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 -4/5 -7/10 -19/30 -3/5 -1/2 -9/20 -17/40 -5/12 -2/5 -39/100 -3/8 -11/30 -7/20 -3/10 -5/18 -11/40 -1/4 -3/14 -1/5 -3/16 -7/40 -1/6 -3/20 0/1 1/7 1/6 1/5 3/14 2/9 1/4 5/18 2/7 3/10 1/3 11/30 3/8 2/5 5/12 7/16 1/2 5/9 3/5 19/30 2/3 7/10 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -6/7 -3/2 -5/6 0/1 -14/17 1/0 -9/11 1/0 -4/5 1/0 -15/19 1/0 -11/14 -2/1 -7/9 1/0 -3/4 -1/1 -8/11 -1/2 -13/18 0/1 -5/7 -1/2 -7/10 0/1 -9/13 1/0 -2/3 1/0 -9/14 -2/3 -7/11 -1/2 -19/30 -1/2 -12/19 -1/2 -5/8 -1/1 -1/3 -13/21 -1/2 -8/13 -1/4 -11/18 0/1 -3/5 0/1 -13/22 0/1 -23/39 1/2 -10/17 1/2 -17/29 1/2 -7/12 1/1 -18/31 1/0 -11/19 1/0 -15/26 0/1 -4/7 1/0 -9/16 -1/1 1/1 -23/41 1/0 -14/25 1/0 -5/9 1/0 -1/2 0/1 -5/11 1/2 -9/20 1/1 -4/9 1/0 -7/16 -1/1 1/1 -17/39 1/0 -10/23 1/0 -3/7 1/0 -17/40 -1/1 1/1 -14/33 1/0 -11/26 0/1 -8/19 1/0 -21/50 1/0 -13/31 1/0 -5/12 -1/1 -12/29 -1/2 -7/17 -1/2 -2/5 0/1 -9/23 1/4 -16/41 1/4 -39/100 1/3 -23/59 1/2 -7/18 0/1 -5/13 1/4 -8/21 1/2 -3/8 1/3 1/1 -7/19 1/2 -11/30 1/2 -15/41 1/2 -4/11 1/2 -5/14 2/3 -6/17 5/6 -7/20 1/1 -1/3 1/0 -3/10 0/1 -5/17 1/2 -12/41 1/2 -7/24 1/3 1/1 -2/7 1/2 -5/18 0/1 -8/29 1/2 -11/40 1/3 1/1 -3/11 1/2 -1/4 1/1 -2/9 1/0 -3/14 2/1 -4/19 1/0 -1/5 1/0 -4/21 1/0 -3/16 -1/1 1/1 -2/11 1/0 -3/17 1/0 -7/40 -1/1 1/1 -4/23 1/0 -1/6 0/1 -2/13 3/4 -3/20 1/1 -1/7 3/2 0/1 1/0 1/7 -3/2 1/6 0/1 3/17 1/0 2/11 1/0 1/5 1/0 4/19 1/0 3/14 -2/1 2/9 1/0 1/4 -1/1 3/11 -1/2 5/18 0/1 2/7 -1/2 3/10 0/1 4/13 1/0 1/3 1/0 5/14 -2/3 4/11 -1/2 11/30 -1/2 7/19 -1/2 3/8 -1/1 -1/3 8/21 -1/2 5/13 -1/4 7/18 0/1 2/5 0/1 9/22 0/1 16/39 1/2 7/17 1/2 12/29 1/2 5/12 1/1 13/31 1/0 8/19 1/0 11/26 0/1 3/7 1/0 7/16 -1/1 1/1 18/41 1/0 11/25 1/0 4/9 1/0 1/2 0/1 6/11 1/2 11/20 1/1 5/9 1/0 9/16 -1/1 1/1 22/39 1/0 13/23 1/0 4/7 1/0 23/40 -1/1 1/1 19/33 1/0 15/26 0/1 11/19 1/0 29/50 1/0 18/31 1/0 7/12 -1/1 17/29 -1/2 10/17 -1/2 3/5 0/1 14/23 1/4 25/41 1/4 61/100 1/3 36/59 1/2 11/18 0/1 8/13 1/4 13/21 1/2 5/8 1/3 1/1 12/19 1/2 19/30 1/2 26/41 1/2 7/11 1/2 9/14 2/3 11/17 5/6 13/20 1/1 2/3 1/0 7/10 0/1 12/17 1/2 29/41 1/2 17/24 1/3 1/1 5/7 1/2 13/18 0/1 21/29 1/2 29/40 1/3 1/1 8/11 1/2 3/4 1/1 7/9 1/0 11/14 2/1 15/19 1/0 4/5 1/0 17/21 1/0 13/16 -1/1 1/1 9/11 1/0 14/17 1/0 33/40 -1/1 1/1 19/23 1/0 5/6 0/1 11/13 3/4 17/20 1/1 6/7 3/2 1/1 1/0 1/0 -1/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(99,86,160,139) (-1/1,-6/7) -> (8/13,13/21) Hyperbolic