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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -14/3 -4/1 -18/5 -10/3 -30/11 -8/3 -5/2 -12/5 -16/7 -2/1 -5/3 -30/19 -10/7 -5/4 0/1 1/1 5/4 10/7 3/2 30/19 5/3 2/1 20/9 40/17 12/5 5/2 100/39 8/3 30/11 20/7 3/1 10/3 7/2 18/5 40/11 11/3 4/1 9/2 14/3 5/1 16/3 40/7 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 1/0 -6/1 -2/1 1/0 -17/3 -11/6 -11/2 -3/2 -5/1 -1/1 -19/4 1/0 -14/3 -2/1 -1/1 -9/2 1/0 -4/1 -1/1 -11/3 -1/2 -18/5 0/1 1/0 -7/2 1/0 -10/3 -1/1 -13/4 -3/4 -3/1 1/0 -14/5 -1/1 -2/3 -11/4 -1/2 -30/11 -1/2 -19/7 -1/2 -8/3 0/1 -21/8 1/0 -13/5 1/0 -18/7 0/1 1/0 -5/2 -1/1 -22/9 -2/3 -1/2 -39/16 -1/2 -17/7 -1/4 -29/12 1/0 -12/5 -1/1 -31/13 -1/2 -19/8 -1/2 -26/11 -1/3 0/1 -7/3 1/0 -16/7 0/1 -41/18 1/0 -25/11 -1/1 -9/4 1/0 -2/1 -1/1 0/1 -11/6 1/0 -20/11 -1/1 -9/5 -1/2 -16/9 0/1 -39/22 1/0 -23/13 -3/2 -7/4 -1/2 -40/23 0/1 -33/19 1/4 -26/15 0/1 1/1 -19/11 1/0 -50/29 1/0 -31/18 1/0 -12/7 -1/1 -29/17 -1/2 -17/10 1/2 -5/3 -1/1 -23/14 -5/8 -41/25 -1/2 -100/61 -1/2 -59/36 -1/2 -18/11 -1/2 0/1 -13/8 -1/2 -21/13 -1/2 -8/5 0/1 -19/12 1/0 -30/19 1/0 -41/26 1/0 -11/7 1/0 -14/9 -2/1 -1/1 -17/11 -5/4 -20/13 -1/1 -3/2 -1/2 -10/7 -1/1 -17/12 -5/6 -41/29 -3/4 -24/17 -2/3 -7/5 -1/2 -18/13 -1/2 0/1 -29/21 -1/4 -40/29 0/1 -11/8 1/0 -4/3 -1/1 -9/7 -1/2 -14/11 -1/1 -2/3 -19/15 -1/2 -5/4 -1/1 -21/17 -3/4 -16/13 -2/3 -11/9 -3/4 -17/14 -11/16 -40/33 -2/3 -23/19 -13/20 -6/5 -2/3 -1/2 -13/11 -1/2 -20/17 -1/2 -7/6 -1/2 -1/1 -1/2 0/1 0/1 1/1 1/2 7/6 1/2 6/5 1/2 2/3 17/14 11/16 11/9 3/4 5/4 1/1 19/15 1/2 14/11 2/3 1/1 9/7 1/2 4/3 1/1 11/8 1/0 18/13 0/1 1/2 7/5 1/2 10/7 1/1 13/9 3/2 3/2 1/2 14/9 1/1 2/1 11/7 1/0 30/19 1/0 19/12 1/0 8/5 0/1 21/13 1/2 13/8 1/2 18/11 0/1 1/2 5/3 1/1 22/13 2/1 1/0 39/23 1/0 17/10 -1/2 29/17 1/2 12/7 1/1 31/18 1/0 19/11 1/0 26/15 -1/1 0/1 7/4 1/2 16/9 0/1 41/23 1/2 25/14 1/1 9/5 1/2 2/1 0/1 1/1 11/5 1/2 20/9 1/1 9/4 1/0 16/7 0/1 39/17 1/2 23/10 3/4 7/3 1/0 40/17 0/1 33/14 1/6 26/11 0/1 1/3 19/8 1/2 50/21 1/2 31/13 1/2 12/5 1/1 29/12 1/0 17/7 1/4 5/2 1/1 23/9 5/2 41/16 1/0 100/39 1/0 59/23 1/0 18/7 0/1 1/0 13/5 1/0 21/8 1/0 8/3 0/1 19/7 1/2 30/11 1/2 41/15 1/2 11/4 1/2 14/5 2/3 1/1 17/6 5/6 20/7 1/1 3/1 1/0 10/3 1/1 17/5 5/4 41/12 3/2 24/7 2/1 7/2 1/0 18/5 0/1 1/0 29/8 -1/2 40/11 0/1 11/3 1/2 4/1 1/1 9/2 1/0 14/3 1/1 2/1 19/4 1/0 5/1 1/1 21/4 3/2 16/3 2/1 11/2 3/2 17/3 11/6 40/7 2/1 23/4 13/6 6/1 2/1 1/0 13/2 1/0 20/3 1/0 7/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,160,8,61) (-7/1,1/0) -> (13/5,21/8) Hyperbolic Matrix(19,120,-16,-101) (-7/1,-6/1) -> (-6/5,-13/11) Hyperbolic Matrix(41,240,-34,-199) (-6/1,-17/3) -> (-23/19,-6/5) Hyperbolic Matrix(121,680,50,281) (-17/3,-11/2) -> (29/12,17/7) Hyperbolic Matrix(59,320,-26,-141) (-11/2,-5/1) -> (-25/11,-9/4) Hyperbolic Matrix(141,680,-62,-299) (-5/1,-19/4) -> (-41/18,-25/11) Hyperbolic Matrix(161,760,68,321) (-19/4,-14/3) -> (26/11,19/8) Hyperbolic