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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -9/2 -4/1 -3/1 -9/4 -2/1 -3/2 -4/3 -1/1 -6/7 -3/4 -2/3 -3/5 0/1 1/2 6/11 3/5 2/3 3/4 1/1 6/5 5/4 4/3 3/2 12/7 7/4 2/1 24/11 9/4 12/5 5/2 11/4 3/1 7/2 4/1 9/2 5/1 11/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -11/2 -1/1 -7/8 -5/1 -4/5 -2/3 -9/2 -1/1 -2/3 -1/2 -13/3 -2/3 0/1 -4/1 -1/1 -15/4 -2/3 -11/3 -2/3 0/1 -7/2 -1/1 -2/3 -10/3 -2/3 -3/1 -2/3 0/1 -14/5 -2/3 -25/9 -2/3 -4/7 -36/13 -1/2 -11/4 -2/3 -1/2 -8/3 -1/2 -5/2 -1/2 0/1 -12/5 0/1 -7/3 -2/3 0/1 -9/4 0/1 -11/5 -2/3 0/1 -2/1 0/1 -13/7 0/1 2/1 -24/13 1/0 -11/6 -1/1 1/0 -9/5 0/1 -7/4 0/1 1/0 -12/7 0/1 -5/3 0/1 2/1 -3/2 -1/1 0/1 1/0 -7/5 0/1 2/1 -18/13 1/0 -11/8 -2/1 1/0 -4/3 1/0 -5/4 -2/1 -1/1 -6/5 -1/1 -7/6 -1/1 0/1 -1/1 -2/1 0/1 -6/7 1/0 -5/6 -2/1 1/0 -4/5 1/0 -3/4 -2/1 -8/11 -3/2 -13/18 -5/4 -1/1 -5/7 -2/1 0/1 -12/17 -2/1 -7/10 -2/1 -1/1 -2/3 -2/1 -11/17 -2/1 -4/3 -9/14 -2/1 -3/2 -1/1 -7/11 -2/1 0/1 -12/19 -2/1 -5/8 -2/1 -1/1 -8/13 1/0 -11/18 -3/1 -5/2 -3/5 -2/1 -7/12 -2/1 -7/4 -18/31 -7/4 -11/19 -2/1 -12/7 -4/7 -5/3 -9/16 -8/5 -5/9 -8/5 -14/9 -6/11 -3/2 -1/2 -3/2 -1/1 0/1 -1/1 1/2 -1/1 -3/4 6/11 -3/4 5/9 -14/19 -8/11 4/7 -5/7 11/19 -12/17 -2/3 7/12 -7/10 -2/3 3/5 -2/3 5/8 -1/1 -2/3 7/11 -2/3 0/1 2/3 -2/3 7/10 -1/1 -2/3 5/7 -2/3 0/1 3/4 -2/3 7/9 -2/3 -8/13 11/14 -3/5 -7/12 4/5 -1/2 1/1 -2/3 0/1 6/5 -1/1 5/4 -1/1 -2/3 9/7 -2/3 13/10 -5/8 -3/5 4/3 -1/2 3/2 -1/1 -1/2 0/1 8/5 -1/2 13/8 -2/3 -1/2 18/11 -1/2 5/3 -2/5 0/1 17/10 -1/3 0/1 12/7 0/1 7/4 -1/2 0/1 9/5 0/1 11/6 -1/1 -1/2 2/1 0/1 13/6 1/1 1/0 24/11 1/0 11/5 -2/1 0/1 9/4 0/1 16/7 1/1 7/3 -2/1 0/1 19/8 -1/1 0/1 12/5 0/1 5/2 0/1 1/0 18/7 1/0 31/12 -2/1 1/0 13/5 -2/1 0/1 8/3 1/0 11/4 -2/1 1/0 3/1 -2/1 0/1 13/4 -2/1 1/0 36/11 1/0 23/7 -4/1 -2/1 10/3 -2/1 7/2 -2/1 -1/1 18/5 -1/1 11/3 -2/1 0/1 15/4 -2/1 4/1 -1/1 13/3 -2/1 0/1 9/2 -2/1 -1/1 1/0 23/5 -2/1 0/1 14/3 -2/1 5/1 -2/1 -4/3 11/2 -7/6 -1/1 6/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(23,132,4,23) (-6/1,-11/2) -> (11/2,6/1) Hyperbolic Matrix(25,132,32,169) (-11/2,-5/1) -> (7/9,11/14) Hyperbolic Matrix(23,108,-36,-169) (-5/1,-9/2) -> (-9/14,-7/11) Hyperbolic Matrix(49,216,-76,-335) (-9/2,-13/3) -> (-11/17,-9/14) Hyperbolic Matrix(23,96,40,167) (-13/3,-4/1) -> (4/7,11/19) Hyperbolic Matrix(73,276,32,121) (-4/1,-15/4) -> (9/4,16/7) Hyperbolic Matrix(71,264,32,119) (-15/4,-11/3) -> (11/5,9/4) Hyperbolic Matrix(23,84,-20,-73) (-11/3,-7/2) -> (-7/6,-1/1) Hyperbolic