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 -6/7 -5/1 -3/4 -9/2 -1/1 -13/3 -3/4 -4/1 -1/2 -15/4 0/1 -11/3 1/0 -7/2 -1/1 -1/2 -10/3 -1/2 -3/1 -1/2 1/0 -14/5 -1/2 -25/9 -1/2 -36/13 0/1 -11/4 -1/2 0/1 -8/3 -1/2 -5/2 -1/1 0/1 -12/5 0/1 -7/3 1/0 -9/4 0/1 -11/5 1/0 -2/1 1/0 -13/7 1/0 -24/13 -2/1 -11/6 -2/1 -1/1 -9/5 1/0 -7/4 -2/1 -1/1 -12/7 -1/1 -5/3 1/0 -3/2 -1/1 -7/5 -1/2 -18/13 0/1 -11/8 0/1 1/0 -4/3 1/0 -5/4 -2/1 -1/1 -6/5 -1/1 -7/6 -1/1 -1/2 -1/1 1/0 -6/7 -2/1 -5/6 -2/1 -5/3 -4/5 -3/2 -3/4 -2/1 -8/11 -7/4 -13/18 -12/7 -5/3 -5/7 -3/2 -12/17 -5/3 -7/10 -5/3 -3/2 -2/3 -3/2 -11/17 -3/2 -9/14 -7/5 -7/11 -11/8 -12/19 -4/3 -5/8 -4/3 -1/1 -8/13 -3/2 -11/18 -2/1 -1/1 -3/5 -3/2 -7/12 -7/5 -4/3 -18/31 -4/3 -11/19 -5/4 -4/7 -3/2 -9/16 -4/3 -5/9 -3/2 -6/11 -4/3 -1/2 -4/3 -1/1 0/1 -1/1 1/2 -1/1 -4/5 6/11 -4/5 5/9 -3/4 4/7 -3/4 11/19 -5/6 7/12 -4/5 -7/9 3/5 -3/4 5/8 -1/1 -4/5 7/11 -11/14 2/3 -3/4 7/10 -3/4 -5/7 5/7 -3/4 3/4 -2/3 7/9 -1/2 11/14 -1/1 -4/5 4/5 -3/4 1/1 -1/2 6/5 -1/1 5/4 -1/1 -2/3 9/7 -1/2 13/10 -1/1 0/1 4/3 -1/2 3/2 -1/1 8/5 -3/4 13/8 -3/4 -2/3 18/11 -2/3 5/3 -1/2 17/10 -1/1 -1/2 12/7 -1/1 7/4 -1/1 -2/3 9/5 -1/2 11/6 -1/1 -2/3 2/1 -1/2 13/6 -1/3 0/1 24/11 0/1 11/5 -1/2 9/4 0/1 16/7 -1/2 7/3 -1/2 19/8 -1/5 0/1 12/5 0/1 5/2 -1/1 0/1 18/7 0/1 31/12 0/1 1/5 13/5 1/2 8/3 1/0 11/4 0/1 1/0 3/1 -1/2 1/0 13/4 0/1 1/0 36/11 0/1 23/7 1/0 10/3 1/0 7/2 -1/1 1/0 18/5 -1/1 11/3 -1/2 15/4 0/1 4/1 1/0 13/3 -3/2 9/2 -1/1 23/5 1/0 14/3 1/0 5/1 -3/2 11/2 -6/5 -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(13,12,-12,-11) Matrix(25,132,32,169) -> Matrix(3,2,-2,-1) Matrix(23,108,-36,-169) -> Matrix(17,10,-12,-7) Matrix(49,216,-76,-335) -> Matrix(31,24,-22,-17) Matrix(23,96,40,167) -> Matrix(1,2,-2,-3) Matrix(73,276,32,121) -> Matrix(1,0,0,1) Matrix(71,264,32,119) -> Matrix(1,0,-2,1) Matrix(23,84,-20,-73) -> Matrix(1,0,0,1) Matrix(25,84,36,121) -> Matrix(13,8,-18,-11) Matrix(23,72,-8,-25) -> Matrix(1,0,0,1) Matrix(241,672,52,145) -> Matrix(1,0,2,1) Matrix(359,996,164,455) -> Matrix(1,0,0,1) Matrix(313,864,96,265) -> Matrix(1,0,2,1) Matrix(71,192,44,119) -> Matrix(1,2,-2,-3) Matrix(23,60,-28,-73) -> Matrix(7,2,-4,-1) Matrix(49,120,20,49) -> Matrix(1,0,0,1) Matrix(71,168,-112,-265) -> Matrix(11,-4,-8,3) Matrix(47,108,-84,-193) -> Matrix(3,-4,-2,3) Matrix(119,264,32,71) -> Matrix(1,0,-2,1) Matrix(23,48,-12,-25) -> Matrix(1,-2,0,1) Matrix(551,1020,168,311) -> Matrix(1,2,0,1) Matrix(313,576,144,265) -> Matrix(1,2,-4,-7) Matrix(119,216,92,167) -> Matrix(1,2,-2,-3) Matrix(47,84,80,143) -> Matrix(3,10,-4,-13) Matrix(97,168,56,97) -> Matrix(3,4,-4,-5) Matrix(71,120,-100,-169) -> Matrix(