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: 40 Genus: 29 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -5/1 -4/1 -10/3 -30/11 -8/3 -5/3 -4/3 0/1 1/1 20/17 5/4 4/3 3/2 20/13 5/3 9/5 20/11 2/1 5/2 8/3 30/11 20/7 3/1 10/3 7/2 18/5 40/11 15/4 4/1 13/3 40/9 14/3 5/1 11/2 6/1 20/3 15/2 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 0/1 -7/1 -1/2 -20/3 0/1 -13/2 -1/1 0/1 -6/1 -1/1 0/1 -11/2 -5/7 -2/3 -5/1 -1/2 -14/3 -3/7 -2/5 -23/5 -7/18 -32/7 -2/5 -9/2 -4/11 -1/3 -13/3 -3/10 -4/1 0/1 -15/4 -1/1 0/1 -11/3 1/0 -18/5 -1/1 0/1 -7/2 -1/1 -2/3 -17/5 -7/12 -10/3 -1/2 -3/1 -1/2 -20/7 -1/3 -17/6 -1/3 0/1 -14/5 -1/3 0/1 -11/4 -1/3 0/1 -30/11 -1/2 -19/7 -5/12 -8/3 -1/3 -21/8 -3/7 -2/5 -34/13 -4/11 -1/3 -13/5 -1/4 -31/12 -6/17 -1/3 -18/7 -1/3 0/1 -5/2 -1/3 0/1 -2/1 -1/3 0/1 -15/8 -1/3 0/1 -13/7 -1/2 -11/6 -1/9 0/1 -20/11 0/1 -9/5 1/4 -25/14 0/1 1/1 -16/9 0/1 -23/13 3/2 -7/4 -1/1 0/1 -19/11 -3/4 -50/29 -1/2 -31/18 -1/3 0/1 -12/7 0/1 -17/10 -1/1 -2/3 -5/3 -1/2 -23/14 -1/3 0/1 -18/11 -2/5 -1/3 -13/8 -1/3 0/1 -34/21 -1/3 0/1 -55/34 -1/3 0/1 -21/13 1/0 -8/5 -1/3 -19/12 -2/7 -3/11 -30/19 -1/4 -41/26 -4/17 -3/13 -11/7 -1/4 -25/16 -1/5 0/1 -14/9 -1/5 0/1 -17/11 -1/10 -20/13 0/1 -3/2 -1/1 0/1 -13/9 -5/8 -10/7 -1/2 -7/5 -1/2 -32/23 -2/5 -57/41 -3/8 -25/18 -2/5 -1/3 -18/13 -2/5 -1/3 -29/21 -3/8 -40/29 -1/3 -11/8 -1/3 0/1 -26/19 -1/3 0/1 -15/11 -1/2 -19/14 -7/17 -2/5 -4/3 -1/3 -21/16 -1/7 0/1 -17/13 1/0 -13/10 -2/5 -1/3 -22/17 -4/11 -1/3 -31/24 -10/29 -1/3 -40/31 -1/3 -9/7 -3/10 -23/18 -1/3 0/1 -14/11 -1/3 0/1 -5/4 -1/3 0/1 -6/5 -1/3 0/1 -13/11 -5/14 -20/17 -1/3 -7/6 -1/3 -4/13 -8/7 -2/7 -9/8 -3/11 -4/15 -1/1 -1/4 0/1 0/1 1/1 1/2 7/6 4/5 1/1 20/17 1/1 13/11 5/4 6/5 0/1 1/1 5/4 0/1 1/1 14/11 0/1 1/1 9/7 3/4 13/10 1/1 2/1 4/3 1/1 15/11 1/0 11/8 0/1 1/1 18/13 1/1 2/1 25/18 1/1 2/1 7/5 1/0 10/7 1/0 13/9 -5/2 16/11 -1/1 3/2 -1/1 0/1 20/13 0/1 17/11 1/8 14/9 0/1 1/3 11/7 1/2 41/26 3/7 4/9 30/19 1/2 19/12 3/5 2/3 8/5 1/1 21/13 -1/2 34/21 0/1 1/1 13/8 0/1 1/1 31/19 5/6 18/11 1/1 2/1 5/3 1/0 17/10 -2/1 -1/1 29/17 -1/2 41/24 -2/5 -1/3 12/7 0/1 31/18 0/1 1/1 19/11 -3/2 26/15 -1/1 0/1 7/4 -1/1 0/1 16/9 0/1 9/5 -1/6 20/11 0/1 11/6 0/1 1/7 13/7 1/0 15/8 0/1 1/1 2/1 0/1 1/1 5/2 0/1 1/1 18/7 0/1 1/1 13/5 1/2 21/8 2/1 3/1 8/3 1/1 19/7 5/2 49/18 4/1 5/1 30/11 1/0 11/4 0/1 1/1 14/5 0/1 1/1 17/6 0/1 1/1 20/7 1/1 3/1 1/0 10/3 1/0 17/5 -7/2 7/2 -2/1 -1/1 25/7 1/0 18/5 -1/1 0/1 29/8 -2/1 -1/1 40/11 -1/1 11/3 -1/2 26/7 -1/1 0/1 15/4 -1/1 0/1 34/9 -1/1 0/1 19/5 -1/4 4/1 0/1 17/4 0/1 1/3 13/3 3/4 22/5 2/3 1/1 31/7 9/10 40/9 1/1 9/2 1/1 4/3 32/7 2/1 23/5 7/4 14/3 2/1 3/1 19/4 2/1 3/1 5/1 1/0 16/3 -3/1 27/5 -9/4 11/2 -2/1 -5/3 17/3 -5/4 6/1 -1/1 0/1 13/2 -1/1 0/1 20/3 0/1 7/1 1/0 15/2 -1/1 0/1 23/3 -1/2 8/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(31,260,18,151) (-8/1,1/0) -> (12/7,31/18) Hyperbolic Matrix(39,280,-28,-201) (-8/1,-7/1) -> (-7/5,-32/23) Hyperbolic Matrix(41,280,6,41) (-7/1,-20/3) -> (20/3,7/1) Hyperbolic