Matrix(19,16,-120,-101) (-6/7,-5/6) -> (-1/6,-2/13) Hyperbolic Matrix(41,34,-240,-199) (-5/6,-14/17) -> (-4/23,-1/6) Hyperbolic Matrix(399,328,680,559) (-14/17,-9/11) -> (17/29,10/17) Hyperbolic Matrix(179,146,-320,-261) (-9/11,-4/5) -> (-14/25,-5/9) Hyperbolic Matrix(381,302,-680,-539) (-4/5,-15/19) -> (-23/41,-14/25) Hyperbolic Matrix(439,346,760,599) (-15/19,-11/14) -> (15/26,11/19) Hyperbolic Matrix(179,140,280,219) (-11/14,-7/9) -> (7/11,9/14) Hyperbolic Matrix(21,16,80,61) (-7/9,-3/4) -> (1/4,3/11) Hyperbolic Matrix(19,14,80,59) (-3/4,-8/11) -> (2/9,1/4) Hyperbolic Matrix(199,144,-720,-521) (-8/11,-13/18) -> (-5/18,-8/29) Hyperbolic Matrix(139,100,360,259) (-13/18,-5/7) -> (5/13,7/18) Hyperbolic Matrix(139,98,-200,-141) (-5/7,-7/10) -> (-7/10,-9/13) Parabolic Matrix(139,96,-320,-221) (-9/13,-2/3) -> (-10/23,-3/7) Hyperbolic Matrix(99,64,-280,-181) (-2/3,-9/14) -> (-5/14,-6/17) Hyperbolic Matrix(219,140,280,179) (-9/14,-7/11) -> (7/9,11/14) Hyperbolic Matrix(621,394,-1480,-939) (-7/11,-19/30) -> (-21/50,-13/31) Hyperbolic Matrix(639,404,-1520,-961) (-19/30,-12/19) -> (-8/19,-21/50) Hyperbolic Matrix(121,76,320,201) (-12/19,-5/8) -> (3/8,8/21) Hyperbolic Matrix(119,74,320,199) (-5/8,-13/21) -> (7/19,3/8) Hyperbolic Matrix(139,86,160,99) (-13/21,-8/13) -> (6/7,1/1) Hyperbolic Matrix(101,62,360,221) (-8/13,-11/18) -> (5/18,2/7) Hyperbolic Matrix(119,72,-200,-121) (-11/18,-3/5) -> (-3/5,-13/22) Parabolic Matrix(779,460,-2000,-1181) (-13/22,-23/39) -> (-23/59,-7/18) Hyperbolic Matrix(961,566,1360,801) (-23/39,-10/17) -> (12/17,29/41) Hyperbolic Matrix(559,328,680,399) (-10/17,-17/29) -> (9/11,14/17) Hyperbolic Matrix(301,176,720,421) (-17/29,-7/12) -> (5/12,13/31) Hyperbolic Matrix(299,174,720,419) (-7/12,-18/31) -> (12/29,5/12) Hyperbolic Matrix(541,314,-1480,-859) (-18/31,-11/19) -> (-15/41,-4/11) Hyperbolic Matrix(599,346,760,439) (-11/19,-15/26) -> (11/14,15/19) Hyperbolic Matrix(441,254,-1040,-599) (-15/26,-4/7) -> (-14/33,-11/26) Hyperbolic Matrix(81,46,-280,-159) (-4/7,-9/16) -> (-7/24,-2/7) Hyperbolic Matrix(1161,652,1640,921) (-9/16,-23/41) -> (29/41,17/24) Hyperbolic Matrix(19,10,-40,-21) (-5/9,-1/2) -> (-1/2,-5/11) Parabolic Matrix(221,100,400,181) (-5/11,-9/20) -> (11/20,5/9) Hyperbolic Matrix(219,98,400,179) (-9/20,-4/9) -> (6/11,11/20) Hyperbolic Matrix(59,26,-320,-141) (-4/9,-7/16) -> (-3/16,-2/11) Hyperbolic Matrix(779,340,960,419) (-7/16,-17/39) -> (17/21,13/16) Hyperbolic