Matrix(25,84,36,121) (-7/2,-10/3) -> (2/3,7/10) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(241,672,52,145) (-14/5,-25/9) -> (23/5,14/3) Hyperbolic Matrix(359,996,164,455) (-25/9,-36/13) -> (24/11,11/5) Hyperbolic Matrix(313,864,96,265) (-36/13,-11/4) -> (13/4,36/11) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(23,60,-28,-73) (-8/3,-5/2) -> (-5/6,-4/5) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,-112,-265) (-12/5,-7/3) -> (-7/11,-12/19) Hyperbolic Matrix(47,108,-84,-193) (-7/3,-9/4) -> (-9/16,-5/9) Hyperbolic Matrix(119,264,32,71) (-9/4,-11/5) -> (11/3,15/4) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(551,1020,168,311) (-13/7,-24/13) -> (36/11,23/7) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(119,216,92,167) (-11/6,-9/5) -> (9/7,13/10) Hyperbolic Matrix(47,84,80,143) (-9/5,-7/4) -> (7/12,3/5) 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(47,60,-76,-97) (-4/3,-5/4) -> (-5/8,-8/13) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(95,84,-164,-145) (-1/1,-6/7) -> (-18/31,-11/19) Hyperbolic Matrix(143,120,56,47) (-6/7,-5/6) -> (5/2,18/7) Hyperbolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(215,156,164,119) (-8/11,-13/18) -> (13/10,4/3) Hyperbolic Matrix(167,120,32,23) (-13/18,-5/7) -> (5/1,11/2) Hyperbolic Matrix(409,288,240,169) (-12/17,-7/10) -> (17/10,12/7) Hyperbolic Matrix(121,84,36,25) (-7/10,-2/3) -> (10/3,7/2) Hyperbolic Matrix(239,156,72,47) (-2/3,-11/17) -> (23/7,10/3) Hyperbolic Matrix(457,288,192,121) (-12/19,-5/8) -> (19/8,12/5) Hyperbolic Matrix(215,132,272,167) (-8/13,-11/18) -> (11/14,4/5) Hyperbolic Matrix(217,132,120,73) (-11/18,-3/5) -> (9/5,11/6) Hyperbolic Matrix(143,84,80,47) (-3/5,-7/12) -> (7/4,9/5) Hyperbolic Matrix(1177,684,456,265) (-7/12,-18/31) -> (18/7,31/12) Hyperbolic Matrix(167,96,40,23) (-11/19,-4/7) -> (4/1,13/3) Hyperbolic Matrix(169,96,44,25) (-4/7,-9/16) -> (15/4,4/1) 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(745,-432,288,-167) (11/19,7/12) -> (31/12,13/5) Hyperbolic Matrix(97,-60,76,-47) (3/5,5/8) -> (5/4,9/7) Hyperbolic Matrix(265,-168,112,-71) (5/8,7/11) -> (7/3,19/8) Hyperbolic Matrix(169,-108,36,-23) (7/11,2/3) -> (14/3,5/1) Hyperbolic