Matrix(79,520,12,79) (-20/3,-13/2) -> (13/2,20/3) Hyperbolic Matrix(81,520,50,321) (-13/2,-6/1) -> (34/21,13/8) Hyperbolic Matrix(89,500,-34,-191) (-6/1,-11/2) -> (-21/8,-34/13) Hyperbolic Matrix(49,260,-36,-191) (-11/2,-5/1) -> (-15/11,-19/14) Hyperbolic Matrix(71,340,-52,-249) (-5/1,-14/3) -> (-26/19,-15/11) Hyperbolic Matrix(121,560,78,361) (-14/3,-23/5) -> (17/11,14/9) Hyperbolic Matrix(559,2560,-402,-1841) (-23/5,-32/7) -> (-32/23,-57/41) Hyperbolic Matrix(311,1420,182,831) (-32/7,-9/2) -> (41/24,12/7) Hyperbolic Matrix(119,520,-46,-201) (-9/2,-13/3) -> (-13/5,-31/12) Hyperbolic Matrix(71,300,-40,-169) (-13/3,-4/1) -> (-16/9,-23/13) Hyperbolic Matrix(89,340,-50,-191) (-4/1,-15/4) -> (-25/14,-16/9) Hyperbolic Matrix(119,440,-76,-281) (-15/4,-11/3) -> (-11/7,-25/16) Hyperbolic Matrix(199,720,-144,-521) (-11/3,-18/5) -> (-18/13,-29/21) Hyperbolic Matrix(151,540,-92,-329) (-18/5,-7/2) -> (-23/14,-18/11) Hyperbolic Matrix(81,280,-46,-159) (-7/2,-17/5) -> (-23/13,-7/4) Hyperbolic Matrix(89,300,62,209) (-17/5,-10/3) -> (10/7,13/9) Hyperbolic Matrix(31,100,22,71) (-10/3,-3/1) -> (7/5,10/7) Hyperbolic Matrix(41,120,14,41) (-3/1,-20/7) -> (20/7,3/1) Hyperbolic Matrix(239,680,84,239) (-20/7,-17/6) -> (17/6,20/7) Hyperbolic Matrix(191,540,110,311) (-17/6,-14/5) -> (26/15,7/4) Hyperbolic Matrix(159,440,-116,-321) (-14/5,-11/4) -> (-11/8,-26/19) Hyperbolic Matrix(329,900,208,569) (-11/4,-30/11) -> (30/19,19/12) 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(199,520,168,439) (-34/13,-13/5) -> (13/11,6/5) Hyperbolic Matrix(481,1240,-372,-959) (-31/12,-18/7) -> (-22/17,-31/24) Hyperbolic Matrix(71,180,28,71) (-18/7,-5/2) -> (5/2,18/7) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(31,60,16,31) (-2/1,-15/8) -> (15/8,2/1) Hyperbolic Matrix(631,1180,-454,-849) (-15/8,-13/7) -> (-57/41,-25/18) Hyperbolic Matrix(249,460,-190,-351) (-13/7,-11/6) -> (-21/16,-17/13) Hyperbolic Matrix(241,440,132,241) (-11/6,-20/11) -> (20/11,11/6) Hyperbolic Matrix(199,360,110,199) (-20/11,-9/5) -> (9/5,20/11) Hyperbolic Matrix(569,1020,-352,-631) (-9/5,-25/14) -> (-55/34,-21/13) Hyperbolic Matrix(289,500,-226,-391) (-7/4,-19/11) -> (-9/7,-23/18) Hyperbolic Matrix(1729,2980,1096,1889) (-50/29,-31/18) -> (41/26,30/19) Hyperbolic Matrix(151,260,18,31) (-31/18,-12/7) -> (8/1,1/0) Hyperbolic Matrix(129,220,-112,-191) (-12/7,-17/10) -> (-7/6,-8/7) Hyperbolic Matrix(119,200,-72,-121) (-17/10,-5/3) -> (-5/3,-23/14) Parabolic Matrix(319,520,-246,-401) (-18/11,-13/8) -> (-13/10,-22/17) Hyperbolic Matrix(321,520,50,81) (-13/8,-34/21) -> (6/1,13/2) Hyperbolic Matrix(1471,2380,390,631) (-34/21,-55/34) -> (15/4,34/9) 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(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(1711,2700,628,991) (-30/19,-41/26) -> (49/18,30/11) Hyperbolic Matrix(1041,1640,610,961) (-41/26,-11/7) -> (29/17,41/24) Hyperbolic Matrix(551,860,148,231) (-25/16,-14/9) -> (26/7,15/4) Hyperbolic Matrix(361,560,78,121) (-14/9,-17/11) -> (23/5,14/3) Hyperbolic