Matrix(1121,488,1840,801) (-17/39,-10/23) -> (14/23,25/41) Hyperbolic Matrix(921,392,1600,681) (-3/7,-17/40) -> (23/40,19/33) Hyperbolic Matrix(919,390,1600,679) (-17/40,-14/33) -> (4/7,23/40) Hyperbolic Matrix(161,68,760,321) (-11/26,-8/19) -> (4/19,3/14) Hyperbolic Matrix(421,176,720,301) (-13/31,-5/12) -> (7/12,17/29) Hyperbolic Matrix(419,174,720,299) (-5/12,-12/29) -> (18/31,7/12) Hyperbolic Matrix(121,50,680,281) (-12/29,-7/17) -> (3/17,2/11) Hyperbolic Matrix(79,32,-200,-81) (-7/17,-2/5) -> (-2/5,-9/23) Parabolic Matrix(1039,406,1840,719) (-9/23,-16/41) -> (22/39,13/23) Hyperbolic Matrix(6101,2380,10000,3901) (-16/41,-39/100) -> (61/100,36/59) Hyperbolic Matrix(6099,2378,10000,3899) (-39/100,-23/59) -> (25/41,61/100) Hyperbolic Matrix(259,100,360,139) (-7/18,-5/13) -> (5/7,13/18) Hyperbolic Matrix(21,8,160,61) (-5/13,-8/21) -> (0/1,1/7) Hyperbolic Matrix(201,76,320,121) (-8/21,-3/8) -> (5/8,12/19) Hyperbolic Matrix(199,74,320,119) (-3/8,-7/19) -> (13/21,5/8) Hyperbolic Matrix(659,242,-1800,-661) (-7/19,-11/30) -> (-11/30,-15/41) Parabolic Matrix(61,22,280,101) (-4/11,-5/14) -> (3/14,2/9) Hyperbolic Matrix(261,92,400,141) (-6/17,-7/20) -> (13/20,2/3) Hyperbolic Matrix(259,90,400,139) (-7/20,-1/3) -> (11/17,13/20) Hyperbolic Matrix(59,18,-200,-61) (-1/3,-3/10) -> (-3/10,-5/17) Parabolic Matrix(559,164,1360,399) (-5/17,-12/41) -> (16/39,7/17) Hyperbolic Matrix(719,210,1640,479) (-12/41,-7/24) -> (7/16,18/41) Hyperbolic Matrix(221,62,360,101) (-2/7,-5/18) -> (11/18,8/13) Hyperbolic Matrix(1161,320,1600,441) (-8/29,-11/40) -> (29/40,8/11) Hyperbolic Matrix(1159,318,1600,439) (-11/40,-3/11) -> (21/29,29/40) Hyperbolic Matrix(61,16,80,21) (-3/11,-1/4) -> (3/4,7/9) Hyperbolic Matrix(59,14,80,19) (-1/4,-2/9) -> (8/11,3/4) Hyperbolic Matrix(101,22,280,61) (-2/9,-3/14) -> (5/14,4/11) Hyperbolic Matrix(321,68,760,161) (-3/14,-4/19) -> (8/19,11/26) Hyperbolic Matrix(39,8,-200,-41) (-4/19,-1/5) -> (-1/5,-4/21) Parabolic Matrix(541,102,960,181) (-4/21,-3/16) -> (9/16,22/39) Hyperbolic Matrix(281,50,680,121) (-2/11,-3/17) -> (7/17,12/29) Hyperbolic Matrix(1321,232,1600,281) (-3/17,-7/40) -> (33/40,19/23) Hyperbolic Matrix(1319,230,1600,279) (-7/40,-4/23) -> (14/17,33/40) Hyperbolic Matrix(341,52,400,61) (-2/13,-3/20) -> (17/20,6/7) Hyperbolic Matrix(339,50,400,59) (-3/20,-1/7) -> (11/13,17/20) Hyperbolic Matrix(61,8,160,21) (-1/7,0/1) -> (8/21,5/13) Hyperbolic