3,8,-2,-5) Matrix(23,36,-16,-25) -> Matrix(1,2,-2,-3) Matrix(95,132,172,239) -> Matrix(11,4,-14,-5) Matrix(313,432,192,265) -> Matrix(3,-2,-4,3) Matrix(97,132,36,49) -> Matrix(1,0,0,1) Matrix(47,60,-76,-97) -> Matrix(3,2,-2,-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(5,14,-4,-11) Matrix(143,120,56,47) -> Matrix(1,2,-4,-7) Matrix(47,36,-64,-49) -> Matrix(3,8,-2,-5) Matrix(215,156,164,119) -> Matrix(7,12,-10,-17) Matrix(167,120,32,23) -> Matrix(11,18,-8,-13) Matrix(409,288,240,169) -> Matrix(1,2,-4,-7) Matrix(121,84,36,25) -> Matrix(5,8,-2,-3) Matrix(239,156,72,47) -> Matrix(7,10,2,3) Matrix(457,288,192,121) -> Matrix(3,4,-16,-21) Matrix(215,132,272,167) -> Matrix(5,6,-6,-7) Matrix(217,132,120,73) -> Matrix(3,4,-4,-5) Matrix(143,84,80,47) -> Matrix(7,10,-12,-17) Matrix(1177,684,456,265) -> Matrix(3,4,20,27) Matrix(167,96,40,23) -> Matrix(1,2,-2,-3) Matrix(169,96,44,25) -> Matrix(3,4,2,3) Matrix(217,120,132,73) -> Matrix(7,10,-12,-17) Matrix(23,12,44,23) -> Matrix(7,8,-8,-9) Matrix(1,0,4,1) -> Matrix(7,8,-8,-9) Matrix(193,-108,84,-47) -> Matrix(5,4,-14,-11) Matrix(745,-432,288,-167) -> Matrix(5,4,16,13) Matrix(97,-60,76,-47) -> Matrix(3,2,-2,-1) Matrix(265,-168,112,-71) -> Matrix(5,4,-24,-19) Matrix(169,-108,36,-23) -> Matrix(13,10,-4,-3) Matrix(169,-120,100,-71) -> Matrix(11,8,-18,-13) Matrix(49,-36,64,-47) -> Matrix(11,8,-18,-13) Matrix(73,-60,28,-23) -> Matrix(3,2,4,3) Matrix(73,-84,20,-23) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,2,-2,-3) Matrix(25,-48,12,-23) -> Matrix(3,2,-8,-5) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(73,-324,16,-71) -> Matrix(1,2,-2,-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): 18 Degree of the the map X: 18 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: 20 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -3/2 -1/1 -3/4 0/1 1/2 3/5 2/3 3/4 1/1 6/5 3/2 12/7 2/1 3/1 7/2 4/1 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 -6/7 -5/1 -3/4 -4/1 -1/2 -7/2 -1/1 -1/2 -10/3 -1/2 -3/1 -1/2 1/0 -2/1 1/0 -9/5 1/0 -7/4 -2/1 -1/1 -12/7 -1/1 -5/3 1/0 -3/2 -1/1 -7/5 -1/2 -11/8 0/1 1/0 -4/3 1/0 -5/4 -2/1 -1/1 -6/5 -1/1 -7/6 -1/1 -1/2 -1/1 1/0 -4/5 -3/2 -3/4 -2/1 -8/11 -7/4 -5/7 -3/2 -12/17 -5/3 -7/10 -5/3 -3/2 -2/3 -3/2 -3/5 -3/2 -7/12 -7/5 -4/3 -11/19 -5/4 -4/7 -3/2 -1/2 -4/3 -1/1 0/1 -1/1 1/2 -1/1 -4/5 4/7 -3/4 7/12 -4/5 -7/9 3/5 -3/4 2/3 -3/4 7/10 -3/4 -5/7 5/7 -3/4 3/4 -2/3 7/9 -1/2 11/14 -1/1 -4/5 4/5 -3/4 1/1 -1/2 6/5 -1/1 5/4 -1/1 -2/3 4/3 -1/2 3/2 -1/1 8/5 -3/4 5/3 -1/2 17/10 -1/1 -1/2 12/7 -1/1 7/4 -1/1 -2/3 9/5 -1/2 2/1 -1/2 3/1 -1/2 1/0 10/3 1/0 7/2 -1/1 1/0 18/5 -1/1 11/3 -1/2 4/1 1/0 5/1 -3/2 11/2 -6/5 -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(5,24,6,29) (-5/1,-4/1) -> (4/5,1/1) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(25,84,36,121) (-7/2,-10/3) -> (2/3,7/10) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) 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(43,60,-38,-53) (-7/5,-11/8) -> (-7/6,-1/1) Hyperbolic Matrix(19,24,34,43) (-4/3,-5/4) -> (1/2,4/7) 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(29,24,6,5) (-1/1,-4/5) -> (4/1,5/1) Hyperbolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(149,108,-258,-187) (-8/11,-5/7) -> (-11/19,-4/7) 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(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(143,84,80,47) (-3/5,-7/12) -> (7/4,9/5) Hyperbolic Matrix(269,156,50,29) (-7/12,-11/19) -> (5/1,11/2) Hyperbolic Matrix(43,24,34,19) (-4/7,-1/2) -> (5/4,4/3) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(125,-72,158,-91) (4/7,7/12) -> (11/14,4/5) 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,-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(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(23,132,4,23) -> Matrix(13,12,-12,-11) Matrix(25,132,32,169) -> Matrix(3,2,-2,-1) Matrix(5,24,6,29) -> Matrix(7,5,-10,-7) Matrix(19,72,-14,-53) -> Matrix(1,1,-2,-1) Matrix(25,84,36,121) -> Matrix(13,8,-18,-11) Matrix(19,60,6,19) -> Matrix(1,1,-2,-1) Matrix(5,12,2,5) -> Matrix(1,1,-2,-1) Matrix(19,36,10,19) -> Matrix(1,3,-2,-5) Matrix(47,84,80,143) -> Matrix(3,10,-4,-13) Matrix(97,168,56,97) -> Matrix(3,4,-4,-5) Matrix(71,120,-100,-169) -> Matrix(3,8,-2,-5) Matrix(23,36,-16,-25) -> Matrix(1,2,-2,-3) Matrix(43,60,-38,-53) -> Matrix(1,1,-2,-1) Matrix(19,24,34,43) -> Matrix(3,7,-4,-9) Matrix(49,60,40,49) -> Matrix(3,4,-4,-5) Matrix(143,168,40,47) -> Matrix(3,2,-2,-1) Matrix(29,24,6,5) -> Matrix(3,5,-2,-3) Matrix(47,36,-64,-49) -> Matrix(3,8,-2,-5) Matrix(149,108,-258,-187) -> Matrix(7,13,-6,-11) Matrix(409,288,240,169) -> Matrix(1,2,-4,-7) Matrix(121,84,36,25) -> Matrix(5,8,-2,-3) Matrix(19,12,30,19) -> Matrix(11,15,-14,-19) Matrix(143,84,80,47) -> Matrix(7,10,-12,-17) Matrix(269,156,50,29) -> Matrix(13,17,-10,-13) Matrix(43,24,34,19) -> Matrix(5,7,-8,-11) Matrix(1,0,4,1) -> Matrix(7,8,-8,-9) Matrix(125,-72,158,-91) -> Matrix(11,9,-16,-13) Matrix(169,-120,100,-71) -> Matrix(11,8,-18,-13) Matrix(49,-36,64,-47) -> Matrix(11,8,-18,-13) Matrix(73,-84,20,-23) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,2,-2,-3) Matrix(67,-108,18,-29) -> Matrix(1,1,-4,-3) 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): 18 ----------------------------------------------------------------------- 