Matrix(441,680,286,441) (-17/11,-20/13) -> (20/13,17/11) Hyperbolic Matrix(79,120,52,79) (-20/13,-3/2) -> (3/2,20/13) Hyperbolic Matrix(151,220,-116,-169) (-3/2,-13/9) -> (-17/13,-13/10) Hyperbolic Matrix(209,300,62,89) (-13/9,-10/7) -> (10/3,17/5) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(649,900,468,649) (-25/18,-18/13) -> (18/13,25/18) 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(119,160,-90,-121) (-19/14,-4/3) -> (-4/3,-21/16) Parabolic Matrix(1239,1600,278,359) (-31/24,-40/31) -> (40/9,9/2) Hyperbolic Matrix(1241,1600,280,361) (-40/31,-9/7) -> (31/7,40/9) Hyperbolic Matrix(439,560,156,199) (-23/18,-14/11) -> (14/5,17/6) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(151,180,26,31) (-6/5,-13/11) -> (17/3,6/1) Hyperbolic Matrix(441,520,374,441) (-13/11,-20/17) -> (20/17,13/11) Hyperbolic Matrix(239,280,204,239) (-20/17,-7/6) -> (7/6,20/17) Hyperbolic Matrix(319,360,70,79) (-8/7,-9/8) -> (9/2,32/7) Hyperbolic Matrix(129,140,82,89) (-9/8,-1/1) -> (11/7,41/26) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(191,-220,112,-129) (1/1,7/6) -> (17/10,29/17) Hyperbolic Matrix(391,-500,226,-289) (14/11,9/7) -> (19/11,26/15) Hyperbolic Matrix(401,-520,246,-319) (9/7,13/10) -> (13/8,31/19) Hyperbolic Matrix(169,-220,116,-151) (13/10,4/3) -> (16/11,3/2) Hyperbolic Matrix(191,-260,36,-49) (4/3,15/11) -> (5/1,16/3) Hyperbolic Matrix(249,-340,52,-71) (15/11,11/8) -> (19/4,5/1) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(201,-280,28,-39) (25/18,7/5) -> (7/1,15/2) Hyperbolic Matrix(441,-640,82,-119) (13/9,16/11) -> (16/3,27/5) Hyperbolic Matrix(281,-440,76,-119) (14/9,11/7) -> (11/3,26/7) Hyperbolic Matrix(841,-1360,222,-359) (21/13,34/21) -> (34/9,19/5) Hyperbolic Matrix(759,-1240,172,-281) (31/19,18/11) -> (22/5,31/7) Hyperbolic Matrix(329,-540,92,-151) (18/11,5/3) -> (25/7,18/5) Hyperbolic Matrix(271,-460,76,-129) (5/3,17/10) -> (7/2,25/7) Hyperbolic Matrix(881,-1520,324,-559) (31/18,19/11) -> (19/7,49/18) Hyperbolic Matrix(169,-300,40,-71) (7/4,16/9) -> (4/1,17/4) Hyperbolic Matrix(191,-340,50,-89) (16/9,9/5) -> (19/5,4/1) Hyperbolic Matrix(239,-440,44,-81) (11/6,13/7) -> (27/5,11/2) Hyperbolic Matrix(289,-540,38,-71) (13/7,15/8) -> (15/2,23/3) Hyperbolic Matrix(201,-520,46,-119) (18/7,13/5) -> (13/3,22/5) Hyperbolic Matrix(191,-500,34,-89) (13/5,21/8) -> (11/2,17/3) Hyperbolic Matrix(151,-420,32,-89) (11/4,14/5) -> (14/3,19/4) Hyperbolic Matrix(111,-380,26,-89) (17/5,7/2) -> (17/4,13/3) Hyperbolic Matrix(201,-920,26,-119) (32/7,23/5) -> (23/3,8/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(31,260,18,151) -> Matrix(1,0,2,1) Matrix(39,280,-28,-201) -> Matrix(3,2,-8,-5) Matrix(41,280,6,41) -> Matrix(1,0,2,1) Matrix(79,520,12,79) -> Matrix(1,0,0,1) Matrix(81,520,50,321) -> Matrix(1,0,2,1) Matrix(89,500,-34,-191) -> Matrix(5,4,-14,-11) Matrix(49,260,-36,-191) -> Matrix(7,4,-16,-9) Matrix(71,340,-52,-249) -> Matrix(5,2,-8,-3) Matrix(121,560,78,361) -> Matrix(5,2,22,9) Matrix(559,2560,-402