Matrix(61,280,22,101) (-14/3,-9/2) -> (11/4,14/5) Hyperbolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(199,720,-144,-521) (-11/3,-18/5) -> (-18/13,-29/21) Hyperbolic Matrix(101,360,62,221) (-18/5,-7/2) -> (13/8,18/11) Hyperbolic Matrix(59,200,-18,-61) (-7/2,-10/3) -> (-10/3,-13/4) Parabolic Matrix(99,320,-56,-181) (-13/4,-3/1) -> (-23/13,-7/4) Hyperbolic Matrix(99,280,-64,-181) (-3/1,-14/5) -> (-14/9,-17/11) Hyperbolic Matrix(101,280,22,61) (-14/5,-11/4) -> (9/2,14/3) Hyperbolic Matrix(541,1480,-314,-859) (-11/4,-30/11) -> (-50/29,-31/18) Hyperbolic Matrix(559,1520,-324,-881) (-30/11,-19/7) -> (-19/11,-50/29) Hyperbolic Matrix(119,320,74,199) (-19/7,-8/3) -> (8/5,21/13) Hyperbolic Matrix(121,320,76,201) (-8/3,-21/8) -> (19/12,8/5) Hyperbolic Matrix(61,160,8,21) (-21/8,-13/5) -> (7/1,1/0) Hyperbolic Matrix(139,360,100,259) (-13/5,-18/7) -> (18/13,7/5) Hyperbolic Matrix(79,200,-32,-81) (-18/7,-5/2) -> (-5/2,-22/9) Parabolic Matrix(819,2000,-500,-1221) (-22/9,-39/16) -> (-59/36,-18/11) Hyperbolic Matrix(559,1360,164,399) (-39/16,-17/7) -> (17/5,41/12) Hyperbolic Matrix(281,680,50,121) (-17/7,-29/12) -> (11/2,17/3) Hyperbolic Matrix(299,720,174,419) (-29/12,-12/5) -> (12/7,31/18) Hyperbolic Matrix(301,720,176,421) (-12/5,-31/13) -> (29/17,12/7) Hyperbolic Matrix(621,1480,-394,-939) (-31/13,-19/8) -> (-41/26,-11/7) Hyperbolic Matrix(321,760,68,161) (-19/8,-26/11) -> (14/3,19/4) Hyperbolic Matrix(441,1040,-254,-599) (-26/11,-7/3) -> (-33/19,-26/15) Hyperbolic Matrix(121,280,-86,-199) (-7/3,-16/7) -> (-24/17,-7/5) Hyperbolic Matrix(719,1640,210,479) (-16/7,-41/18) -> (41/12,24/7) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(219,400,98,179) (-11/6,-20/11) -> (20/9,9/4) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(179,320,-146,-261) (-9/5,-16/9) -> (-16/13,-11/9) Hyperbolic Matrix(541,960,102,181) (-16/9,-39/22) -> (21/4,16/3) Hyperbolic Matrix(1039,1840,406,719) (-39/22,-23/13) -> (23/9,41/16) Hyperbolic Matrix(919,1600,390,679) (-7/4,-40/23) -> (40/17,33/14) Hyperbolic Matrix(921,1600,392,681) (-40/23,-33/19) -> (7/3,40/17) Hyperbolic Matrix(439,760,346,599) (-26/15,-19/11) -> (19/15,14/11) Hyperbolic Matrix(419,720,174,299) (-31/18,-12/7) -> (12/5,29/12) Hyperbolic Matrix(421,720,176,301) (-12/7,-29/17) -> (31/13,12/5) Hyperbolic Matrix(399,680,328,559) (-29/17,-17/10) -> (17/14,11/9) Hyperbolic Matrix(119,200,-72,-121) (-17/10,-5/3) -> (-5/3,-23/14) Parabolic Matrix(1121,1840,488,801) (-23/14,-41/25) -> (39/17,23/10) Hyperbolic Matrix(6099,10000,2378,3899) (-41/25,-100/61) -> (100/39,59/23) Hyperbolic Matrix(6101,10000,2380,3901) (-100/61,-59/36) -> (41/16,100/39) Hyperbolic Matrix(221,360,62,101) (-18/11,-13/8) -> (7/2,18/5) Hyperbolic Matrix(99,160,86,139) (-13/8,-21/13) -> (1/1,7/6) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(1139,1800,-722,-1141) (-19/12,-30/19) -> (-30/19,-41/26) Parabolic Matrix(179,280,140,219) (-11/7,-14/9) -> (14/11,9/7) Hyperbolic