,0/1) -> (2/3,7/10) Hyperbolic Matrix(53,-8,20,-3) (0/1,1/5) -> (13/5,8/3) Hyperbolic Matrix(53,-12,84,-19) (1/5,1/4) -> (5/8,7/11) Hyperbolic Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(53,-20,8,-3) (3/8,2/5) -> (6/1,1/0) Hyperbolic Matrix(33,-14,26,-11) (2/5,1/2) -> (5/4,4/3) Hyperbolic Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(209,-128,338,-207) (3/5,8/13) -> (8/13,13/21) Parabolic Matrix(169,-108,36,-23) (7/11,2/3) -> (14/3,5/1) Hyperbolic Matrix(351,-248,92,-65) (12/17,5/7) -> (19/5,4/1) Hyperbolic Matrix(87,-64,34,-25) (5/7,3/4) -> (5/2,13/5) Hyperbolic Matrix(33,-26,14,-11) (3/4,1/1) -> (7/3,5/2) Hyperbolic Matrix(45,-62,8,-11) (4/3,7/5) -> (5/1,6/1) Hyperbolic Matrix(303,-428,80,-113) (7/5,17/12) -> (15/4,19/5) Hyperbolic Matrix(53,-84,12,-19) (3/2,8/5) -> (4/1,9/2) Hyperbolic Matrix(209,-338,128,-207) (8/5,13/8) -> (13/8,18/11) Parabolic Matrix(119,-212,32,-57) (7/4,9/5) -> (11/3,15/4) Hyperbolic Matrix(183,-334,40,-73) (9/5,11/6) -> (9/2,23/5) Hyperbolic Matrix(157,-568,34,-123) (18/5,11/3) -> (23/5,14/3) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,42,1) Matrix(47,22,32,15) -> Matrix(79,-2,672,-17) Matrix(55,24,-204,-89) -> Matrix(149,-4,4284,-115) Matrix(49,20,22,9) -> Matrix(71,-2,462,-13) Matrix(31,12,-168,-65) -> Matrix(137,-4,3528,-103) Matrix(339,130,206,79) -> Matrix(545,-16,4326,-127) Matrix(437,166,308,117) -> Matrix(745,-22,6468,-191) Matrix(17,6,48,17) -> Matrix(65,-2,1008,-31) Matrix(45,14,16,5) -> Matrix(61,-2,336,-11) Matrix(265,74,376,105) -> Matrix(641,-22,7896,-271) Matrix(289,80,466,129) -> Matrix(755,-26,9786,-337) Matrix(177,46,50,13) -> Matrix(227,-8,1050,-37) Matrix(29,6,24,5) -> Matrix(53,-2,504,-19) Matrix(133,24,72,13) -> Matrix(205,-8,1512,-59) Matrix(19,2,28,3) -> Matrix(47,-2,588,-25) Matrix(53,-8,20,-3) -> Matrix(73,-4,420,-23) Matrix(53,-12,84,-19) -> Matrix(137,-8,1764,-103) Matrix(41,-12,24,-7) -> Matrix(65,-4,504,-31) Matrix(53,-20,8,-3) -> Matrix(61,-4,168,-11) Matrix(33,-14,26,-11) -> Matrix(59,-4,546,-37) Matrix(41,-24,12,-7) -> Matrix(53,-4,252,-19) Matrix(209,-128,338,-207) -> Matrix(547,-42,7098,-545) Matrix(169,-108,36,-23) -> Matrix(205,-16,756,-59) Matrix(351,-248,92,-65) -> Matrix(443,-36,1932,-157) Matrix(87,-64,34,-25) -> Matrix(121,-10,714,-59) Matrix(33,-26,14,-11) -> Matrix(47,-4,294,-25) Matrix(45,-62,8,-11) -> Matrix(53,-6,168,-19) Matrix(303,-428,80,-113) -> Matrix(383,-44,1680,-193) Matrix(53,-84,12,-19) -> Matrix(65,-8,252,-31) Matrix(209,-338,128,-207) -> Matrix(337,-42,2688,-335) Matrix(119,-212,32,-57) -> Matrix(151,-20,672,-89) Matrix(183,-334,40,-73) -> Matrix(223,-30,840,-113) Matrix(157,-568,34,-123) -> Matrix(191,-42,714,-157) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 12 Genus: 11 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 32 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,20,11,22,31,26,10,9,14,29,30,24,27,25,18,5,17,21,7,2)(3,12,8,23,28,13,4)(15,19,16); (1,4,16,21,23,29,14,13,6,19,8,7,11,3,10,18,32,22,24,12,5)(9,15,30,31,17,20,25)(26,28,27); (1,2,8,24,15,4,14,25,28,21,31,32,18,20,13,26,30,23,19,9,3)(5,10,27,22,7,16,6)(11,17,12)) ----------------------------------------------------------------------- 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): 32 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 2 Number of equivalence classes of cusps: 4 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 2/1 5/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 1/21 1/3 4/63 2/5 1/15 1/2 1/14 1/1 2/21 4/3 1/9 3/2 5/42 2/1 1/7 5/2 1/6 3/1 4/21 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(4,-1,13,-3) (0/1,2/7) -> (1/5,1/3) Elliptic Matrix(27,-10,19,-7) (1/3,2/5) -> (4/3,3/2) Hyperbolic Matrix(20,-9,9,-4) (2/5,1/2) -> (2/1,5/2) Hyperbolic Matrix(8,-5,5,-3) (1/2,1/1) -> (3/2,2/1) Hyperbolic Matrix(14,-17,5,-6) (1/1,4/3) -> (5/2,3/1) Hyperbolic Matrix(4,-13,1,-3) (3/1,5/1) -> (7/2,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,21,1) Matrix(4,-1,13,-3) -> Matrix(17,-1,273,-16) Matrix(27,-10,19,-7) -> Matrix(46,-3,399,-26) Matrix(20,-9,9,-4) -> Matrix(29,-2,189,-13) Matrix(8,-5,5,-3) -> Matrix(13,-1,105,-8) Matrix(14,-17,5,-6) -> Matrix(19,-2,105,-11) Matrix(4,-13,1,-3) -> Matrix(5,-1,21,-4) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 32 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 2 Number of equivalence classes of cusps: 4 Genus: 1 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 -1/1 0/1 21 1 1/1 2/21 1 21 4/3 1/9 7 3 2/1 1/7 3 7 5/2 1/6 7 3 3/1 4/21 1 21 1/0 1/0 1 21 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(14,-17,5,-6) (1/1,4/3) -> (5/2,3/1) Hyperbolic Matrix(5,-8,3,-5) (4/3,2/1) -> (4/3,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(4,-13,1,-3) (3/1,5/1) -> (7/2,1/0) Elliptic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,21,-1) (-1/1,1/1) -> (0/1,2/21) Matrix(14,-17,5,-6) -> Matrix(19,-2,105,-11) Matrix(5,-8,3,-5) -> Matrix(8,-1,63,-8) (4/3,2/1) -> (1/9,1/7) Matrix(9,-20,4,-9) -> Matrix(13,-2,84,-13) (2/1,5/2) -> (1/7,1/6) Matrix(4,-13,1,-3) -> Matrix(5,-1,21,-4) (1/5,1/3).(0/1,2/9).(1/6,1/4) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.