,-1841) -> Matrix(21,8,-50,-19) Matrix(311,1420,182,831) -> Matrix(5,2,-18,-7) Matrix(119,520,-46,-201) -> Matrix(7,2,-18,-5) Matrix(71,300,-40,-169) -> Matrix(1,0,4,1) Matrix(89,340,-50,-191) -> Matrix(1,0,2,1) Matrix(119,440,-76,-281) -> Matrix(1,0,-4,1) Matrix(199,720,-144,-521) -> Matrix(3,2,-8,-5) Matrix(151,540,-92,-329) -> Matrix(3,2,-8,-5) Matrix(81,280,-46,-159) -> Matrix(3,2,-2,-1) Matrix(89,300,62,209) -> Matrix(11,6,-2,-1) Matrix(31,100,22,71) -> Matrix(5,2,2,1) Matrix(41,120,14,41) -> Matrix(5,2,2,1) Matrix(239,680,84,239) -> Matrix(1,0,4,1) Matrix(191,540,110,311) -> Matrix(1,0,2,1) Matrix(159,440,-116,-321) -> Matrix(1,0,0,1) Matrix(329,900,208,569) -> Matrix(3,2,4,3) Matrix(559,1520,-324,-881) -> Matrix(9,4,-16,-7) Matrix(119,320,74,199) -> Matrix(5,2,2,1) Matrix(121,320,76,201) -> Matrix(1,0,4,1) Matrix(199,520,168,439) -> Matrix(11,4,8,3) Matrix(481,1240,-372,-959) -> Matrix(13,4,-36,-11) Matrix(71,180,28,71) -> Matrix(1,0,4,1) Matrix(9,20,4,9) -> Matrix(1,0,4,1) Matrix(31,60,16,31) -> Matrix(1,0,4,1) Matrix(631,1180,-454,-849) -> Matrix(7,2,-18,-5) Matrix(249,460,-190,-351) -> Matrix(1,0,2,1) Matrix(241,440,132,241) -> Matrix(1,0,16,1) Matrix(199,360,110,199) -> Matrix(1,0,-10,1) Matrix(569,1020,-352,-631) -> Matrix(1,0,-4,1) Matrix(289,500,-226,-391) -> Matrix(1,0,-2,1) Matrix(1729,2980,1096,1889) -> Matrix(9,4,20,9) Matrix(151,260,18,31) -> Matrix(1,0,2,1) Matrix(129,220,-112,-191) -> Matrix(1,2,-4,-7) Matrix(119,200,-72,-121) -> Matrix(3,2,-8,-5) Matrix(319,520,-246,-401) -> Matrix(7,2,-18,-5) Matrix(321,520,50,81) -> Matrix(1,0,2,1) Matrix(1471,2380,390,631) -> Matrix(1,0,2,1) Matrix(199,320,74,119) -> Matrix(5,2,2,1) Matrix(201,320,76,121) -> Matrix(1,0,4,1) Matrix(569,900,208,329) -> Matrix(7,2,-4,-1) Matrix(1711,2700,628,991) -> Matrix(33,8,4,1) Matrix(1041,1640,610,961) -> Matrix(9,2,-14,-3) Matrix(551,860,148,231) -> Matrix(1,0,4,1) Matrix(361,560,78,121) -> Matrix(13,2,6,1) Matrix(441,680,286,441) -> Matrix(1,0,18,1) Matrix(79,120,52,79) -> Matrix(1,0,0,1) Matrix(151,220,-116,-169) -> Matrix(3,2,-8,-5) Matrix(209,300,62,89) -> Matrix(11,6,-2,-1) Matrix(71,100,22,31) -> Matrix(5,2,2,1) Matrix(649,900,468,649) -> Matrix(11,4,8,3) Matrix(1159,1600,318,439) -> Matrix(11,4,-14,-5) Matrix(1161,1600,320,441) -> Matrix(7,2,-4,-1) Matrix(119,160,-90,-121) -> Matrix(5,2,-18,-7) Matrix(1239,1600,278,359) -> Matrix(41,14,38,13) Matrix(1241,1600,280,361) -> Matrix(37,12,40,13) Matrix(439,560,156,199) -> Matrix(1,0,4,1) Matrix(111,140,88,111) -> Matrix(1,0,4,1) Matrix(49,60,40,49) -> Matrix(1,0,4,1) Matrix(151,180,26,31) -> Matrix(1,0,2,1) Matrix(441,520,374,441) -> Matrix(29,10,26,9) Matrix(239,280,204,239) -> Matrix(25,8,28,9) Matrix(319,360,70,79) -> Matrix(29,8,18,5) Matrix(129,140,82,89) -> Matrix(1,0,6,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(191,-220,112,-129) -> Matrix(3,-2,-4,3) Matrix(391,-500,226,-289) -> Matrix(1,0,-2,1) Matrix(401,-520,246,-319) -> Matrix(1,-2,2,-3) Matrix(169,