Matrix(259,400,90,139) (-17/11,-20/13) -> (20/7,3/1) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) Hyperbolic Matrix(139,200,-98,-141) (-3/2,-10/7) -> (-10/7,-17/12) Parabolic Matrix(961,1360,566,801) (-17/12,-41/29) -> (39/23,17/10) Hyperbolic Matrix(1161,1640,652,921) (-41/29,-24/17) -> (16/9,41/23) Hyperbolic Matrix(259,360,100,139) (-7/5,-18/13) -> (18/7,13/5) Hyperbolic Matrix(1159,1600,318,439) (-29/21,-40/29) -> (40/11,11/3) Hyperbolic Matrix(1161,1600,320,441) (-40/29,-11/8) -> (29/8,40/11) Hyperbolic Matrix(59,80,14,19) (-11/8,-4/3) -> (4/1,9/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(219,280,140,179) (-9/7,-14/11) -> (14/9,11/7) Hyperbolic Matrix(599,760,346,439) (-14/11,-19/15) -> (19/11,26/15) Hyperbolic Matrix(159,200,-128,-161) (-19/15,-5/4) -> (-5/4,-21/17) Parabolic Matrix(779,960,340,419) (-21/17,-16/13) -> (16/7,39/17) Hyperbolic Matrix(559,680,328,399) (-11/9,-17/14) -> (17/10,29/17) Hyperbolic Matrix(1319,1600,230,279) (-17/14,-40/33) -> (40/7,23/4) Hyperbolic Matrix(1321,1600,232,281) (-40/33,-23/19) -> (17/3,40/7) Hyperbolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(139,160,86,99) (-7/6,-1/1) -> (21/13,13/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(199,-240,34,-41) (6/5,17/14) -> (23/4,6/1) Hyperbolic Matrix(261,-320,146,-179) (11/9,5/4) -> (25/14,9/5) Hyperbolic Matrix(539,-680,302,-381) (5/4,19/15) -> (41/23,25/14) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(221,-320,96,-139) (13/9,3/2) -> (23/10,7/3) Hyperbolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(939,-1480,394,-621) (11/7,30/19) -> (50/21,31/13) Hyperbolic Matrix(961,-1520,404,-639) (30/19,19/12) -> (19/8,50/21) Hyperbolic Matrix(121,-200,72,-119) (18/11,5/3) -> (5/3,22/13) Parabolic Matrix(1181,-2000,460,-779) (22/13,39/23) -> (59/23,18/7) Hyperbolic Matrix(859,-1480,314,-541) (31/18,19/11) -> (41/15,11/4) Hyperbolic Matrix(599,-1040,254,-441) (26/15,7/4) -> (33/14,26/11) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(41,-200,8,-39) (19/4,5/1) -> (5/1,21/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,160,8,61) -> Matrix(1,2,0,1) Matrix(19,120,-16,-101) -> Matrix(1,4,-2,-7) Matrix(41,240,-34,-199) -> Matrix(1,4,-2,-7) Matrix(121,680,50,281) -> Matrix(1,2,-2,-3) Matrix(59,320,-26,-141) -> Matrix(1,2,-2,-3) Matrix(141,680,-62,-299) -> Matrix(1,0,0,1) Matrix(161,760,68,321) -> Matrix(1,2,2,5) Matrix(61,280,22,101) -> Matrix(1,0,2,1) Matrix(19,80,14,59) -> Matrix(1,2,0,1) Matrix(21,80,16,61) -> Matrix(3,2,4,3) Matrix(199,720,-144,-521) -> Matrix(1,0,-2,1) Matrix(101,360,62,221) -> Matrix(1,0,2,1) Matrix(59,200,-18,-61) -> Matrix(3,4,-4,-5) Matrix(99,320,-56,-181) -> Matrix(3,2,-2,-1) Matrix(99,280,-64,-181) -> Matrix(5,4,-4,-3) Matrix(101,280,22,61) -> Matrix(1,0,2,1) Matrix(541,1480,-314,-859) -> Matrix(7,4,-2,-1) Matrix(559,1520,-324,-881) -> Matrix(1,0,2,1) Matrix(119,320,74,199) -> Matrix(1,0,4,1) Matrix(121,320,76,201) -> Matrix(1,0,0,1) Matrix(61,160,8,21) -> Matrix(1,2,0,1) Matrix(139,360,100,259) -> Matrix(1,0,2,1) Matrix(79,200,-32,-81) -> Matrix(1,2,-2,-3) Matrix(819,2000,-500,-1221) -> Matrix(3,2,-8,-5) Matrix(559,1360,164,399) -> Matrix(11,4,8,3) Matrix(281,680,50,121) -> Matrix(3,-2,2,-1) Matrix(299,720,174,419) -> Matrix(1,2,0,1) Matrix(301,720,176,421) -> Matrix(3,2,4,3) Matrix(621,1480,-394,-939) -> Matrix(7,4,-2,-1) Matrix(321,760,68,161) -> Matrix(5,2,2,1) Matrix(441,1040,-254,-599) -> Matrix(1,0,4,1) Matrix(121,280,-86,-199) -> Matrix(1,2,-2,-3) Matrix(719,1640,210,479) -> Matrix(3,-2,2,-1) Matrix(19,40,-10,-21) -> Matrix(1,0,0,1) Matrix(219,400,98,179) -> Matrix(1,2,0,1) Matrix(221,400,100,181) -> Matrix(3,2,4,3) Matrix(179,320,-146,-261) -> Matrix(1,2,-2,-3) Matrix(541,960,102,181) -> Matrix(3,-2,2,-1) Matrix(1039,1840,406,719) -> Matrix(1,4,0,1) Matrix(919,1600,390,679) -> Matrix(1,0,8,1) Matrix(921,1600,392,681) -> Matrix(1,0,-4,1) Matrix(439,760,346,599) -> Matrix(1,-2,2,-3) Matrix(419,720,174,299) -> Matrix(1,2,0,1) Matrix(421,720,176,301) -> Matrix(3,2,4,3) Matrix(399,680,328,559) -> Matrix(7,2,10,3) Matrix(119,200,-72,-121) -> Matrix(1,2,-2,-3) Matrix(1121,1840,488,801) -> Matrix(7,4,12,7) Matrix(6099,10000,2378,3899) -> Matrix(27,14,-2,-1) Matrix(6101,10000,2380,3901) -> Matrix(29,14,2,1) Matrix(221,360,62,101) -> Matrix(1,0,2,1) Matrix(99,160,86,139) -> Matrix(3,2,4,3) Matrix(199,320,74,119) -> Matrix(1,0,4,1) Matrix(201,320,76,121) -> Matrix(1,0,0,1) Matrix(1139,1800,-722,-1141) -> Matrix(1,-4,0,1) Matrix(179,280,140,219) -> Matrix(1,0,2,1) Matrix(259,400,90,139) -> Matrix(5,6,4,5) Matrix(261,400,92,141) -> Matrix(7,6,8,7) Matrix(139,200,-98,-141) -> Matrix(3,4,-4,-5) Matrix(961,1360,566,801) -> Matrix(5,4,-4,-3) Matrix(1161,1640,652,921) -> Matrix(3,2,10,7) Matrix(259,360,100,139) -> Matrix(1,0,2,1) Matrix(1159,1600,318,439) -> Matrix(1,0,6,1) Matrix(1161,1600,320,441) -> Matrix(1,0,-2,1) Matrix(59,80,14,19) -> Matrix(1,2,0,1) Matrix(61,80,16,21) -> Matrix(3,2,4,3) Matrix(219,280,140,179) -> Matrix(1,0,2,1) Matrix(599,760,346,439) -> Matrix(3,2,-2,-1) Matrix(159,200,-128,-161) -> Matrix(1,2,-2,-3) Matrix(779,960,340,419) -> Matrix(3,2,10,7) Matrix(559,680,328,399) -> Matrix(3,2,10,7) Matrix(1319,1600,230,279) -> Matrix(71,48,34,23) Matrix(1321,1600,232,281) -> Matrix(73,48,38,25) Matrix(339,400,50,59) -> Matrix(3,2,-2,-1) Matrix(341,400,52,61) -> Matrix(5,2,2,1) Matrix(139,160,86,99) -> Matrix(3,2,4,3) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(101,-120,16,-19) -> Matrix(7,-4,2,-1) Matrix(199,-240,34,-41) -> Matrix(7,-4,2,-1) Matrix(261,-320,146,-179) -> Matrix(3,-2,2,-1) Matrix(539,-680,302,-381) -> Matrix(1,0,0,1) Matrix(521,-720,144,-199) -> Matrix(1,0,-2,1) Matrix(141,-200,98,-139) -> Matrix(5,-4,4,-3) Matrix(221,-320,96,-139) -> Matrix(1,-2,2,-3) Matrix(181,-280,64,-99) -> Matrix(3,-4,4,-5) Matrix(939,-1480,394,-621) -> Matrix(1,-4,2,-7) Matrix(961,-1520,404,-639) -> Matrix(1,0,2,1) Matrix(121,-200,72,-119