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 -1/1 -4/9 -7/17 -3/8 -1/3 -2/7 -3/13 -1/7 0/1 1/4 1/3 3/7 1/2 11/19 2/3 1/1 11/9 23/17 7/5 41/29 3/2 61/39 31/19 5/3 19/11 9/5 13/7 2/1 7/3 17/7 5/2 13/5 8/3 11/4 3/1 17/5 31/9 7/2 11/3 4/1 13/3 23/5 33/7 5/1 17/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 -1/2 1/0 -5/11 -2/1 -9/20 -5/2 -4/9 -1/1 -3/7 -1/1 -5/12 -1/2 -7/17 -1/1 -9/22 -3/4 -2/5 -1/2 -7/18 -3/4 -5/13 -1/1 -1/3 -8/21 -1/2 -3/8 0/1 -10/27 1/0 -7/19 -1/1 -11/30 -1/2 -4/11 -1/2 -9/25 -1/3 -1/5 -14/39 -1/5 -5/14 -1/8 -1/3 0/1 -5/16 -1/6 -4/13 1/8 -7/23 1/5 1/3 -10/33 1/4 -3/10 1/2 -8/27 1/2 -5/17 1/1 -7/24 1/0 -2/7 0/1 -5/18 1/2 -3/11 1/3 1/1 -4/15 3/4 -1/4 1/2 -3/13 1/1 -5/22 1/0 -7/31 -1/1 1/1 -2/9 1/2 -1/5 1/1 -1/6 1/1 -3/19 1/1 5/3 -2/13 5/2 -1/7 2/1 0/1 1/0 1/6 1/0 1/5 -2/1 3/14 1/0 2/9 -5/2 1/4 -1/1 4/15 1/0 7/26 1/0 3/11 -2/1 5/18 -3/2 2/7 -5/4 3/10 1/0 1/3 -1/1 4/11 -1/1 3/8 -3/2 5/13 -4/5 12/31 -3/4 7/18 -2/3 2/5 -1/2 3/7 -1/1 4/9 -3/4 1/2 -1/2 5/9 0/1 4/7 -3/4 11/19 -1/2 7/12 -1/2 3/5 -1/1 -1/3 11/18 -3/4 19/31 -1/2 8/13 -1/2 13/21 0/1 5/8 -1/2 12/19 -5/14 7/11 -2/7 2/3 0/1 9/13 2/1 16/23 1/0 7/10 1/0 19/27 0/1 12/17 1/2 17/24 1/0 5/7 -1/1 13/18 -5/8 8/11 -1/2 11/15 0/1 3/4 -1/2 7/9 -1/3 -1/5 25/32 -1/4 18/23 -1/4 11/14 -1/5 4/5 -1/8 1/1 0/1 6/5 -1/6 11/9 0/1 16/13 1/22 5/4 1/8 9/7 1/5 1/3 22/17 1/4 13/10 1/4 4/3 1/2 23/17 1/3 1/1 19/14 1/2 15/11 0/1 11/8 1/2 29/21 1/2 18/13 5/8 25/18 1/2 7/5 1/1 24/17 1/0 41/29 1/0 17/12 -1/2 27/19 0/1 37/26 1/2 10/7 1/0 3/2 0/1 14/9 1/4 25/16 1/4 61/39 1/4 36/23 1/4 11/7 2/7 8/5 1/2 29/18 1/2 21/13 0/1 13/8 1/2 31/19 1/2 49/30 1/2 18/11 3/4 5/3 1/3 1/1 17/10 1/2 12/7 1/2 19/11 1/2 26/15 1/2 7/4 3/4 16/9 1/1 25/14 1/0 9/5 0/1 11/6 1/0 13/7 0/1 2/1 1/2 7/3 1/1 12/5 1/0 41/17 4/1 29/12 1/0 46/19 -1/1 17/7 -1/1 1/1 5/2 1/2 18/7 2/3 31/12 3/4 44/17 3/4 101/39 3/4 57/22 3/4 13/5 4/5 21/8 11/12 29/11 1/1 8/3 3/2 19/7 0/1 30/11 1/2 41/15 3/5 1/1 11/4 1/1 3/1 1/1 10/3 1/0 17/5 2/3 24/7 9/10 31/9 1/1 7/2 5/4 18/5 3/2 47/13 2/1 29/8 1/0 11/3 2/1 37/10 1/0 63/17 1/1 3/1 26/7 1/0 15/4 1/0 4/1 1/1 17/4 3/2 30/7 3/2 13/3 1/1 5/3 9/2 5/2 23/5 1/1 14/3 1/0 33/7 1/1 52/11 5/4 19/4 3/2 5/1 2/1 11/2 1/0 17/3 1/0 23/4 1/0 6/1 1/0 7/1 1/1 3/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(185,86,114,53) (-1/2,-5/11) -> (21/13,13/8) Hyperbolic Matrix(727,328,512,231) (-5/11,-9/20) -> (17/12,27/19) Hyperbolic Matrix(325,146,-906,-407) (-9/20,-4/9) -> (-14/39,-5/14) Hyperbolic Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(33,14,106,45) (-3/7,-5/12) -> (3/10,1/3) Hyperbolic Matrix(237,98,-578,-239) (-5/12,-7/17) -> (-7/17,-9/22) Parabolic Matrix(235,96,-776,-317) (-9/22,-2/5) -> (-10/33,-3/10) Hyperbolic Matrix(359,140,100,39) (-2/5,-7/18) -> (7/2,18/5) Hyperbolic Matrix(197,76,324,125) (-7/18,-5/13) -> (3/5,11/18) Hyperbolic Matrix(225,86,34,13) (-5/13,-8/21) -> (6/1,7/1) Hyperbolic