-220,116,-151) -> Matrix(1,-2,0,1) Matrix(191,-260,36,-49) -> Matrix(1,-4,0,1) Matrix(249,-340,52,-71) -> Matrix(1,2,0,1) Matrix(521,-720,144,-199) -> Matrix(1,-2,0,1) Matrix(201,-280,28,-39) -> Matrix(1,-2,0,1) Matrix(441,-640,82,-119) -> Matrix(5,8,-2,-3) Matrix(281,-440,76,-119) -> Matrix(1,0,-4,1) Matrix(841,-1360,222,-359) -> Matrix(1,0,-2,1) Matrix(759,-1240,172,-281) -> Matrix(3,-4,4,-5) Matrix(329,-540,92,-151) -> Matrix(1,-2,0,1) Matrix(271,-460,76,-129) -> Matrix(1,0,0,1) Matrix(881,-1520,324,-559) -> Matrix(1,4,0,1) Matrix(169,-300,40,-71) -> Matrix(1,0,4,1) Matrix(191,-340,50,-89) -> Matrix(1,0,2,1) Matrix(239,-440,44,-81) -> Matrix(9,-2,-4,1) Matrix(289,-540,38,-71) -> Matrix(1,0,-2,1) Matrix(201,-520,46,-119) -> Matrix(1,-2,2,-3) Matrix(191,-500,34,-89) -> Matrix(3,-4,-2,3) Matrix(151,-420,32,-89) -> Matrix(1,2,0,1) Matrix(111,-380,26,-89) -> Matrix(1,2,2,5) Matrix(201,-920,26,-119) -> 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): 30 Degree of the the map X: 30 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): 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: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -8/3 -5/3 0/1 1/1 5/4 3/2 20/13 5/3 9/5 20/11 2/1 5/2 8/3 30/11 3/1 10/3 4/1 13/3 5/1 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 -5/1 -1/2 -14/3 -3/7 -2/5 -9/2 -4/11 -1/3 -13/3 -3/10 -4/1 0/1 -3/1 -1/2 -20/7 -1/3 -17/6 -1/3 0/1 -14/5 -1/3 0/1 -11/4 -1/3 0/1 -30/11 -1/2 -19/7 -5/12 -8/3 -1/3 -21/8 -3/7 -2/5 -34/13 -4/11 -1/3 -13/5 -1/4 -18/7 -1/3 0/1 -5/2 -1/3 0/1 -2/1 -1/3 0/1 -13/7 -1/2 -11/6 -1/9 0/1 -20/11 0/1 -9/5 1/4 -7/4 -1/1 0/1 -5/3 -1/2 -13/8 -1/3 0/1 -21/13 1/0 -8/5 -1/3 -19/12 -2/7 -3/11 -30/19 -1/4 -11/7 -1/4 -3/2 -1/1 0/1 -13/9 -5/8 -10/7 -1/2 -7/5 -1/2 -4/3 -1/3 -17/13 1/0 -13/10 -2/5 -1/3 -9/7 -3/10 -23/18 -1/3 0/1 -14/11 -1/3 0/1 -5/4 -1/3 0/1 -6/5 -1/3 0/1 -13/11 -5/14 -20/17 -1/3 -7/6 -1/3 -4/13 -1/1 -1/4 0/1 0/1 1/1 1/2 6/5 0/1 1/1 5/4 0/1 1/1 14/11 0/1 1/1 9/7 3/4 4/3 1/1 3/2 -1/1 0/1 20/13 0/1 17/11 1/8 14/9 0/1 1/3 11/7 1/2 30/19 1/2 19/12 3/5 2/3 8/5 1/1 5/3 1/0 12/7 0/1 19/11 -3/2 26/15 -1/1 0/1 7/4 -1/1 0/1 16/9 0/1 9/5 -1/6 20/11 0/1 11/6 0/1 1/7 2/1 0/1 1/1 5/2 0/1 1/1 18/7 0/1 1/1 13/5 1/2 21/8 2/1 3/1 8/3 1/1 19/7 5/2 30/11 1/0 11/4 0/1 1/1 3/1 1/0 10/3 1/0 17/5 -7/2 7/2 -2/1 -1/1 4/1 0/1 17/4 0/1 1/3 13/3 3/4 22/5 2/3 1/1 9/2 1/1 4/3 5/1 1/0 11/2 -2/1 -5/3 17/3 -5/4 6/1 -1/1 0/1 13/2 -1/1 0/1 20/3 0/1 7/1 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,80,-4,-29) (-6/1,1/0) -> (-14/5,-11/4) Hyperbolic Matrix(19,100,-4,-21) (-6/1,-5/1) -> (-5/1,-14/3) Parabolic Matrix(131,600,-50,-229) (-14/3,-9/2) -> (-21/8,-34/13) Hyperbolic Matrix(59,260,-32,-141) (-9/2,-13/3) -> (-13/7,-11/6) Hyperbolic Matrix(29,120,-22,-91) (-13/3,-4/1) -> (-4/3,-17/13) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(69,200,10,29) (-3/1,-20/7) -> (20/3,7/1) Hyperbolic Matrix(211,600,32,91) (-20/7,-17/6) -> (13/2,20/3) Hyperbolic