) -> Matrix(3,-2,2,-1) Matrix(1181,-2000,460,-779) -> Matrix(1,-2,0,1) Matrix(859,-1480,314,-541) -> Matrix(1,-4,2,-7) Matrix(599,-1040,254,-441) -> Matrix(1,0,4,1) Matrix(159,-280,46,-81) -> Matrix(3,-2,2,-1) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(141,-320,26,-59) -> Matrix(3,-2,2,-1) Matrix(81,-200,32,-79) -> Matrix(3,-2,2,-1) Matrix(661,-1800,242,-659) -> Matrix(9,-4,16,-7) Matrix(61,-200,18,-59) -> Matrix(5,-4,4,-3) Matrix(41,-200,8,-39) -> Matrix(3,-2,2,-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): 32 Degree of the the map X: 32 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to 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): 288 Minimal number of generators: 49 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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 5/4 4/3 10/7 30/19 8/5 5/3 12/7 2/1 20/9 40/17 12/5 5/2 100/39 8/3 30/11 20/7 3/1 10/3 24/7 18/5 40/11 4/1 9/2 14/3 5/1 16/3 40/7 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/2 7/6 1/2 6/5 1/2 2/3 17/14 11/16 11/9 3/4 5/4 1/1 19/15 1/2 14/11 2/3 1/1 9/7 1/2 4/3 1/1 11/8 1/0 18/13 0/1 1/2 7/5 1/2 10/7 1/1 13/9 3/2 3/2 1/2 14/9 1/1 2/1 11/7 1/0 30/19 1/0 19/12 1/0 8/5 0/1 21/13 1/2 13/8 1/2 18/11 0/1 1/2 5/3 1/1 22/13 2/1 1/0 39/23 1/0 17/10 -1/2 29/17 1/2 12/7 1/1 31/18 1/0 19/11 1/0 26/15 -1/1 0/1 7/4 1/2 16/9 0/1 41/23 1/2 25/14 1/1 9/5 1/2 2/1 0/1 1/1 11/5 1/2 20/9 1/1 9/4 1/0 16/7 0/1 39/17 1/2 23/10 3/4 7/3 1/0 40/17 0/1 33/14 1/6 26/11 0/1 1/3 19/8 1/2 50/21 1/2 31/13 1/2 12/5 1/1 29/12 1/0 17/7 1/4 5/2 1/1 23/9 5/2 41/16 1/0 100/39 1/0 59/23 1/0 18/7 0/1 1/0 13/5 1/0 21/8 1/0 8/3 0/1 19/7 1/2 30/11 1/2 41/15 1/2 11/4 1/2 14/5 2/3 1/1 17/6 5/6 20/7 1/1 3/1 1/0 10/3 1/1 17/5 5/4 41/12 3/2 24/7 2/1 7/2 1/0 18/5 0/1 1/0 29/8 -1/2 40/11 0/1 11/3 1/2 4/1 1/1 9/2 1/0 14/3 1/1 2/1 19/4 1/0 5/1 1/1 21/4 3/2 16/3 2/1 11/2 3/2 17/3 11/6 40/7 2/1 23/4 13/6 6/1 2/1 1/0 13/2 1/0 20/3 1/0 7/1 1/0 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(139,-160,53,-61) (1/1,7/6) -> (13/5,21/8) Hyperbolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(199,-240,34,-41) (6/5,17/14) -> (23/4,6/1) Hyperbolic Matrix(559,-680,231,-281) (17/14,11/9) -> (29/12,17/7) Hyperbolic Matrix(261,-320,146,-179) (11/9,5/4) -> (25/14,9/5) Hyperbolic Matrix(539,-680,302,-381) (5/4,19/15) -> (41/23,25/14) Hyperbolic Matrix(599,-760,253,-321) (19/15,14/11) -> (26/11,19/8) Hyperbolic Matrix(219,-280,79,-101) (14/11,9/7) -> (11/4,14/5) Hyperbolic Matrix(61,-80,45,-59) (9/7,4/3) -> (4/3,11/8) Parabolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(259,-360,159,-221) (18/13,7/5) -> (13/8,18/11) Hyperbolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(221,-320,96,-139) (13/9,3/2) -> (23/10,7/3) Hyperbolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(179,-280,39,-61) (14/9,11/7) -> (9/2,14/3) Hyperbolic Matrix(939,-1480,394,-621) (11/7,30/19) -> (50/21,31/13) Hyperbolic