Matrix(169,-120,100,-71) (7/10,5/7) -> (5/3,17/10) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(73,-60,28,-23) (4/5,1/1) -> (13/5,8/3) Hyperbolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(73,-324,16,-71) (13/3,9/2) -> (9/2,23/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(23,132,4,23) -> Matrix(15,14,-14,-13) Matrix(25,132,32,169) -> Matrix(17,14,-28,-23) Matrix(23,108,-36,-169) -> Matrix(5,4,-4,-3) Matrix(49,216,-76,-335) -> Matrix(5,4,-4,-3) Matrix(23,96,40,167) -> Matrix(3,-2,-4,3) Matrix(73,276,32,121) -> Matrix(3,2,4,3) Matrix(71,264,32,119) -> Matrix(3,2,-2,-1) Matrix(23,84,-20,-73) -> Matrix(3,2,-2,-1) Matrix(25,84,36,121) -> Matrix(1,0,0,1) Matrix(23,72,-8,-25) -> Matrix(1,0,0,1) Matrix(241,672,52,145) -> Matrix(7,4,-2,-1) Matrix(359,996,164,455) -> Matrix(7,4,-2,-1) Matrix(313,864,96,265) -> Matrix(7,4,-2,-1) Matrix(71,192,44,119) -> Matrix(1,0,0,1) Matrix(23,60,-28,-73) -> Matrix(3,2,-2,-1) Matrix(49,120,20,49) -> Matrix(1,0,2,1) Matrix(71,168,-112,-265) -> Matrix(3,2,-2,-1) Matrix(47,108,-84,-193) -> Matrix(19,8,-12,-5) Matrix(119,264,32,71) -> Matrix(3,2,-2,-1) Matrix(23,48,-12,-25) -> Matrix(1,0,2,1) Matrix(551,1020,168,311) -> Matrix(1,-4,0,1) Matrix(313,576,144,265) -> Matrix(1,2,0,1) Matrix(119,216,92,167) -> Matrix(5,2,-8,-3) Matrix(47,84,80,143) -> Matrix(7,-2,-10,3) Matrix(97,168,56,97) -> Matrix(1,0,-2,1) Matrix(71,120,-100,-169) -> Matrix(1,-2,0,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(95,132,172,239) -> Matrix(3,-14,-4,19) Matrix(313,432,192,265) -> Matrix(1,4,-2,-7) Matrix(97,132,36,49) -> Matrix(1,0,0,1) Matrix(47,60,-76,-97) -> Matrix(1,0,0,1) Matrix(49,60,40,49) -> Matrix(3,4,-4,-5) Matrix(143,168,40,47) -> Matrix(3,2,-2,-1) Matrix(95,84,-164,-145) -> Matrix(7,2,-4,-1) Matrix(143,120,56,47) -> Matrix(1,2,0,1) Matrix(47,36,-64,-49) -> Matrix(3,8,-2,-5) Matrix(215,156,164,119) -> Matrix(7,10,-12,-17) Matrix(167,120,32,23) -> Matrix(3,2,-2,-1) Matrix(409,288,240,169) -> Matrix(1,2,-4,-7) Matrix(121,84,36,25) -> Matrix(1,0,0,1) Matrix(239,156,72,47) -> Matrix(5,8,-2,-3) Matrix(457,288,192,121) -> Matrix(1,2,-2,-3) Matrix(215,132,272,167) -> Matrix(1,6,-2,-11) Matrix(217,132,120,73) -> Matrix(1,2,0,1) Matrix(143,84,80,47) -> Matrix(1,2,-6,-11) Matrix(1177,684,456,265) -> Matrix(9,16,-4,-7) Matrix(167,96,40,23) -> Matrix(1,2,-4,-7) Matrix(169,96,44,25) -> Matrix(11,18,-8,-13) Matrix(217,120,132,73) -> Matrix(9,14,-20,-31) Matrix(23,12,44,23) -> Matrix(5,6,-6,-7) Matrix(1,0,4,1) -> Matrix(5,6,-6,-7) Matrix(193,-108,84,-47) -> Matrix(11,8,4,3) Matrix(745,-432,288,-167) -> Matrix(17,12,-10,-7) Matrix(97,-60,76,-47) -> Matrix(1,0,0,1) Matrix(265,-168,112,-71) -> Matrix(3,2,-2,-1) Matrix(169,-108,36,-23) -> Matrix(5,4,-4,-3) Matrix(169,-120,100,-71) -> Matrix(3,2,-8,-5) Matrix(49,-36,64,-47) -> Matrix(11,8,-18,-13) Matrix(73,-60,28,-23) -> Matrix(3,2,-2,-1) Matrix(73,-84,20,-23) -> Matrix(3,2,-2,-1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,2,1) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(73,-324,16,-71) -> Matrix(1,0,0,1) 