Matrix(163,62,418,159) (-8/21,-3/8) -> (7/18,2/5) Hyperbolic Matrix(285,106,734,273) (-3/8,-10/27) -> (12/31,7/18) Hyperbolic Matrix(887,328,192,71) (-10/27,-7/19) -> (23/5,14/3) Hyperbolic Matrix(473,174,666,245) (-7/19,-11/30) -> (17/24,5/7) Hyperbolic Matrix(945,346,254,93) (-11/30,-4/11) -> (26/7,15/4) Hyperbolic Matrix(127,46,-566,-205) (-4/11,-9/25) -> (-7/31,-2/9) Hyperbolic Matrix(1813,652,748,269) (-9/25,-14/39) -> (46/19,17/7) Hyperbolic Matrix(29,10,-90,-31) (-5/14,-1/3) -> (-1/3,-5/16) Parabolic Matrix(321,100,260,81) (-5/16,-4/13) -> (16/13,5/4) Hyperbolic Matrix(85,26,-546,-167) (-4/13,-7/23) -> (-3/19,-2/13) Hyperbolic Matrix(1119,340,260,79) (-7/23,-10/33) -> (30/7,13/3) Hyperbolic Matrix(229,68,852,253) (-3/10,-8/27) -> (4/15,7/26) Hyperbolic Matrix(597,176,424,125) (-8/27,-5/17) -> (7/5,24/17) Hyperbolic Matrix(593,174,426,125) (-5/17,-7/24) -> (25/18,7/5) Hyperbolic Matrix(649,188,252,73) (-7/24,-2/7) -> (18/7,31/12) Hyperbolic Matrix(359,100,140,39) (-2/7,-5/18) -> (5/2,18/7) Hyperbolic Matrix(277,76,164,45) (-5/18,-3/11) -> (5/3,17/10) Hyperbolic Matrix(273,74,166,45) (-3/11,-4/15) -> (18/11,5/3) Hyperbolic Matrix(83,22,298,79) (-4/15,-1/4) -> (5/18,2/7) Hyperbolic Matrix(77,18,-338,-79) (-1/4,-3/13) -> (-3/13,-5/22) Parabolic Matrix(929,210,1190,269) (-5/22,-7/31) -> (7/9,25/32) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(73,14,26,5) (-1/5,-1/6) -> (11/4,3/1) Hyperbolic Matrix(455,74,166,27) (-1/6,-3/19) -> (41/15,11/4) Hyperbolic Matrix(331,50,470,71) (-2/13,-1/7) -> (19/27,12/17) Hyperbolic Matrix(69,8,112,13) (-1/7,0/1) -> (8/13,13/21) Hyperbolic Matrix(89,-12,52,-7) (0/1,1/6) -> (17/10,12/7) Hyperbolic Matrix(85,-16,16,-3) (1/6,1/5) -> (5/1,11/2) Hyperbolic Matrix(165,-34,34,-7) (1/5,3/14) -> (19/4,5/1) Hyperbolic Matrix(477,-104,344,-75) (3/14,2/9) -> (18/13,25/18) Hyperbolic Matrix(115,-26,146,-33) (2/9,1/4) -> (11/14,4/5) Hyperbolic Matrix(237,-62,302,-79) (1/4,4/15) -> (18/23,11/14) Hyperbolic Matrix(693,-188,188,-51) (7/26,3/11) -> (11/3,37/10) Hyperbolic Matrix(517,-142,142,-39) (3/11,5/18) -> (29/8,11/3) Hyperbolic Matrix(155,-46,246,-73) (2/7,3/10) -> (5/8,12/19) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(1427,-552,592,-229) (5/13,12/31) -> (12/5,41/17) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(125,-56,96,-43) (4/9,1/2) -> (13/10,4/3) Hyperbolic Matrix(117,-64,64,-35) (1/2,5/9) -> (9/5,11/6) Hyperbolic Matrix(205,-116,76,-43) (5/9,4/7) -> (8/3,19/7) Hyperbolic Matrix(545,-314,394,-227) (4/7,11/19) -> (29/21,18/13) Hyperbolic Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(89,-52,12,-7) (7/12,3/5) -> (7/1,1/0) Hyperbolic Matrix(1265,-774,894,-547) (11/18,19/31) -> (41/29,17/12) Hyperbolic Matrix(1277,-784,904,-555) (19/31,8/13) -> (24/17,41/29) Hyperbolic Matrix(777,-482,482,-299) (13/21,5/8) -> (29/18,21/13) Hyperbolic