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 4 2 1/2 (-1/1,-4/5) 0 12 4/7 -3/4 1 6 7/12 (-4/5,-7/9) 0 12 3/5 -3/4 1 4 2/3 -3/4 3 6 7/10 (-3/4,-5/7) 0 12 5/7 -3/4 1 12 3/4 -2/3 2 4 7/9 -1/2 1 12 11/14 (-1/1,-4/5) 0 12 4/5 -3/4 1 6 1/1 -1/2 1 12 6/5 -1/1 3 2 5/4 (-1/1,-2/3) 0 12 4/3 -1/2 1 6 3/2 -1/1 2 4 8/5 -3/4 1 6 5/3 -1/2 1 12 17/10 (-1/1,-1/2) 0 12 12/7 -1/1 1 2 7/4 (-1/1,-2/3) 0 12 9/5 -1/2 1 4 2/1 -1/2 1 6 3/1 (-1/2,1/0) 0 4 10/3 1/0 3 6 7/2 (-1/1,1/0) 0 12 18/5 -1/1 3 2 11/3 -1/2 1 12 4/1 1/0 1 6 5/1 -3/2 1 12 11/2 (-6/5,-1/1) 0 12 6/1 -1/1 6 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(43,-24,34,-19) (1/2,4/7) -> (5/4,4/3) Glide Reflection Matrix(125,-72,158,-91) (4/7,7/12) -> (11/14,4/5) Hyperbolic Matrix(143,-84,80,-47) (7/12,3/5) -> (7/4,9/5) Glide Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(121,-84,36,-25) (2/3,7/10) -> (10/3,7/2) Glide Reflection 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(169,-132,32,-25) (7/9,11/14) -> (5/1,11/2) Glide Reflection Matrix(29,-24,6,-5) (4/5,1/1) -> (4/1,5/1) Glide Reflection 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(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(239,-408,140,-239) (17/10,12/7) -> (17/10,12/7) Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection 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(9,8,-10,-9) (0/1,1/2) -> (-1/1,-4/5) Matrix(43,-24,34,-19) -> Matrix(9,7,-14,-11) Matrix(125,-72,158,-91) -> Matrix(11,9,-16,-13) -3/4 Matrix(143,-84,80,-47) -> Matrix(13,10,-22,-17) Matrix(19,-12,30,-19) -> Matrix(19,15,-24,-19) (3/5,2/3) -> (-5/6,-3/4) Matrix(121,-84,36,-25) -> Matrix(11,8,-4,-3) Matrix(169,-120,100,-71) -> Matrix(11,8,-18,-13) -2/3 Matrix(49,-36,64,-47) -> Matrix(11,8,-18,-13) -2/3 Matrix(169,-132,32,-25) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(29,-24,6,-5) -> Matrix(7,5,-4,-3) Matrix(73,-84,20,-23) -> Matrix(1,0,0,1) Matrix(49,-60,40,-49) -> Matrix(5,4,-6,-5) (6/5,5/4) -> (-1/1,-2/3) Matrix(25,-36,16,-23) -> Matrix(1,2,-2,-3) -1/1 Matrix(67,-108,18,-29) -> Matrix(1,1,-4,-3) -1/2 Matrix(239,-408,140,-239) -> Matrix(3,2,-4,-3) (17/10,12/7) -> (-1/1,-1/2) Matrix(97,-168,56,-97) -> Matrix(5,4,-6,-5) (12/7,7/4) -> (-1/1,-2/3) Matrix(19,-36,10,-19) -> Matrix(5,3,-8,-5) (9/5,2/1) -> (-3/4,-1/2) Matrix(5,-12,2,-5) -> Matrix(1,1,0,-1) (2/1,3/1) -> (-1/2,1/0) Matrix(19,-60,6,-19) -> Matrix(1,1,0,-1) (3/1,10/3) -> (-1/2,1/0) Matrix(71,-252,20,-71) -> Matrix(1,2,0,-1) (7/2,18/5) -> (-1/1,1/0) Matrix(23,-132,4,-23) -> Matrix(11,12,-10,-11) (11/2,6/1) -> (-6/5,-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.