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 Degree of the the map X: 16 Degree of the the map Y: 64 Permutation triple for Y: ((1,2)(3,10,34,61,35,11)(4,16,51,52,17,5)(6,21,38,62,43,22)(7,27,60,47,28,8)(9,26,56,20,15,32)(12,39,50,23,57,40)(13,44,25,24,45,14)(18,37,30,29,55,19)(31,46)(33,53)(36,49)(41,42)(48,59)(54,58)(63,64); (1,5,19,46,45,61,63,60,57,48,20,6)(2,8,30,59,44,62,64,51,39,31,9,3)(4,14,47,15)(7,25,52,26)(10,32,58,50,16,49,43,13,42,29,28,33)(11,37,38,12)(17,53,21,56,41,40,27,36,35,24,54,18)(22,55,34,23); (1,3,12,41,13,4)(2,6,23,58,24,7)(5,18)(8,29)(9,10)(11,36,16,15,48,30)(14,46,39,38,53,28)(17,25,59,57,34,33)(19,22,49,27,26,31)(20,21)(32,47,63,62,37,54)(35,45)(40,60)(42,56,52,64,61,55)(43,44)(50,51)) ----------------------------------------------------------------------- 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): 192 Minimal number of generators: 33 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: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -3/1 -2/1 -3/2 -1/1 -3/4 -3/5 0/1 1/2 3/5 3/4 1/1 6/5 5/4 3/2 2/1 12/5 5/2 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -5/1 -4/5 -2/3 -4/1 -1/1 -3/1 -2/3 0/1 -8/3 -1/2 -5/2 -1/2 0/1 -12/5 0/1 -7/3 -2/3 0/1 -2/1 0/1 -7/4 0/1 1/0 -12/7 0/1 -5/3 0/1 2/1 -3/2 -1/1 0/1 1/0 -7/5 0/1 2/1 -4/3 1/0 -5/4 -2/1 -1/1 -6/5 -1/1 -1/1 -2/1 0/1 -6/7 1/0 -5/6 -2/1 1/0 -4/5 1/0 -3/4 -2/1 -8/11 -3/2 -5/7 -2/1 0/1 -2/3 -2/1 -5/8 -2/1 -1/1 -3/5 -2/1 -7/12 -2/1 -7/4 -4/7 -5/3 -5/9 -8/5 -14/9 -6/11 -3/2 -1/2 -3/2 -1/1 0/1 -1/1 1/2 -1/1 -3/4 5/9 -14/19 -8/11 4/7 -5/7 3/5 -2/3 5/8 -1/1 -2/3 2/3 -2/3 5/7 -2/3 0/1 3/4 -2/3 7/9 -2/3 -8/13 4/5 -1/2 1/1 -2/3 0/1 6/5 -1/1 5/4 -1/1 -2/3 9/7 -2/3 4/3 -1/2 3/2 -1/1 -1/2 0/1 8/5 -1/2 5/3 -2/5 0/1 12/7 0/1 7/4 -1/2 0/1 2/1 0/1 7/3 -2/1 0/1 12/5 0/1 5/2 0/1 1/0 3/1 -2/1 0/1 7/2 -2/1 -1/1 18/5 -1/1 11/3 -2/1 0/1 4/1 -1/1 5/1 -2/1 -4/3 11/2 -7/6 -1/1 6/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(11,60,-20,-109) (-6/1,-5/1) -> (-5/9,-6/11) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(23,60,-28,-73) (-8/3,-5/2) -> (-5/6,-4/5) 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(11,24,16,35) (-7/3,-2/1) -> (2/3,5/7) Hyperbolic