Matrix(961,-1520,404,-639) (30/19,19/12) -> (19/8,50/21) Hyperbolic Matrix(201,-320,125,-199) (19/12,8/5) -> (8/5,21/13) Parabolic Matrix(99,-160,13,-21) (21/13,13/8) -> (7/1,1/0) Hyperbolic Matrix(121,-200,72,-119) (18/11,5/3) -> (5/3,22/13) Parabolic Matrix(1181,-2000,460,-779) (22/13,39/23) -> (59/23,18/7) Hyperbolic Matrix(801,-1360,235,-399) (39/23,17/10) -> (17/5,41/12) Hyperbolic Matrix(399,-680,71,-121) (17/10,29/17) -> (11/2,17/3) Hyperbolic Matrix(421,-720,245,-419) (29/17,12/7) -> (12/7,31/18) Parabolic Matrix(859,-1480,314,-541) (31/18,19/11) -> (41/15,11/4) Hyperbolic Matrix(439,-760,93,-161) (19/11,26/15) -> (14/3,19/4) Hyperbolic Matrix(599,-1040,254,-441) (26/15,7/4) -> (33/14,26/11) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(921,-1640,269,-479) (16/9,41/23) -> (41/12,24/7) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(181,-400,81,-179) (11/5,20/9) -> (20/9,9/4) Parabolic Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(419,-960,79,-181) (16/7,39/17) -> (21/4,16/3) Hyperbolic Matrix(801,-1840,313,-719) (39/17,23/10) -> (23/9,41/16) Hyperbolic Matrix(681,-1600,289,-679) (7/3,40/17) -> (40/17,33/14) Parabolic Matrix(301,-720,125,-299) (31/13,12/5) -> (12/5,29/12) Parabolic Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(3901,-10000,1521,-3899) (41/16,100/39) -> (100/39,59/23) Parabolic Matrix(139,-360,39,-101) (18/7,13/5) -> (7/2,18/5) Hyperbolic Matrix(121,-320,45,-119) (21/8,8/3) -> (8/3,19/7) Parabolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(141,-400,49,-139) (17/6,20/7) -> (20/7,3/1) Parabolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(441,-1600,121,-439) (29/8,40/11) -> (40/11,11/3) Parabolic Matrix(21,-80,5,-19) (11/3,4/1) -> (4/1,9/2) Parabolic Matrix(41,-200,8,-39) (19/4,5/1) -> (5/1,21/4) Parabolic Matrix(281,-1600,49,-279) (17/3,40/7) -> (40/7,23/4) Parabolic Matrix(61,-400,9,-59) (13/2,20/3) -> (20/3,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(139,-160,53,-61) -> Matrix(3,-2,2,-1) Matrix(101,-120,16,-19) -> Matrix(7,-4,2,-1) Matrix(199,-240,34,-41) -> Matrix(7,-4,2,-1) Matrix(559,-680,231,-281) -> Matrix(3,-2,-4,3) Matrix(261,-320,146,-179) -> Matrix(3,-2,2,-1) Matrix(539,-680,302,-381) -> Matrix(1,0,0,1) Matrix(599,-760,253,-321) -> Matrix(3,-2,8,-5) Matrix(219,-280,79,-101) -> Matrix(1,0,0,1) Matrix(61,-80,45,-59) -> Matrix(3,-2,2,-1) Matrix(521,-720,144,-199) -> Matrix(1,0,-2,1) Matrix(259,-360,159,-221) -> Matrix(1,0,0,1) Matrix(141,-200,98,-139) -> Matrix(5,-4,4,-3) Matrix(221,-320,96,-139) -> Matrix(1,-2,2,-3) Matrix(181,-280,64,-99) -> Matrix(3,-4,4,-5) Matrix(179,-280,39,-61) -> Matrix(1,0,0,1) Matrix(939,-1480,394,-621) -> Matrix(1,-4,2,-7) Matrix(961,-1520,404,-639) -> Matrix(1,0,2,1) Matrix(201,-320,125,-199) -> Matrix(1,0,2,1) Matrix(99,-160,13,-21) -> Matrix(3,-2,2,-1) Matrix(121,-200,72,-119) -> Matrix(3,-2,2,-1) Matrix(1181,-2000,460,-779) -> Matrix(1,-2,0,1) Matrix(801,-1360,235,-399) -> Matrix(3,4,2,3) Matrix