Matrix(587,-372,172,-109) (12/19,7/11) -> (17/5,24/7) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(721,-500,460,-319) (9/13,16/23) -> (36/23,11/7) Hyperbolic Matrix(755,-526,966,-673) (16/23,7/10) -> (25/32,18/23) Hyperbolic Matrix(1269,-892,892,-627) (7/10,19/27) -> (27/19,37/26) Hyperbolic Matrix(1265,-894,774,-547) (12/17,17/24) -> (49/30,18/11) Hyperbolic Matrix(477,-344,104,-75) (5/7,13/18) -> (9/2,23/5) Hyperbolic Matrix(545,-394,314,-227) (13/18,8/11) -> (26/15,7/4) Hyperbolic Matrix(659,-482,242,-177) (8/11,11/15) -> (19/7,30/11) Hyperbolic Matrix(345,-254,254,-187) (11/15,3/4) -> (19/14,15/11) Hyperbolic Matrix(113,-86,46,-35) (3/4,7/9) -> (17/7,5/2) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(199,-242,162,-197) (6/5,11/9) -> (11/9,16/13) Parabolic Matrix(115,-146,26,-33) (5/4,9/7) -> (13/3,9/2) Hyperbolic Matrix(907,-1172,332,-429) (9/7,22/17) -> (30/11,41/15) Hyperbolic Matrix(967,-1254,374,-485) (22/17,13/10) -> (31/12,44/17) Hyperbolic Matrix(631,-850,170,-229) (4/3,23/17) -> (63/17,26/7) Hyperbolic Matrix(1511,-2048,408,-553) (23/17,19/14) -> (37/10,63/17) Hyperbolic Matrix(347,-474,194,-265) (15/11,11/8) -> (25/14,9/5) Hyperbolic Matrix(1485,-2114,314,-447) (37/26,10/7) -> (52/11,19/4) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(633,-988,148,-231) (14/9,25/16) -> (17/4,30/7) Hyperbolic Matrix(3839,-6002,1482,-2317) (25/16,61/39) -> (101/39,57/22) Hyperbolic Matrix(4039,-6320,1560,-2441) (61/39,36/23) -> (44/17,101/39) Hyperbolic Matrix(155,-246,46,-73) (11/7,8/5) -> (10/3,17/5) Hyperbolic Matrix(223,-358,38,-61) (8/5,29/18) -> (23/4,6/1) Hyperbolic Matrix(1179,-1922,722,-1177) (13/8,31/19) -> (31/19,49/30) Parabolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(905,-1614,374,-667) (16/9,25/14) -> (29/12,46/19) Hyperbolic Matrix(485,-894,134,-247) (11/6,13/7) -> (47/13,29/8) Hyperbolic Matrix(173,-328,48,-91) (13/7,2/1) -> (18/5,47/13) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(1125,-2714,434,-1047) (41/17,29/12) -> (57/22,13/5) Hyperbolic Matrix(325,-854,94,-247) (21/8,29/11) -> (31/9,7/2) Hyperbolic Matrix(357,-944,104,-275) (29/11,8/3) -> (24/7,31/9) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(463,-2178,98,-461) (14/3,33/7) -> (33/7,52/11) Parabolic Matrix(103,-578,18,-101) (11/2,17/3) -> (17/3,23/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(185,86,114,53) -> Matrix(1,2,2,5) Matrix(727,328,512,231) -> Matrix(1,2,0,1) Matrix(325,146,-906,-407) -> Matrix(1,2,-6,-11) Matrix(37,16,104,45) -> Matrix(1,2,-2,-3) Matrix(33,14,106,45) -> Matrix(3,2,-2,-1) Matrix(237,98,-578,-239) -> Matrix(1,2,-2,-3) Matrix(235,96,-776,-317) -> Matrix(3,2,10,7) Matrix(359,140,100,39) -> Matrix(1,2,0,1) Matrix(197,76,324,125) -> Matrix(1,0,0,1) Matrix(225,86,34,13) -> Matrix(5,2,2,1) Matrix(163,62,418,159) -> Matrix(5,2,-8,-3) Matrix(285,106,734,273) -> Matrix(3,-2,-4,3) Matrix(887,328,192,71) -> Matrix(1,2,0,1) Matrix(473,174,666,245) -> Matrix(3,2,-2,-1) Matrix(945,