Matrix(191,540,110,311) (-17/6,-14/5) -> (26/15,7/4) Hyperbolic Matrix(329,900,208,569) (-11/4,-30/11) -> (30/19,19/12) Hyperbolic Matrix(419,1140,154,419) (-30/11,-19/7) -> (19/7,30/11) Hyperbolic Matrix(141,380,82,221) (-19/7,-8/3) -> (12/7,19/11) Hyperbolic Matrix(121,320,76,201) (-8/3,-21/8) -> (19/12,8/5) Hyperbolic Matrix(291,760,188,491) (-34/13,-13/5) -> (17/11,14/9) Hyperbolic Matrix(101,260,-54,-139) (-13/5,-18/7) -> (-2/1,-13/7) Hyperbolic Matrix(71,180,28,71) (-18/7,-5/2) -> (5/2,18/7) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(241,440,132,241) (-11/6,-20/11) -> (20/11,11/6) Hyperbolic Matrix(199,360,110,199) (-20/11,-9/5) -> (9/5,20/11) Hyperbolic Matrix(101,180,-78,-139) (-9/5,-7/4) -> (-13/10,-9/7) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(371,600,-290,-469) (-13/8,-21/13) -> (-9/7,-23/18) 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(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(419,660,266,419) (-30/19,-11/7) -> (11/7,30/19) Hyperbolic Matrix(51,80,-44,-69) (-11/7,-3/2) -> (-7/6,-1/1) Hyperbolic Matrix(151,220,-116,-169) (-3/2,-13/9) -> (-17/13,-13/10) Hyperbolic Matrix(209,300,62,89) (-13/9,-10/7) -> (10/3,17/5) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(251,320,40,51) (-23/18,-14/11) -> (6/1,13/2) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(151,180,26,31) (-6/5,-13/11) -> (17/3,6/1) Hyperbolic Matrix(509,600,330,389) (-13/11,-20/17) -> (20/13,17/11) Hyperbolic Matrix(171,200,112,131) (-20/17,-7/6) -> (3/2,20/13) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(69,-80,44,-51) (1/1,6/5) -> (14/9,11/7) Hyperbolic Matrix(391,-500,226,-289) (14/11,9/7) -> (19/11,26/15) Hyperbolic Matrix(139,-180,78,-101) (9/7,4/3) -> (16/9,9/5) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(169,-300,40,-71) (7/4,16/9) -> (4/1,17/4) Hyperbolic Matrix(141,-260,32,-59) (11/6,2/1) -> (22/5,9/2) Hyperbolic Matrix(201,-520,46,-119) (18/7,13/5) -> (13/3,22/5) Hyperbolic Matrix(191,-500,34,-89) (13/5,21/8) -> (11/2,17/3) Hyperbolic Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic Matrix(111,-380,26,-89) (17/5,7/2) -> (17/4,13/3) Hyperbolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,80,-4,-29) -> Matrix(1,1,-4,-3) Matrix(19,100,-4,-21) -> Matrix(5,3,-12,-7) Matrix(131,600,-50,-229) -> Matrix(13,5,-34,-13) Matrix(59,260,-32,-141) -> Matrix(3,1,-16,-5) Matrix(29,120,-22,-91) -> Matrix(3,1,-10,-3) Matrix(11,40,-8,-29) -> Matrix(1,1,-4,-3) Matrix(69,200,10,29) -> Matrix(3,1,2,1) Matrix(211,600,32,91) -> Matrix(3,1,-4,-1) Matrix(191,540,110,311) -> Matrix(1,0,2,1) Matrix(329,900,208,569) -> Matrix(3,2,4,3) Matrix(419,1140,154,419) -> Matrix(11,5,2,1) Matrix(141,380,82,221) -> Matrix(3,1,2,1) Matrix(121,320,76,201) -> Matrix(1,0,4,1) Matrix(291,760,188,491) -> Matrix(3,1,20,7) Matrix(101,260,-54,-139) -> Matrix(3,1,-10,-3) Matrix(71,180,28,71) -> Matrix(1,0,4,1) Matrix(9,20,4,9) -> Matrix(1,0,4,1) Matrix(241,440,132,241) -> Matrix(1,0,16,1) Matrix(199,360,110,199) -> Matrix(1,0,-10,1) Matrix(101,180,-78,-139) -> Matrix(1,-1,-2,3) Matrix(59,100,-36,-61) -> Matrix(1,1,-4,-3) Matrix(371,600,-290,-469) -> Matrix(3,1,-10,-3) Matrix(199,320,74,119) -> Matrix(5,2,2,1) Matrix(201,320,76,121) -> Matrix(1,0,4,1) Matrix(569,900,208,329) -> Matrix(7,2,-4,-1) Matrix(419,660,266,419) -> Matrix(5,1,14,3) Matrix(51,80,-44,-69) -> Matrix(5,1,-16,-3) Matrix(151,220,-116,-169) -> Matrix(3,2,-8,-5) Matrix(209,300,62,89) -> Matrix(11,6,-2,-1) Matrix(71,100,22,31) -> Matrix(5,2,2,1) Matrix(251,320,40,51) -> Matrix(3,1,-4,-1) Matrix(111,140,88,111) -> Matrix(1,0,4,1) Matrix(49,60,40,49) -> Matrix(1,0,4,1) Matrix(151,180,26,31) -> Matrix(1,0,2,1) Matrix(509,600,330,389) -> Matrix(3,1,38,13) Matrix(171,200,112,131) -> Matrix(3,1,-16,-5) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(69,-80,44,-51) -> Matrix(1,-1,4,-3) Matrix(391,-500,226,-289) -> Matrix(1,0,-2,1) Matrix(139,-180,78,-101) -> Matrix(1,-1,-2,3) Matrix(29,-40,8,-11) -> Matrix(1,-1,0,1) Matrix(61,-100,36,-59) -> Matrix(1,-1,0,1) Matrix(169,-300,40,-71) -> Matrix(1,0,4,1) Matrix(141,-260,32,-59) -> Matrix(3,-1,4,-1) Matrix(201,-520,46,-119) -> Matrix(1,-2,2,-3) Matrix(191,-500,34,-89) -> Matrix(3,-4,-2,3) Matrix(29,-80,4,-11) -> Matrix(1,-1,0,1) Matrix(111,-380,26,-89) -> Matrix(1,2,2,5) Matrix(21,-100,4,-19) -> Matrix(1,-3,0,1) 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): 30 ----------------------------------------------------------------------- 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 3 1 1/1 1/2 1 20 6/5 (0/1,1/1) 0 10 5/4 (0/1,1/1) 0 4 14/11 (0/1,1/1) 0 10 9/7 3/4 1 20 4/3 1/1 1 5 3/2 (-1/1,0/1) 0 20 20/13 0/1 9 1 17/11 1/8 1 20 14/9 (0/1,1/3) 0 10 11/7 1/2 1 20 30/19 1/2 5 2 19/12 (3/5,2/3) 0 20 8/5 1/1 1 5 5/3 1/0 1 4 12/7 0/1 1 5 19/11 -3/2 1 20 26/15 (-1/1,0/1) 0 10 7/4 (-1/1,0/1) 0 20 16/9 0/1 1 5 9/5 -1/6 1 20 20/11 0/1 13 1 11/6 (0/1,1/7) 0 20 2/1 (0/1,1/1) 0 10 5/2 (0/1,1/1) 0 4 18/7 (0/1,1/1) 0 10 13/5 1/2 1 20 21/8 (2/1,3/1) 0 20 8/3 1/1 1 5 19/7 5/2 1 20 30/11 1/0 5 2 11/4 (0/1,1/1) 0 20 3/1 1/0 1 20 10/3 1/0 4 2 17/5 -7/2 1 20 7/2 (-2/1,-1/1) 0 20 4/1 0/1 1 5 17/4 (0/1,1/3) 0 20 13/3 3/4 1 20 22/5 (2/3,1/1) 0 10 9/2 (1/1,4/3) 0 20 5/1 1/0 3 4 11/2 (-2/1,-5/3) 0 20 17/3 -5/4 1 20 6/1 (-1/1,0/1) 0 10 13/2 (-1/1,0/1) 0 20 20/3 0/1 1 1 7/1 1/0 1 20 1/0 (-1/1,0/1) 0 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(69,-80,44,-51) (1/1,6/5) -> (14/9,11/7) Hyperbolic Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(111,-140,88,-111) (5/4,14/11) -> (5/4,14/11) Reflection Matrix(391,-500,226,-289) (14/11,9/7) -> (19/11,26/15) Hyperbolic Matrix(139,-180,78,-101) (9/7,4/3) -> (16/9,9/5) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(79,-120,52,-79) (3/2,20/13) -> (3/2,20/13) Reflection Matrix(441,-680,286,-441) (20/13,17/11) -> (20/13,17/11) Reflection Matrix(219,-340,38,-59) (17/11,14/9) -> (17/3,6/1) Glide Reflection Matrix(419,-660,266,-419) (11/7,30/19) -> (11/7,30/19) Reflection Matrix(569,-900,208,-329) (30/19,19/12) -> (30/11,11/4) Glide