Matrix(101,-16,120,-19) (1/7,1/6) -> (5/6,11/13) Hyperbolic Matrix(199,-34,240,-41) (1/6,3/17) -> (19/23,5/6) Hyperbolic Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(299,-62,680,-141) (1/5,4/19) -> (18/41,11/25) Hyperbolic Matrix(521,-144,720,-199) (3/11,5/18) -> (13/18,21/29) Hyperbolic Matrix(61,-18,200,-59) (2/7,3/10) -> (3/10,4/13) Parabolic Matrix(181,-56,320,-99) (4/13,1/3) -> (13/23,4/7) Hyperbolic Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(859,-314,1480,-541) (4/11,11/30) -> (29/50,18/31) Hyperbolic Matrix(881,-324,1520,-559) (11/30,7/19) -> (11/19,29/50) Hyperbolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(1221,-500,2000,-819) (9/22,16/39) -> (36/59,11/18) Hyperbolic Matrix(939,-394,1480,-621) (13/31,8/19) -> (26/41,7/11) Hyperbolic Matrix(599,-254,1040,-441) (11/26,3/7) -> (19/33,15/26) Hyperbolic Matrix(199,-86,280,-121) (3/7,7/16) -> (17/24,5/7) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(261,-146,320,-179) (5/9,9/16) -> (13/16,9/11) Hyperbolic Matrix(121,-72,200,-119) (10/17,3/5) -> (3/5,14/23) Parabolic Matrix(1141,-722,1800,-1139) (12/19,19/30) -> (19/30,26/41) Parabolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(161,-128,200,-159) (15/19,4/5) -> (4/5,17/21) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(99,86,160,139) -> Matrix(1,2,2,5) Matrix(19,16,-120,-101) -> Matrix(1,0,2,1) Matrix(41,34,-240,-199) -> Matrix(1,0,0,1) Matrix(399,328,680,559) -> Matrix(1,0,-2,1) Matrix(179,146,-320,-261) -> Matrix(1,0,0,1) Matrix(381,302,-680,-539) -> Matrix(1,2,0,1) Matrix(439,346,760,599) -> Matrix(1,2,0,1) Matrix(179,140,280,219) -> Matrix(1,0,2,1) Matrix(21,16,80,61) -> Matrix(1,2,-2,-3) Matrix(19,14,80,59) -> Matrix(3,2,-2,-1) Matrix(199,144,-720,-521) -> Matrix(1,0,4,1) Matrix(139,100,360,259) -> Matrix(1,0,-2,1) Matrix(139,98,-200,-141) -> Matrix(1,0,2,1) Matrix(139,96,-320,-221) -> Matrix(1,0,0,1) Matrix(99,64,-280,-181) -> Matrix(5,4,6,5) Matrix(219,140,280,179) -> Matrix(1,0,2,1) Matrix(621,394,-1480,-939) -> Matrix(7,4,-2,-1) Matrix(639,404,-1520,-961) -> Matrix(5,2,2,1) Matrix(121,76,320,201) -> Matrix(1,0,0,1) Matrix(119,74,320,199) -> Matrix(1,0,0,1) Matrix(139,86,160,99) -> Matrix(5,2,2,1) Matrix(101,62,360,221) -> Matrix(1,0,2,1) Matrix(119,72,-200,-121) -> Matrix(1,0,6,1) Matrix(779,460,-2000,-1181) -> Matrix(1,0,0,1) Matrix(961,566,1360,801) -> Matrix(1,0,0,1) Matrix(559,328,680,399) -> Matrix(1,0,-2,1) Matrix(301,176,720,421) -> Matrix(3,-2,2,-1) Matrix(299,174,720,419) -> Matrix(1,-2,2,-3) Matrix(541,314,-1480,-859) -> Matrix(1,-4,2,-7) Matrix(599,346,760,439) -> Matrix(1,2,0,1) Matrix(441,254,-1040,-599) -> Matrix(1,0,0,1) Matrix(81,46,-280,-159) -> Matrix(1,0,2,1) Matrix(1161,652,1640,921) -> Matrix(1,0,2,1) Matrix(19,10,-40,-21) -> Matrix(1,0,2,1) Matrix(221,100,400,181) -> Matrix(3,-2,2,-1) Matrix(219,98,400,179) -> Matrix(1,-2,2,-3) Matrix(59,26,-320,-141) -> Matrix(1,0,0,1) Matrix(779,340,960,419) -> Matrix(1,0,0,1) Matrix(1121,488,1840,801) -> Matrix(1,0,4,1) Matrix(921,392,1600,681) -> Matrix(1,0,0,1) Matrix(919,390,1600,679) -> Matrix(1,0,0,1) Matrix(161,68,760,321) -> Matrix(1,-2,0,1) Matrix(421,176,720,301) -> Matrix(1,2,-2,-3) Matrix(419,174,720,299) -> Matrix(3,2,-2,-1) Matrix(121,50,680,281) -> Matrix(1,0,2,1) Matrix(79,32,-200,-81) -> Matrix(1,0,6,1) Matrix(1039,406,1840,719) -> Matrix(1,0,-4,1) Matrix(6101,2380,10000,3901) -> Matrix(7,-2,18,-5) Matrix(6099,2378,10000,3899) -> Matrix(5,-2,18,-7) Matrix(259,100,360,139) -> Matrix(1,0,-2,1) Matrix(21,8,160,61) -> Matrix(5,-2,-2,1) Matrix(201,76,320,121) -> Matrix(1,0,0,1) Matrix(199,74,320,119) -> Matrix(1,0,0,1) Matrix(659,242,-1800,-661) -> Matrix(13,-6,24,-11) Matrix(61,22,280,101) -> Matrix(1,0,-2,1) Matrix(261,92,400,141) -> Matrix(7,-6,6,-5) Matrix(259,90,400,139) -> Matrix(5,-6,6,-7) Matrix(59,18,-200,-61) -> Matrix(1,0,2,1) Matrix(559,164,1360,399) -> Matrix(1,0,0,1) Matrix(719,210,1640,479) -> Matrix(1,0,-2,1) Matrix(221,62,360,101) -> Matrix(1,0,2,1) Matrix(1161,320,1600,441) -> Matrix(1,0,0,1) Matrix(1159,318,1600,439) -> Matrix(1,0,0,1) Matrix(61,16,80,21) -> Matrix(3,-2,2,-1) Matrix(59,14,80,19) -> Matrix(1,-2,2,-3) Matrix(101,22,280,61) -> Matrix(1,0,-2,1) Matrix(321,68,760,161) -> Matrix(1,-2,0,1) Matrix(39,8,-200,-41) -> Matrix(1,-2,0,1) Matrix(541,102,960,181) -> Matrix(1,0,0,1) Matrix(281,50,680,121) -> Matrix(1,0,2,1) Matrix(1321,232,1600,281) -> Matrix(1,0,0,1) Matrix(1319,230,1600,279) -> Matrix(1,0,0,1) Matrix(341,52,400,61) -> Matrix(7,-6,6,-5) Matrix(339,50,400,59) -> Matrix(5,-6,6,-7) Matrix(61,8,160,21) -> Matrix(1,-2,-2,5) Matrix(101,-16,120,-19) -> Matrix(1,0,2,1) Matrix(199,-34,240,-41) -> Matrix(1,0,0,1) Matrix(141,-26,320,-59) -> Matrix(1,0,0,1) Matrix(299,-62,680,-141) -> Matrix(1,2,0,1) Matrix(521,-144,720,-199) -> Matrix(1,0,4,1) Matrix(61,-18,200,-59) -