346,254,93) -> Matrix(5,2,2,1) Matrix(127,46,-566,-205) -> Matrix(1,0,4,1) Matrix(1813,652,748,269) -> Matrix(1,0,4,1) Matrix(29,10,-90,-31) -> Matrix(1,0,2,1) Matrix(321,100,260,81) -> Matrix(1,0,14,1) Matrix(85,26,-546,-167) -> Matrix(11,-2,6,-1) Matrix(1119,340,260,79) -> Matrix(11,-2,6,-1) Matrix(229,68,852,253) -> Matrix(5,-2,-2,1) Matrix(597,176,424,125) -> Matrix(3,-2,2,-1) Matrix(593,174,426,125) -> Matrix(1,-2,2,-3) Matrix(649,188,252,73) -> Matrix(3,2,4,3) Matrix(359,100,140,39) -> Matrix(5,-2,8,-3) Matrix(277,76,164,45) -> Matrix(1,0,0,1) Matrix(273,74,166,45) -> Matrix(1,0,0,1) Matrix(83,22,298,79) -> Matrix(1,-2,0,1) Matrix(77,18,-338,-79) -> Matrix(3,-2,2,-1) Matrix(929,210,1190,269) -> Matrix(1,0,-4,1) Matrix(77,16,24,5) -> Matrix(3,-2,2,-1) Matrix(73,14,26,5) -> Matrix(1,-2,2,-3) Matrix(455,74,166,27) -> Matrix(3,-4,4,-5) Matrix(331,50,470,71) -> Matrix(1,-2,0,1) Matrix(69,8,112,13) -> Matrix(1,-2,-2,5) Matrix(89,-12,52,-7) -> Matrix(1,2,2,5) Matrix(85,-16,16,-3) -> Matrix(1,4,0,1) Matrix(165,-34,34,-7) -> Matrix(3,4,2,3) Matrix(477,-104,344,-75) -> Matrix(1,0,2,1) Matrix(115,-26,146,-33) -> Matrix(1,2,-6,-11) Matrix(237,-62,302,-79) -> Matrix(1,0,-4,1) Matrix(693,-188,188,-51) -> Matrix(1,4,0,1) Matrix(517,-142,142,-39) -> Matrix(3,4,2,3) Matrix(155,-46,246,-73) -> Matrix(1,0,-2,1) Matrix(205,-76,116,-43) -> Matrix(1,0,2,1) Matrix(377,-144,144,-55) -> Matrix(9,8,10,9) Matrix(1427,-552,592,-229) -> Matrix(11,8,4,3) Matrix(43,-18,98,-41) -> Matrix(1,2,-2,-3) Matrix(125,-56,96,-43) -> Matrix(3,2,10,7) Matrix(117,-64,64,-35) -> Matrix(1,0,2,1) Matrix(205,-116,76,-43) -> Matrix(1,0,2,1) Matrix(545,-314,394,-227) -> Matrix(7,4,12,7) Matrix(557,-324,404,-235) -> Matrix(1,0,4,1) Matrix(89,-52,12,-7) -> Matrix(5,2,2,1) Matrix(1265,-774,894,-547) -> Matrix(3,2,-2,-1) Matrix(1277,-784,904,-555) -> Matrix(5,2,2,1) Matrix(777,-482,482,-299) -> Matrix(1,0,4,1) Matrix(587,-372,172,-109) -> Matrix(13,4,16,5) Matrix(49,-32,72,-47) -> Matrix(1,0,4,1) Matrix(721,-500,460,-319) -> Matrix(1,-4,4,-15) Matrix(755,-526,966,-673) -> Matrix(1,2,-4,-7) Matrix(1269,-892,892,-627) -> Matrix(1,0,2,1) Matrix(1265,-894,774,-547) -> Matrix(1,-2,2,-3) Matrix(477,-344,104,-75) -> Matrix(1,0,2,1) Matrix(545,-394,314,-227) -> Matrix(7,4,12,7) Matrix(659,-482,242,-177) -> Matrix(1,0,4,1) Matrix(345,-254,254,-187) -> Matrix(1,0,4,1) Matrix(113,-86,46,-35) -> Matrix(1,0,4,1) Matrix(11,-10,10,-9) -> Matrix(1,0,2,1) Matrix(199,-242,162,-197) -> Matrix(1,0,28,1) Matrix(115,-146,26,-33) -> Matrix(11,-2,6,-1) Matrix(907,-1172,332,-429) -> Matrix(9,-2,14,-3) Matrix(967,-1254,374,-485) -> Matrix(29,-8,40,-11) Matrix(631,-850,170,-229) -> Matrix(3,-2,2,-1) Matrix(1511,-2048,408,-553) -> Matrix(3,-2,2,-1) Matrix(347,-474,194,-265) -> Matrix(1,0,-2,1) Matrix(1485,-2114,314,-447) -> Matrix(5,-4,4,-3) Matrix(49,-72,32,-47) -> Matrix(1,0,4,1) Matrix(633,-988,148,-231) -> Matrix(35,-