Matrix(13,24,20,37) (-2/1,-7/4) -> (5/8,2/3) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,-64,-83) (-4/3,-5/4) -> (-7/12,-4/7) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(157,132,44,37) (-6/7,-5/6) -> (7/2,18/5) Hyperbolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(83,60,148,107) (-8/11,-5/7) -> (5/9,4/7) Hyperbolic 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(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(85,48,108,61) (-4/7,-5/9) -> (7/9,4/5) Hyperbolic Matrix(133,72,24,13) (-6/11,-1/2) -> (11/2,6/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(109,-60,20,-11) (1/2,5/9) -> (5/1,11/2) Hyperbolic Matrix(83,-48,64,-37) (4/7,3/5) -> (9/7,4/3) Hyperbolic Matrix(97,-60,76,-47) (3/5,5/8) -> (5/4,9/7) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(59,-48,16,-13) (4/5,1/1) -> (11/3,4/1) Hyperbolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(11,60,-20,-109) -> Matrix(23,20,-15,-13) Matrix(13,60,8,37) -> Matrix(3,2,-5,-3) Matrix(11,36,-4,-13) -> Matrix(3,2,-5,-3) Matrix(23,60,-28,-73) -> Matrix(3,2,-2,-1) Matrix(49,120,20,49) -> Matrix(1,0,2,1) Matrix(61,144,36,85) -> Matrix(1,0,-1,1) Matrix(11,24,16,35) -> Matrix(3,2,-5,-3) Matrix(13,24,20,37) -> Matrix(1,-2,-1,3) Matrix(97,168,56,97) -> Matrix(1,0,-2,1) Matrix(85,144,36,61) -> Matrix(1,0,-1,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(35,48,8,11) -> Matrix(1,2,-1,-1) Matrix(37,48,-64,-83) -> Matrix(5,12,-3,-7) Matrix(49,60,40,49) -> Matrix(3,4,-4,-5) Matrix(11,12,-12,-13) -> Matrix(1,2,-1,-1) Matrix(157,132,44,37) -> Matrix(1,4,-1,-3) Matrix(47,36,-64,-49) -> Matrix(3,8,-2,-5) Matrix(83,60,148,107) -> Matrix(11,14,-15,-19) Matrix(35,24,16,11) -> Matrix(1,2,-1,-1) Matrix(37,24,20,13) -> Matrix(1,2,-3,-5) Matrix(59,36,-100,-61) -> Matrix(5,12,-3,-7) Matrix(85,48,108,61) -> Matrix(11,18,-19,-31) Matrix(133,72,24,13) -> Matrix(15,22,-13,-19) Matrix(1,0,4,1) -> Matrix(5,6,-6,-7) Matrix(109,-60,20,-11) -> Matrix(27,20,-23,-17) Matrix(83,-48,64,-37) -> Matrix(17,12,-27,-19) Matrix(97,-60,76,-47) -> Matrix(1,0,0,1) Matrix(49,-36,64,-47) -> Matrix(11,8,-18,-13) Matrix(59,-48,16,-13) -> Matrix(1,0,1,1) Matrix(73,-84,20,-23) -> Matrix(3,2,-2,-1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(13,-36,4,-11) -> Matrix(1,2,-1,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 ----------------------------------------------------------------------- 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 -1/1 3 2 1/2 (-1/1,-3/4) 0 12 5/9 (-3/4,-11/15).(-14/19,-8/11) 0 12 4/7 -5/7 2 6 3/5 -2/3 3 4 5/8 (-1/1,-2/3) 0 12 2/3 -2/3 1 6 5/7 (-1/1,-1/2).