Reflection Matrix(201,-320,76,-121) (19/12,8/5) -> (21/8,8/3) Glide Reflection Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(221,-380,82,-141) (12/7,19/11) -> (8/3,19/7) Glide Reflection Matrix(219,-380,34,-59) (26/15,7/4) -> (6/1,13/2) Glide Reflection Matrix(169,-300,40,-71) (7/4,16/9) -> (4/1,17/4) Hyperbolic Matrix(199,-360,110,-199) (9/5,20/11) -> (9/5,20/11) Reflection Matrix(241,-440,132,-241) (20/11,11/6) -> (20/11,11/6) Reflection Matrix(141,-260,32,-59) (11/6,2/1) -> (22/5,9/2) Hyperbolic Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(201,-520,46,-119) (18/7,13/5) -> (13/3,22/5) Hyperbolic Matrix(191,-500,34,-89) (13/5,21/8) -> (11/2,17/3) Hyperbolic Matrix(419,-1140,154,-419) (19/7,30/11) -> (19/7,30/11) Reflection Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(101,-340,30,-101) (10/3,17/5) -> (10/3,17/5) Reflection Matrix(111,-380,26,-89) (17/5,7/2) -> (17/4,13/3) Hyperbolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic Matrix(79,-520,12,-79) (13/2,20/3) -> (13/2,20/3) 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(69,-80,44,-51) -> Matrix(1,-1,4,-3) 1/2 Matrix(49,-60,40,-49) -> Matrix(1,0,2,-1) (6/5,5/4) -> (0/1,1/1) Matrix(111,-140,88,-111) -> Matrix(1,0,2,-1) (5/4,14/11) -> (0/1,1/1) Matrix(391,-500,226,-289) -> Matrix(1,0,-2,1) 0/1 Matrix(139,-180,78,-101) -> Matrix(1,-1,-2,3) Matrix(29,-40,8,-11) -> Matrix(1,-1,0,1) 1/0 Matrix(79,-120,52,-79) -> Matrix(-1,0,2,1) (3/2,20/13) -> (-1/1,0/1) Matrix(441,-680,286,-441) -> Matrix(1,0,16,-1) (20/13,17/11) -> (0/1,1/8) Matrix(219,-340,38,-59) -> Matrix(3,-1,-4,1) Matrix(419,-660,266,-419) -> Matrix(3,-1,8,-3) (11/7,30/19) -> (1/4,1/2) Matrix(569,-900,208,-329) -> Matrix(3,-2,-2,1) Matrix(201,-320,76,-121) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(61,-100,36,-59) -> Matrix(1,-1,0,1) 1/0 Matrix(221,-380,82,-141) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(219,-380,34,-59) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(169,-300,40,-71) -> Matrix(1,0,4,1) 0/1 Matrix(199,-360,110,-199) -> Matrix(-1,0,12,1) (9/5,20/11) -> (-1/6,0/1) Matrix(241,-440,132,-241) -> Matrix(1,0,14,-1) (20/11,11/6) -> (0/1,1/7) Matrix(141,-260,32,-59) -> Matrix(3,-1,4,-1) 1/2 Matrix(9,-20,4,-9) -> Matrix(1,0,2,-1) (2/1,5/2) -> (0/1,1/1) Matrix(71,-180,28,-71) -> Matrix(1,0,2,-1) (5/2,18/7) -> (0/1,1/1) Matrix(201,-520,46,-119) -> Matrix(1,-2,2,-3) 1/1 Matrix(191,-500,34,-89) -> Matrix(3,-4,-2,3) Matrix(419,-1140,154,-419) -> Matrix(-1,5,0,1) (19/7,30/11) -> (5/2,1/0) Matrix(29,-80,4,-11) -> Matrix(1,-1,0,1) 1/0 Matrix(19,-60,6,-19) -> Matrix(-1,1,0,1) (3/1,10/3) -> (1/2,1/0) Matrix(101,-340,30,-101) -> Matrix(1,7,0,-1) (10/3,17/5) -> (-7/2,1/0) Matrix(111,-380,26,-89) -> Matrix(1,2,2,5) Matrix(21,-100,4,-19) -> Matrix(1,-3,0,1) 1/0 Matrix(79,-520,12,-79) -> Matrix(-1,0,2,1) (13/2,20/3) -> (-1/1,0/1) Matrix(41,-280,6,-41) -> Matrix(1,0,0,-1) (20/3,7/1) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.