8,22,-5) Matrix(3839,-6002,1482,-2317) -> Matrix(163,-40,216,-53) Matrix(4039,-6320,1560,-2441) -> Matrix(173,-44,232,-59) Matrix(155,-246,46,-73) -> Matrix(1,0,-2,1) Matrix(223,-358,38,-61) -> Matrix(1,0,-2,1) Matrix(1179,-1922,722,-1177) -> Matrix(9,-4,16,-7) Matrix(419,-722,242,-417) -> Matrix(9,-4,16,-7) Matrix(905,-1614,374,-667) -> Matrix(1,-2,0,1) Matrix(485,-894,134,-247) -> Matrix(1,2,0,1) Matrix(173,-328,48,-91) -> Matrix(7,-2,4,-1) Matrix(43,-98,18,-41) -> Matrix(3,-2,2,-1) Matrix(1125,-2714,434,-1047) -> Matrix(3,-16,4,-21) Matrix(325,-854,94,-247) -> Matrix(17,-16,16,-15) Matrix(357,-944,104,-275) -> Matrix(11,-12,12,-13) Matrix(33,-128,8,-31) -> Matrix(3,-2,2,-1) Matrix(463,-2178,98,-461) -> Matrix(5,-4,4,-3) Matrix(103,-578,18,-101) -> Matrix(1,-4,0,1) 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): 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 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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/4 1/3 3/7 11/19 3/5 2/3 5/7 1/1 11/9 23/17 7/5 3/2 61/39 5/3 19/11 13/7 2/1 7/3 17/7 13/5 29/11 3/1 7/2 11/3 4/1 13/3 33/7 5/1 17/3 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 0/1 1/0 1/6 1/0 1/5 -2/1 3/14 1/0 2/9 -5/2 1/4 -1/1 4/15 1/0 3/11 -2/1 2/7 -5/4 1/3 -1/1 3/8 -3/2 5/13 -4/5 2/5 -1/2 3/7 -1/1 4/9 -3/4 1/2 -1/2 5/9 0/1 4/7 -3/4 11/19 -1/2 7/12 -1/2 3/5 -1/1 -1/3 8/13 -1/2 5/8 -1/2 7/11 -2/7 2/3 0/1 9/13 2/1 16/23 1/0 7/10 1/0 12/17 1/2 5/7 -1/1 13/18 -5/8 8/11 -1/2 11/15 0/1 3/4 -1/2 7/9 -1/3 -1/5 18/23 -1/4 11/14 -1/5 4/5 -1/8 1/1 0/1 6/5 -1/6 11/9 0/1 5/4 1/8 9/7 1/5 1/3 22/17 1/4 13/10 1/4 4/3 1/2 23/17 1/3 1/1 19/14 1/2 15/11 0/1 11/8 1/2 29/21 1/2 18/13 5/8 7/5 1/1 17/12 -1/2 10/7 1/0 3/2 0/1 14/9 1/4 25/16 1/4 61/39 1/4 36/23 1/4 11/7 2/7 8/5 1/2 13/8 1/2 5/3 1/3 1/1 12/7 1/2 19/11 1/2 26/15 1/2 7/4 3/4 9/5 0/1 11/6 1/0 13/7 0/1 2/1 1/2 7/3 1/1 12/5 1/0 29/12 1/0 17/7 -1/1 1/1 5/2 1/2 13/5 4/5 21/8 11/12 29/11 1/1 8/3 3/2 3/1 1/1 7/2 5/4 11/3 2/1 15/4 1/0 4/1 1/1 17/4 3/2 13/3 1/1 5/3 9/2 5/2 14/3 1/0 33/7 1/1 19/4 3/2 5/1 2/1 11/2 1/0 17/3 1/0 6/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(86,-13,53,-8) (0/1,1/6) -> (8/5,13/8) Hyperbolic Matrix(85,-16,16,-3) (1/6,1/5) -> (5/1,11/2) Hyperbolic Matrix(165,-34,34,-7) (1/5,3/14) -> (19/4,5/1) Hyperbolic Matrix(328,-71,231,-50) (3/14,2/9) -> (17/12,10/7) Hyperbolic Matrix(115,-26,146,-33) (2/9,1/4) -> (11/14,4/5) Hyperbolic Matrix(237,-62,302,-79) (1/4,4/15) -> (18/23,11/14) Hyperbolic Matrix(346,-93,253,-68) (4/15,3/11) -> (15/11,11/8) Hyperbolic Matrix(140,-39,79,-22) (3/11,2/7) -> (7/4,9/5) Hyperbolic Matrix(16,-5,45,-14) (2/7,1/3) -> (1/3,3/8) Parabolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(100,-39,159,-62) (5/13,2/5) -> (5/8,7/11) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(125,-56,96,-43) (4/9,1/2) -> (13/10,4/3) Hyperbolic Matrix(117,-64,64,-35) (1/2,5/9) -> (9/5,11/6) Hyperbolic