> Matrix(1,0,2,1) Matrix(181,-56,320,-99) -> Matrix(1,0,0,1) Matrix(181,-64,280,-99) -> Matrix(5,4,6,5) Matrix(859,-314,1480,-541) -> Matrix(7,4,-2,-1) Matrix(881,-324,1520,-559) -> Matrix(5,2,2,1) Matrix(81,-32,200,-79) -> Matrix(1,0,6,1) Matrix(1221,-500,2000,-819) -> Matrix(1,0,0,1) Matrix(939,-394,1480,-621) -> Matrix(1,-4,2,-7) Matrix(599,-254,1040,-441) -> Matrix(1,0,0,1) Matrix(199,-86,280,-121) -> Matrix(1,0,2,1) Matrix(21,-10,40,-19) -> Matrix(1,0,2,1) Matrix(261,-146,320,-179) -> Matrix(1,0,0,1) Matrix(121,-72,200,-119) -> Matrix(1,0,6,1) Matrix(1141,-722,1800,-1139) -> Matrix(13,-6,24,-11) Matrix(141,-98,200,-139) -> Matrix(1,0,2,1) Matrix(161,-128,200,-159) -> 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 24 Degree of the the map X: 24 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 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): 144 Minimal number of generators: 25 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: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/6 1/5 1/4 1/3 3/8 2/5 5/12 7/16 1/2 11/20 19/30 7/10 33/40 17/20 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/6 0/1 2/11 1/0 1/5 1/0 1/4 -1/1 1/3 1/0 5/14 -2/3 4/11 -1/2 3/8 -1/1 -1/3 2/5 0/1 5/12 1/1 8/19 1/0 11/26 0/1 3/7 1/0 7/16 -1/1 1/1 11/25 1/0 4/9 1/0 1/2 0/1 6/11 1/2 11/20 1/1 5/9 1/0 9/16 -1/1 1/1 4/7 1/0 11/19 1/0 18/31 1/0 7/12 -1/1 3/5 0/1 5/8 1/3 1/1 12/19 1/2 19/30 1/2 26/41 1/2 7/11 1/2 2/3 1/0 7/10 0/1 12/17 1/2 17/24 1/3 1/1 5/7 1/2 3/4 1/1 4/5 1/0 13/16 -1/1 1/1 9/11 1/0 14/17 1/0 33/40 -1/1 1/1 19/23 1/0 5/6 0/1 11/13 3/4 17/20 1/1 6/7 3/2 1/1 1/0 1/0 -1/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(29,-4,80,-11) (0/1,1/6) -> (5/14,4/11) Hyperbolic Matrix(169,-30,400,-71) (1/6,2/11) -> (8/19,11/26) Hyperbolic Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(31,-7,40,-9) (1/5,1/4) -> (3/4,4/5) Hyperbolic Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic Matrix(169,-60,200,-71) (1/3,5/14) -> (5/6,11/13) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(49,-19,80,-31) (3/8,2/5) -> (3/5,5/8) Hyperbolic Matrix(71,-29,120,-49) (2/5,5/12) -> (7/12,3/5) Hyperbolic Matrix(349,-146,600,-251) (5/12,8/19) -> (18/31,7/12) Hyperbolic Matrix(529,-224,640,-271) (11/26,3/7) -> (19/23,5/6) Hyperbolic Matrix(199,-86,280,-121) (3/7,7/16) -> (17/24,5/7) Hyperbolic