(399,-680,71,-121) -> Matrix(7,-2,4,-1) Matrix(421,-720,245,-419) -> Matrix(3,-2,2,-1) Matrix(859,-1480,314,-541) -> Matrix(1,-4,2,-7) Matrix(439,-760,93,-161) -> Matrix(1,2,0,1) Matrix(599,-1040,254,-441) -> Matrix(1,0,4,1) Matrix(159,-280,46,-81) -> Matrix(3,-2,2,-1) Matrix(921,-1640,269,-479) -> Matrix(7,-2,4,-1) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(181,-400,81,-179) -> Matrix(3,-2,2,-1) Matrix(141,-320,26,-59) -> Matrix(3,-2,2,-1) Matrix(419,-960,79,-181) -> Matrix(7,-2,4,-1) Matrix(801,-1840,313,-719) -> Matrix(7,-4,2,-1) Matrix(681,-1600,289,-679) -> Matrix(1,0,6,1) Matrix(301,-720,125,-299) -> Matrix(3,-2,2,-1) Matrix(81,-200,32,-79) -> Matrix(3,-2,2,-1) Matrix(3901,-10000,1521,-3899) -> Matrix(1,-14,0,1) Matrix(139,-360,39,-101) -> Matrix(1,0,0,1) Matrix(121,-320,45,-119) -> Matrix(1,0,2,1) Matrix(661,-1800,242,-659) -> Matrix(9,-4,16,-7) Matrix(141,-400,49,-139) -> Matrix(7,-6,6,-5) Matrix(61,-200,18,-59) -> Matrix(5,-4,4,-3) Matrix(441,-1600,121,-439) -> Matrix(1,0,4,1) Matrix(21,-80,5,-19) -> Matrix(3,-2,2,-1) Matrix(41,-200,8,-39) -> Matrix(3,-2,2,-1) Matrix(281,-1600,49,-279) -> Matrix(25,-48,12,-23) Matrix(61,-400,9,-59) -> Matrix(1,-2,0,1) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 0/1 2 1 2/1 (0/1,1/1) 0 10 20/9 1/1 2 1 9/4 1/0 1 20 16/7 0/1 2 5 7/3 1/0 1 20 40/17 0/1 6 1 26/11 (0/1,1/3) 0 10 19/8 1/2 1 20 31/13 1/2 1 20 12/5 1/1 2 5 29/12 1/0 1 20 17/7 1/4 1 20 5/2 1/1 1 4 23/9 5/2 1 20 41/16 1/0 1 20 100/39 1/0 14 1 18/7 (0/1,1/0) 0 10 13/5 1/0 1 20 21/8 1/0 1 20 8/3 0/1 2 5 19/7 1/2 1 20 30/11 1/2 4 2 41/15 1/2 1 20 11/4 1/2 1 20 14/5 (2/3,1/1) 0 10 20/7 1/1 6 1 3/1 1/0 1 20 10/3 1/1 4 2 17/5 5/4 1 20 41/12 3/2 1 20 24/7 2/1 2 5 7/2 1/0 1 20 18/5 (0/1,1/0) 0 10 40/11 0/1 4 1 11/3 1/2 1 20 4/1 1/1 2 5 9/2 1/0 1 20 14/3 (1/1,2/1) 0 10 19/4 1/0 1 20 5/1 1/1 1 4 21/4 3/2 1 20 16/3 2/1 2 5 11/2 3/2 1 20 17/3 11/6 1 20 40/7 2/1 12 1 6/1 (2/1,1/0) 0 10 20/3 1/0 2 1 7/1 1/0 1 20 1/0 1/0 1 20 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(19,-40,9,-19) (2/1,20/9) -> (2/1,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(121,-280,35,-81) (16/7,7/3) -> (24/7,7/2) Glide Reflection Matrix(239,-560,102,-239) (7/3,40/17) -> (7/3,40/17) Reflection Matrix(441,-1040,187,-441) (40/17,26/11) -> (40/17,26/11) Reflection Matrix(321,-760,68,-161) (26/11,19/8) -> (14/3,19/4) Glide Reflection Matrix(621,-1480,227,-541) (19/8,31/13) -> (41/15,11/4) Glide Reflection Matrix(301,-720,125,-299) (31/13,12/5) -> (12/5,29/12) Parabolic Matrix(281,-680,50,-121) (29/12,17/7) -> (11/2,17/3) Glide Reflection Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(641,-1640,188,-481) (23/9,41/16) -> (17/5,41/12) Glide Reflection Matrix(3199,-8200,1248,-3199) (41/16,100/39) -> (41/16,100/39) Reflection Matrix(701,-1800,273,-701) (100/39,18/7) -> (100/39,18/7) Reflection Matrix(139,-360,39,-101) (18/7,13/5) -> (7/2,18/5) Hyperbolic Matrix(61,-160,8,-21) (13/5,21/8) -> (7/1,1/0) Glide