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 -7/1 -6/1 -4/1 -7/2 -10/3 -12/5 -7/3 -2/1 -7/4 -42/29 -7/5 -7/6 0/1 1/1 7/6 14/11 7/5 3/2 14/9 7/4 9/5 2/1 7/3 12/5 5/2 28/11 8/3 14/5 3/1 42/13 10/3 7/2 112/31 11/3 56/15 4/1 13/3 84/19 9/2 14/3 5/1 11/2 28/5 6/1 13/2 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/2 -6/1 -4/9 -2/5 -11/2 -7/18 -5/1 -1/3 -14/3 -1/3 -9/2 -5/16 -22/5 -2/7 0/1 -13/3 -1/3 -4/1 -2/7 -15/4 -1/4 -11/3 -1/3 -7/2 -1/4 -17/5 -5/21 -27/8 -11/48 -10/3 -2/9 0/1 -13/4 -1/4 -16/5 -4/17 -3/1 -1/5 -14/5 -1/5 -11/4 -3/16 -19/7 -5/27 -46/17 -2/11 -12/67 -27/10 -7/40 -8/3 0/1 -21/8 -1/5 -13/5 -1/5 -18/7 -4/21 -2/11 -5/2 -1/6 -17/7 -1/3 -29/12 -5/24 -12/5 -2/11 -7/3 -1/6 -16/7 0/1 -25/11 -3/19 -34/15 -2/13 -4/27 -9/4 -1/8 -11/5 -1/7 -13/6 -1/6 -2/1 -2/13 0/1 -11/6 -1/6 -9/5 -1/5 -7/4 -1/7 -19/11 -3/23 -31/18 -1/6 -12/7 -2/15 -41/24 -1/8 -70/41 -1/8 -29/17 -5/41 -17/10 -1/10 -5/3 -1/7 -28/17 -2/15 -23/14 -5/38 -18/11 -2/15 -4/31 -13/8 -1/8 -34/21 -2/15 -4/31 -21/13 -1/8 -29/18 -7/58 -8/5 0/1 -27/17 -7/51 -19/12 -5/38 -11/7 -3/23 -14/9 -1/8 -3/2 -1/8 -16/11 -4/35 -29/20 -11/98 -42/29 -1/9 -13/9 -1/9 -10/7 -2/17 0/1 -7/5 -1/9 -18/13 -6/55 -4/37 -65/47 -25/231 -112/81 -4/37 -47/34 -19/176 -29/21 -13/121 -11/8 -1/10 -26/19 -2/19 0/1 -41/30 -1/10 -56/41 0/1 -15/11 -1/9 -4/3 -2/19 -13/10 -1/10 -22/17 -2/19 0/1 -53/41 -5/47 -84/65 -2/19 -31/24 -5/48 -9/7 -5/49 -14/11 -1/10 -5/4 -1/10 -11/9 -7/73 -28/23 -2/21 -17/14 -11/116 -6/5 -2/21 -4/43 -13/11 -1/11 -7/6 -1/11 -8/7 -4/45 -1/1 -1/13 0/1 0/1 1/1 1/13 7/6 1/11 6/5 4/43 2/21 11/9 7/73 5/4 1/10 14/11 1/10 9/7 5/49 22/17 0/1 2/19 13/10 1/10 4/3 2/19 15/11 1/9 11/8 1/10 7/5 1/9 17/12 5/44 27/19 11/95 10/7 0/1 2/17 13/9 1/9 16/11 4/35 3/2 1/8 14/9 1/8 11/7 3/23 19/12 5/38 46/29 2/15 12/89 27/17 7/51 8/5 0/1 21/13 1/8 13/8 1/8 18/11 4/31 2/15 5/3 1/7 17/10 1/10 29/17 5/41 12/7 2/15 7/4 1/7 16/9 0/1 25/14 3/20 34/19 2/13 4/25 9/5 1/5 11/6 1/6 13/7 1/7 2/1 0/1 2/13 11/5 1/7 9/4 1/8 7/3 1/6 19/8 3/16 31/13 1/7 12/5 2/11 41/17 1/5 70/29 1/5 29/12 5/24 17/7 1/3 5/2 1/6 28/11 2/11 23/9 5/27 18/7 2/11 4/21 13/5 1/5 34/13 2/11 4/21 21/8 1/5 29/11 7/33 8/3 