(-2/3,0/1) 0 12 3/4 -2/3 2 4 7/9 (-2/3,-8/13).(-5/8,-3/5) 0 12 4/5 -1/2 2 6 1/1 (-1/1,-1/2).(-2/3,0/1) 0 12 6/5 -1/1 2 2 5/4 (-1/1,-2/3) 0 12 9/7 -2/3 3 4 4/3 -1/2 2 6 3/2 0 4 8/5 -1/2 2 6 5/3 (-1/2,-1/3).(-2/5,0/1) 0 12 12/7 0/1 1 2 7/4 (-1/2,0/1) 0 12 2/1 0/1 1 6 7/3 (-2/1,0/1).(-1/1,1/0) 0 12 12/5 0/1 1 2 5/2 (0/1,1/0) 0 12 3/1 (-2/1,0/1).(-1/1,1/0) 0 4 7/2 (-2/1,-1/1) 0 12 18/5 -1/1 2 2 11/3 (-2/1,0/1).(-1/1,1/0) 0 12 4/1 -1/1 2 6 5/1 (-2/1,-4/3).(-3/2,-1/1) 0 12 11/2 (-7/6,-1/1) 0 12 6/1 -1/1 7 2 1/0 (-1/1,0/1) 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(109,-60,20,-11) (1/2,5/9) -> (5/1,11/2) Hyperbolic Matrix(85,-48,108,-61) (5/9,4/7) -> (7/9,4/5) Glide Reflection Matrix(83,-48,64,-37) (4/7,3/5) -> (9/7,4/3) Hyperbolic Matrix(97,-60,76,-47) (3/5,5/8) -> (5/4,9/7) Hyperbolic Matrix(37,-24,20,-13) (5/8,2/3) -> (7/4,2/1) Glide Reflection Matrix(35,-24,16,-11) (2/3,5/7) -> (2/1,7/3) Glide Reflection Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(59,-48,16,-13) (4/5,1/1) -> (11/3,4/1) Hyperbolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(37,-60,8,-13) (8/5,5/3) -> (4/1,5/1) Glide Reflection 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(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(23,-132,4,-23) (11/2,6/1) -> (11/2,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,4,-1) -> Matrix(7,6,-8,-7) (0/1,1/2) -> (-1/1,-3/4) Matrix(109,-60,20,-11) -> Matrix(27,20,-23,-17) Matrix(85,-48,108,-61) -> Matrix(25,18,-43,-31) Matrix(83,-48,64,-37) -> Matrix(17,12,-27,-19) -2/3 Matrix(97,-60,76,-47) -> Matrix(1,0,0,1) Matrix(37,-24,20,-13) -> Matrix(3,2,-7,-5) Matrix(35,-24,16,-11) -> Matrix(3,2,-1,-1) Matrix(49,-36,64,-47) -> Matrix(11,8,-18,-13) -2/3 Matrix(59,-48,16,-13) -> Matrix(1,0,1,1) 0/1 Matrix(73,-84,20,-23) -> Matrix(3,2,-2,-1) -1/1 Matrix(49,-60,40,-49) -> Matrix(5,4,-6,-5) (6/5,5/4) -> (-1/1,-2/3) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(37,-60,8,-13) -> Matrix(3,2,-1,-1) Matrix(85,-144,36,-61) -> Matrix(-1,0,3,1) *** -> (-2/3,0/1) Matrix(97,-168,56,-97) -> Matrix(-1,0,4,1) (12/7,7/4) -> (-1/2,0/1) Matrix(49,-120,20,-49) -> Matrix(1,0,0,-1) (12/5,5/2) -> (0/1,1/0) Matrix(13,-36,4,-11) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(71,-252,20,-71) -> Matrix(3,4,-2,-3) (7/2,18/5) -> (-2/1,-1/1) Matrix(23,-132,4,-23) -> Matrix(13,14,-12,-13) (11/2,6/1) -> (-7/6,-1/1) Matrix(-1,12,0,1) -> Matrix(-1,0,2,1) (6/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.