Matrix(140,-79,39,-22) (5/9,4/7) -> (7/2,11/3) Hyperbolic Matrix(545,-314,394,-227) (4/7,11/19) -> (29/21,18/13) Hyperbolic Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(76,-45,125,-74) (7/12,3/5) -> (3/5,8/13) Parabolic Matrix(86,-53,13,-8) (8/13,5/8) -> (6/1,1/0) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(721,-500,460,-319) (9/13,16/23) -> (36/23,11/7) Hyperbolic Matrix(566,-395,235,-164) (16/23,7/10) -> (12/5,29/12) Hyperbolic Matrix(328,-231,71,-50) (7/10,12/17) -> (9/2,14/3) Hyperbolic Matrix(176,-125,245,-174) (12/17,5/7) -> (5/7,13/18) Parabolic Matrix(545,-394,314,-227) (13/18,8/11) -> (26/15,7/4) Hyperbolic Matrix(346,-253,93,-68) (8/11,11/15) -> (11/3,15/4) Hyperbolic Matrix(345,-254,254,-187) (11/15,3/4) -> (19/14,15/11) Hyperbolic Matrix(113,-86,46,-35) (3/4,7/9) -> (17/7,5/2) Hyperbolic Matrix(652,-509,269,-210) (7/9,18/23) -> (29/12,17/7) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(100,-121,81,-98) (6/5,11/9) -> (11/9,5/4) Parabolic Matrix(115,-146,26,-33) (5/4,9/7) -> (13/3,9/2) Hyperbolic Matrix(340,-439,79,-102) (9/7,22/17) -> (17/4,13/3) Hyperbolic Matrix(488,-633,313,-406) (22/17,13/10) -> (14/9,25/16) Hyperbolic Matrix(392,-529,289,-390) (4/3,23/17) -> (23/17,19/14) Parabolic Matrix(176,-245,125,-174) (18/13,7/5) -> (7/5,17/12) Parabolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(2380,-3721,1521,-2378) (25/16,61/39) -> (61/39,36/23) Parabolic Matrix(100,-159,39,-62) (11/7,8/5) -> (5/2,13/5) Hyperbolic Matrix(76,-125,45,-74) (13/8,5/3) -> (5/3,12/7) Parabolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(92,-169,49,-90) (11/6,13/7) -> (13/7,2/1) Parabolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(320,-841,121,-318) (21/8,29/11) -> (29/11,8/3) Parabolic Matrix(16,-45,5,-14) (8/3,3/1) -> (3/1,7/2) Parabolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(232,-1089,49,-230) (14/3,33/7) -> (33/7,19/4) Parabolic Matrix(52,-289,9,-50) (11/2,17/3) -> (17/3,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,0,1) Matrix(86,-13,53,-8) -> Matrix(1,2,2,5) Matrix(85,-16,16,-3) -> Matrix(1,4,0,1) Matrix(165,-34,34,-7) -> Matrix(3,4,2,3) Matrix(328,-71,231,-50) -> Matrix(1,2,0,1) Matrix(115,-26,146,-33) -> Matrix(1,2,-6,-11) Matrix(237,-62,302,-79) -> Matrix(1,0,-4,1) Matrix(346,-93,253,-68) -> Matrix(1,2,2,5) Matrix(140,-39,79,-22) -> Matrix(1,2,0,1) Matrix(16,-5,45,-14) -> Matrix(1,2,-2,-3) Matrix(377,-144,144,-55) -> Matrix(9,8,10,9) Matrix(100,-39,159,-62) -> Matrix(3,2,-8,-5) Matrix(43,-18,98,-41) -> Matrix(1,2,-2,-3) Matrix(125,-56,96,-43) -> Matrix(3,2,10,7) Matrix(117,-64,64,-35) -> Matrix(1,0,2,1) Matrix(140,-79,39,-22) -> Matrix(1,2,0,1) Matrix(545,-314,394,-227) -> Matrix(7,4,12,7) Matrix(557,-324,404,-235) -> Matrix(1,0,4,1) Matrix(76,-45,125,-74) -> Matrix(1,0,0,1) Matrix(86,-53,13,-8) -> Matrix(5,2,2,1) Matrix(49,-32,72,-47) -> Matrix(1,0,4,1) Matrix(721,-500,460