Reflection Matrix(121,-320,45,-119) (21/8,8/3) -> (8/3,19/7) Parabolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(101,-280,22,-61) (11/4,14/5) -> (9/2,14/3) Glide Reflection Matrix(99,-280,35,-99) (14/5,20/7) -> (14/5,20/7) Reflection Matrix(41,-120,14,-41) (20/7,3/1) -> (20/7,3/1) Reflection Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(339,-1160,64,-219) (41/12,24/7) -> (21/4,16/3) Glide Reflection Matrix(199,-720,55,-199) (18/5,40/11) -> (18/5,40/11) Reflection Matrix(241,-880,66,-241) (40/11,11/3) -> (40/11,11/3) Reflection Matrix(21,-80,5,-19) (11/3,4/1) -> (4/1,9/2) Parabolic Matrix(41,-200,8,-39) (19/4,5/1) -> (5/1,21/4) Parabolic Matrix(239,-1360,42,-239) (17/3,40/7) -> (17/3,40/7) Reflection Matrix(41,-240,7,-41) (40/7,6/1) -> (40/7,6/1) Reflection Matrix(19,-120,3,-19) (6/1,20/3) -> (6/1,20/3) Reflection Matrix(41,-280,6,-41) (20/3,7/1) -> (20/3,7/1) 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(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(19,-40,9,-19) -> Matrix(1,0,2,-1) (2/1,20/9) -> (0/1,1/1) Matrix(161,-360,72,-161) -> Matrix(-1,2,0,1) (20/9,9/4) -> (1/1,1/0) Matrix(141,-320,26,-59) -> Matrix(3,-2,2,-1) 1/1 Matrix(121,-280,35,-81) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(239,-560,102,-239) -> Matrix(1,0,0,-1) (7/3,40/17) -> (0/1,1/0) Matrix(441,-1040,187,-441) -> Matrix(1,0,6,-1) (40/17,26/11) -> (0/1,1/3) Matrix(321,-760,68,-161) -> Matrix(5,-2,2,-1) Matrix(621,-1480,227,-541) -> Matrix(7,-4,12,-7) *** -> (1/2,2/3) Matrix(301,-720,125,-299) -> Matrix(3,-2,2,-1) 1/1 Matrix(281,-680,50,-121) -> Matrix(3,2,2,1) Matrix(81,-200,32,-79) -> Matrix(3,-2,2,-1) 1/1 Matrix(641,-1640,188,-481) -> Matrix(3,-10,2,-7) Matrix(3199,-8200,1248,-3199) -> Matrix(-1,14,0,1) (41/16,100/39) -> (7/1,1/0) Matrix(701,-1800,273,-701) -> Matrix(1,0,0,-1) (100/39,18/7) -> (0/1,1/0) Matrix(139,-360,39,-101) -> Matrix(1,0,0,1) Matrix(61,-160,8,-21) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(121,-320,45,-119) -> Matrix(1,0,2,1) 0/1 Matrix(661,-1800,242,-659) -> Matrix(9,-4,16,-7) 1/2 Matrix(101,-280,22,-61) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(99,-280,35,-99) -> Matrix(5,-4,6,-5) (14/5,20/7) -> (2/3,1/1) Matrix(41,-120,14,-41) -> Matrix(-1,2,0,1) (20/7,3/1) -> (1/1,1/0) Matrix(61,-200,18,-59) -> Matrix(5,-4,4,-3) 1/1 Matrix(339,-1160,64,-219) -> Matrix(7,-12,4,-7) *** -> (3/2,2/1) Matrix(199,-720,55,-199) -> Matrix(1,0,0,-1) (18/5,40/11) -> (0/1,1/0) Matrix(241,-880,66,-241) -> Matrix(1,0,4,-1) (40/11,11/3) -> (0/1,1/2) Matrix(21,-80,5,-19) -> Matrix(3,-2,2,-1) 1/1 Matrix(41,-200,8,-39) -> Matrix(3,-2,2,-1) 1/1 Matrix(239,-1360,42,-239) -> Matrix(23,-44,12,-23) (17/3,40/7) -> (11/6,2/1) Matrix(41,-240,7,-41) -> Matrix(-1,4,0,1) (40/7,6/1) -> (2/1,1/0) Matrix(19,-120,3,-19) -> Matrix(-1,4,0,1) (6/1,20/3) -> (2/1,1/0) Matrix(41,-280,6,-41) -> Matrix(-1,2,0,1) (20/3,7/1) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.