Matrix(389,-171,480,-211) (7/16,11/25) -> (4/5,13/16) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(221,-121,400,-219) (6/11,11/20) -> (11/20,5/9) Parabolic Matrix(261,-146,320,-179) (5/9,9/16) -> (13/16,9/11) Hyperbolic Matrix(311,-176,440,-249) (9/16,4/7) -> (12/17,17/24) Hyperbolic Matrix(329,-190,400,-231) (4/7,11/19) -> (9/11,14/17) Hyperbolic Matrix(711,-412,1120,-649) (11/19,18/31) -> (26/41,7/11) Hyperbolic Matrix(1141,-722,1800,-1139) (12/19,19/30) -> (19/30,26/41) Parabolic Matrix(69,-44,80,-51) (7/11,2/3) -> (6/7,1/1) Hyperbolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(1321,-1089,1600,-1319) (14/17,33/40) -> (33/40,19/23) Parabolic Matrix(341,-289,400,-339) (11/13,17/20) -> (17/20,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,0,1) Matrix(29,-4,80,-11) -> Matrix(1,2,-2,-3) Matrix(169,-30,400,-71) -> Matrix(1,0,0,1) Matrix(141,-26,320,-59) -> Matrix(1,0,0,1) Matrix(31,-7,40,-9) -> Matrix(1,2,0,1) Matrix(29,-8,40,-11) -> Matrix(1,0,2,1) Matrix(169,-60,200,-71) -> Matrix(3,2,4,3) Matrix(151,-56,240,-89) -> Matrix(1,0,4,1) Matrix(49,-19,80,-31) -> Matrix(1,0,4,1) Matrix(71,-29,120,-49) -> Matrix(1,0,-2,1) Matrix(349,-146,600,-251) -> Matrix(1,-2,0,1) Matrix(529,-224,640,-271) -> Matrix(1,0,0,1) Matrix(199,-86,280,-121) -> Matrix(1,0,2,1) Matrix(389,-171,480,-211) -> Matrix(1,0,0,1) Matrix(21,-10,40,-19) -> Matrix(1,0,2,1) Matrix(221,-121,400,-219) -> Matrix(3,-2,2,-1) Matrix(261,-146,320,-179) -> Matrix(1,0,0,1) Matrix(311,-176,440,-249) -> Matrix(1,0,2,1) Matrix(329,-190,400,-231) -> Matrix(1,0,0,1) Matrix(711,-412,1120,-649) -> Matrix(1,-2,2,-3) Matrix(1141,-722,1800,-1139) -> Matrix(13,-6,24,-11) Matrix(69,-44,80,-51) -> Matrix(3,-2,2,-1) Matrix(141,-98,200,-139) -> Matrix(1,0,2,1) Matrix(1321,-1089,1600,-1319) -> Matrix(1,0,0,1) Matrix(341,-289,400,-339) -> Matrix(7,-6,6,-5) 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): 6 ----------------------------------------------------------------------- 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 1/0 1 20 1/6 0/1 2 10 2/11 1/0 1 20 1/5 1/0 1 4 1/4 -1/1 2 5 3/10 0/1 2 2 1/3 1/0 1 20 7/20 -1/1 6 1 5/14 -2/3 2 10 4/11 -1/2 1 20 11/30 -1/2 6 2 3/8 (-1/2,0/1) 0 5 2/5 0/1 3 4 5/12 1/1 2 5 21/50 1/0 6 2 8/19 1/0 1 20 11/26 0/1 2 10 17/40 (0/1,1/0) 0 1 3/7 1/0 1 20 7/16 (0/1,1/0) 0 5 11/25 1/0 1 4 4/9 1/0 1 20 9/20 -1/1 2 1 1/2 0/1 2 10 1/0 (0/1,1/0) 0 1 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(29,-4,80,-11) (0/1,1/6) -> (5/14,4/11) Hyperbolic