,-319) -> Matrix(1,-4,4,-15) Matrix(566,-395,235,-164) -> Matrix(1,2,0,1) Matrix(328,-231,71,-50) -> Matrix(1,2,0,1) Matrix(176,-125,245,-174) -> Matrix(1,2,-2,-3) Matrix(545,-394,314,-227) -> Matrix(7,4,12,7) Matrix(346,-253,93,-68) -> Matrix(5,2,2,1) Matrix(345,-254,254,-187) -> Matrix(1,0,4,1) Matrix(113,-86,46,-35) -> Matrix(1,0,4,1) Matrix(652,-509,269,-210) -> Matrix(1,0,4,1) Matrix(11,-10,10,-9) -> Matrix(1,0,2,1) Matrix(100,-121,81,-98) -> Matrix(1,0,14,1) Matrix(115,-146,26,-33) -> Matrix(11,-2,6,-1) Matrix(340,-439,79,-102) -> Matrix(11,-2,6,-1) Matrix(488,-633,313,-406) -> Matrix(7,-2,32,-9) Matrix(392,-529,289,-390) -> Matrix(1,0,0,1) Matrix(176,-245,125,-174) -> Matrix(3,-2,2,-1) Matrix(49,-72,32,-47) -> Matrix(1,0,4,1) Matrix(2380,-3721,1521,-2378) -> Matrix(57,-14,224,-55) Matrix(100,-159,39,-62) -> Matrix(5,-2,8,-3) Matrix(76,-125,45,-74) -> Matrix(1,0,0,1) Matrix(419,-722,242,-417) -> Matrix(9,-4,16,-7) Matrix(92,-169,49,-90) -> Matrix(1,0,2,1) Matrix(43,-98,18,-41) -> Matrix(3,-2,2,-1) Matrix(320,-841,121,-318) -> Matrix(15,-14,14,-13) Matrix(16,-45,5,-14) -> Matrix(3,-2,2,-1) Matrix(33,-128,8,-31) -> Matrix(3,-2,2,-1) Matrix(232,-1089,49,-230) -> Matrix(3,-2,2,-1) Matrix(52,-289,9,-50) -> 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 -1/1 (0/1,1/0) 0 1 1/1 0/1 2 10 11/9 0/1 14 1 5/4 1/8 1 20 9/7 0 5 4/3 1/2 1 20 23/17 (0/1,1/2) 0 1 15/11 0/1 2 10 11/8 1/2 1 20 18/13 5/8 1 20 7/5 1/1 2 5 17/12 -1/2 1 20 10/7 1/0 1 20 3/2 0/1 2 4 14/9 1/4 1 20 25/16 1/4 1 20 61/39 1/4 14 1 11/7 2/7 2 10 8/5 1/2 1 20 13/8 1/2 1 20 5/3 0 5 12/7 1/2 1 20 19/11 1/2 4 2 26/15 1/2 1 20 7/4 3/4 1 20 9/5 0/1 2 10 13/7 0/1 2 1 2/1 1/2 1 20 7/3 1/1 2 2 12/5 1/0 1 20 29/12 1/0 1 20 17/7 0 5 5/2 1/2 1 20 13/5 4/5 2 10 29/11 1/1 14 1 8/3 3/2 1 20 3/1 1/1 2 5 7/2 5/4 1 20 11/3 2/1 2 10 15/4 1/0 1 20 4/1 1/1 1 4 17/4 3/2 1 20 13/3 0 5 9/2 5/2 1 20 14/3 1/0 1 20 33/7 1/1 2 1 5/1 2/1 2 10 17/3 1/0 2 1 6/1 1/0 1 20 1/0 1/0 1 20 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(10,-11,9,-10) (1/1,11/9) -> (1/1,11/9) Reflection Matrix(89,-110,72,-89) (11/9,5/4) -> (11/9,5/4) Reflection Matrix(115,-146,26,-33) (5/4,9/7) -> (13/3,9/2) Hyperbolic Matrix(86,-113,35,-46) (9/7,4/3) -> (17/7,5/2) Glide Reflection Matrix(137,-184,102,-137) (4/3,23/17) -> (4/3,23/17) Reflection Matrix(254,-345,187,-254) (23/17,15/11) -> (23/17,15/11) Reflection Matrix(253,-346,68,-93) (15/11,11/8) -> (11/3,15/4) Glide Reflection Matrix(394,-545,227,-314) (11/8,18/13) -> (26/15,7/4) Glide Reflection Matrix(176,-245,125,-174) (18/13,7/5) -> (7/5,17/12) Parabolic Matrix(231,-328,50,-71) (17/12,10/7) -> (9/2,14/3) Glide Reflection Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(453,-706,188,-293) (14/9,25/16) -> (12/5,29/12) Glide Reflection Matrix(1951,-3050,1248,-1951) (25/16,61/39) -> (25/16,61/39) Reflection Matrix(428,-671,273,-428) (61/39,11/7) -> (61/39,11/7) Reflection Matrix(100,-159,39,-62) (11/7,8/5) -> (5/2,13/5) Hyperbolic Matrix(53,-86,8,-13) (8/5,13/8) -> (6/1,1/0) Glide Reflection Matrix(76,-125,45,-74) (13/8,5/3) -> (5/3,12/7) Parabolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(79,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(64,-117,35,-64) (9/5,13/7) -> (9/5,13/7) Reflection Matrix(27,-52,14,-27) (13/7,2/1) -> (13/7,2/1) Reflection Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(275,-666,64,-155) (29/12,17/7) -> (17/4,13/3) Glide Reflection Matrix(144,-377,55,-144) (13/5,29/11) -> (13/5,29/11) Reflection Matrix(175,-464,66,-175) (29/11,8/3) -> (29/11,8/3) Reflection Matrix(16,-45,5,-14) (8/3,3/1) -> (3/1,7/2) Parabolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(197,-924,42,-197) (14/3,33/7) -> (14/3,33/7) Reflection Matrix(34,-165,7,-34) (33/7,5/1) -> (33/7,5/1) Reflection Matrix(16,-85,3,-16) (5/1,17/3) -> (5/1,17/3) Reflection Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,0,-1) (-1/1,1/1) -> (0/1,1/0) Matrix(10,-11,9,-10) -> Matrix(1,0,2,-1) (1/1,11/9) -> (0/1,1/1) Matrix(89,-110,72,-89) -> Matrix(1,0,16,-1) (11/9,5/4) -> (0/1,1/8) Matrix(115,-146,26,-33) -> Matrix(11,-2,6,-1) Matrix(86,-113,35,-46) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(137,-184,102,-137) -> Matrix(1,0,4,-1) (4/3,23/17) -> (0/1,1/2) Matrix(254,-345,187,-254) -> Matrix(1,0,4,-1) (23/17,15/11) -> (0/1,1/2) Matrix(253,-346,68,-93) -> Matrix(5,-2,2,-1) Matrix(394,-545,227,-314) -> Matrix(7,-4,12,-7) *** -> (1/2,2/3) Matrix(176,-245,125,-174) -> Matrix(3,-2,2,-1) 1/1 Matrix(231,-328,50,-71) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(49,-72,32,-47) -> Matrix(1,0,4,1) 0/1 Matrix(453,-706,188,-293) -> Matrix(9,-2,4,-1) Matrix(1951,-3050,1248,-1951) -> Matrix(41,-10,168,-41) (25/16,61/39) -> (5/21,1/4) Matrix(428,-671,273,-428) -> Matrix(15,-4,56,-15) (61/39,11/7) -> (1/4,2/7) Matrix(100,-159,39,-62) -> Matrix(5,-2,8,-3) 1/2 Matrix(53,-86,8,-13) -> Matrix(5,-2,2,-1) Matrix(76,-125,45,-74) -> Matrix(1,0,0,1) Matrix(419,-722,242,-417) -> Matrix(9,-4,16,-7) 1/2 Matrix(79,-140,22,-39) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(64,-117,35,-64) -> Matrix(1,0,2,-1) (9/5,13/7) -> (0/1,1/1) Matrix(27,-52,14,-27) -> Matrix(1,0,4,-1) (13/7,2/1) -> (0/1,1/2) Matrix(43,-98,18,-41) -> Matrix(3,-2,2,-1) 1/1 Matrix(275,-666,64,-155) -> Matrix(3,2,2,1) Matrix(144,-377,55,-144) -> Matrix(9,-8,10,-9) (13/5,29/11) -> (4/5,1/1) Matrix(175,-464,66,-175) -> Matrix(5,-6,4,-5) (29/11,8/3) -> (1/1,3/2) Matrix(16,-45,5,-14) -> Matrix(3,-2,2,-1) 1/1 Matrix(33,-128,8,-31) -> Matrix(3,-2,2,-1) 1/1 Matrix(197,-924,42,-197) -> Matrix(-1,2,0,1) (14/3,33/7) -> (1/1,1/0) Matrix(34,-165,7,-34) -> Matrix(3,-4,2,-3) (33/7,5/1) -> (1/1,2/1) Matrix(16,-85,3,-16) -> Matrix(-1,4,0,1) (5/1,17/3) -> (2/1,1/0) Matrix(35,-204,6,-35) -> Matrix(-1,2,0,1) (17/3,6/1) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.