Matrix(169,-30,400,-71) (1/6,2/11) -> (8/19,11/26) Hyperbolic Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(11,-3,40,-11) (1/4,3/10) -> (1/4,3/10) Reflection Matrix(49,-15,160,-49) (3/10,5/16) -> (3/10,5/16) Reflection Matrix(69,-22,160,-51) (4/13,1/3) -> (3/7,10/23) Hyperbolic Matrix(41,-14,120,-41) (1/3,7/20) -> (1/3,7/20) Reflection Matrix(99,-35,280,-99) (7/20,5/14) -> (7/20,5/14) Reflection Matrix(471,-172,1120,-409) (4/11,11/30) -> (21/50,8/19) Hyperbolic Matrix(89,-33,240,-89) (11/30,3/8) -> (11/30,3/8) Reflection Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(251,-105,600,-251) (5/12,21/50) -> (5/12,21/50) Reflection Matrix(441,-187,1040,-441) (11/26,17/40) -> (11/26,17/40) Reflection Matrix(239,-102,560,-239) (17/40,3/7) -> (17/40,3/7) Reflection Matrix(209,-91,480,-209) (13/30,7/16) -> (13/30,7/16) Reflection Matrix(351,-154,800,-351) (7/16,11/25) -> (7/16,11/25) Reflection Matrix(161,-72,360,-161) (4/9,9/20) -> (4/9,9/20) Reflection Matrix(19,-9,40,-19) (9/20,1/2) -> (9/20,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(29,-4,80,-11) -> Matrix(1,2,-2,-3) -1/1 Matrix(169,-30,400,-71) -> Matrix(1,0,0,1) Matrix(141,-26,320,-59) -> Matrix(1,0,0,1) Matrix(9,-2,40,-9) -> Matrix(1,2,0,-1) (1/5,1/4) -> (-1/1,1/0) Matrix(11,-3,40,-11) -> Matrix(-1,0,2,1) (1/4,3/10) -> (-1/1,0/1) Matrix(49,-15,160,-49) -> Matrix(1,0,0,-1) (3/10,5/16) -> (0/1,1/0) Matrix(69,-22,160,-51) -> Matrix(1,0,0,1) Matrix(41,-14,120,-41) -> Matrix(1,2,0,-1) (1/3,7/20) -> (-1/1,1/0) Matrix(99,-35,280,-99) -> Matrix(5,4,-6,-5) (7/20,5/14) -> (-1/1,-2/3) Matrix(471,-172,1120,-409) -> Matrix(3,2,-2,-1) -1/1 Matrix(89,-33,240,-89) -> Matrix(-1,0,4,1) (11/30,3/8) -> (-1/2,0/1) Matrix(31,-12,80,-31) -> Matrix(-1,0,4,1) (3/8,2/5) -> (-1/2,0/1) Matrix(49,-20,120,-49) -> Matrix(1,0,2,-1) (2/5,5/12) -> (0/1,1/1) Matrix(251,-105,600,-251) -> Matrix(-1,2,0,1) (5/12,21/50) -> (1/1,1/0) Matrix(441,-187,1040,-441) -> Matrix(1,0,0,-1) (11/26,17/40) -> (0/1,1/0) Matrix(239,-102,560,-239) -> Matrix(1,0,0,-1) (17/40,3/7) -> (0/1,1/0) Matrix(209,-91,480,-209) -> Matrix(1,0,0,-1) (13/30,7/16) -> (0/1,1/0) Matrix(351,-154,800,-351) -> Matrix(1,0,0,-1) (7/16,11/25) -> (0/1,1/0) Matrix(161,-72,360,-161) -> Matrix(1,2,0,-1) (4/9,9/20) -> (-1/1,1/0) Matrix(19,-9,40,-19) -> Matrix(-1,0,2,1) (9/20,1/2) -> (-1/1,0/1) Matrix(-1,1,0,1) -> Matrix(1,0,0,-1) (1/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.