0/1 27/10 7/40 19/7 5/27 11/4 3/16 14/5 1/5 3/1 1/5 16/5 4/17 29/9 11/45 42/13 1/4 13/4 1/4 10/3 0/1 2/9 7/2 1/4 18/5 6/23 4/15 65/18 25/94 112/31 4/15 47/13 19/71 29/8 13/48 11/3 1/3 26/7 0/1 2/7 41/11 1/3 56/15 0/1 15/4 1/4 4/1 2/7 13/3 1/3 22/5 0/1 2/7 53/12 5/18 84/19 2/7 31/7 5/17 9/2 5/16 14/3 1/3 5/1 1/3 11/2 7/18 28/5 2/5 17/3 11/27 6/1 2/5 4/9 13/2 1/2 7/1 1/2 8/1 4/7 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(29,224,-18,-139) (-7/1,1/0) -> (-21/13,-29/18) Hyperbolic Matrix(55,364,-34,-225) (-7/1,-6/1) -> (-34/21,-21/13) Hyperbolic Matrix(29,168,-24,-139) (-6/1,-11/2) -> (-17/14,-6/5) Hyperbolic Matrix(27,140,16,83) (-11/2,-5/1) -> (5/3,17/10) Hyperbolic Matrix(29,140,6,29) (-5/1,-14/3) -> (14/3,5/1) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(113,504,-50,-223) (-9/2,-22/5) -> (-34/15,-9/4) Hyperbolic Matrix(167,728,64,279) (-22/5,-13/3) -> (13/5,34/13) Hyperbolic Matrix(27,112,20,83) (-13/3,-4/1) -> (4/3,15/11) Hyperbolic Matrix(29,112,22,85) (-4/1,-15/4) -> (13/10,4/3) Hyperbolic Matrix(83,308,-38,-141) (-15/4,-11/3) -> (-11/5,-13/6) Hyperbolic Matrix(55,196,-16,-57) (-11/3,-7/2) -> (-7/2,-17/5) Parabolic Matrix(281,952,116,393) (-17/5,-27/8) -> (29/12,17/7) Hyperbolic Matrix(449,1512,-166,-559) (-27/8,-10/3) -> (-46/17,-27/10) Hyperbolic Matrix(111,364,68,223) (-10/3,-13/4) -> (13/8,18/11) Hyperbolic Matrix(253,812,-148,-475) (-13/4,-16/5) -> (-12/7,-41/24) Hyperbolic Matrix(167,532,70,223) (-16/5,-3/1) -> (31/13,12/5) Hyperbolic Matrix(29,84,10,29) (-3/1,-14/5) -> (14/5,3/1) Hyperbolic Matrix(111,308,40,111) (-14/5,-11/4) -> (11/4,14/5) Hyperbolic Matrix(113,308,62,169) (-11/4,-19/7) -> (9/5,11/6) Hyperbolic Matrix(1231,3332,-890,-2409) (-19/7,-46/17) -> (-18/13,-65/47) Hyperbolic Matrix(83,224,10,27) (-27/10,-8/3) -> (8/1,1/0) Hyperbolic Matrix(85,224,-74,-195) (-8/3,-21/8) -> (-7/6,-8/7) Hyperbolic Matrix(139,364,-118,-309) (-21/8,-13/5) -> (-13/11,-7/6) Hyperbolic Matrix(141,364,98,253) (-13/5,-18/7) -> (10/7,13/9) Hyperbolic Matrix(197,504,-120,-307) (-18/7,-5/2) -> (-23/14,-18/11) Hyperbolic Matrix(57,140,46,113) (-5/2,-17/7) -> (11/9,5/4) Hyperbolic Matrix(335,812,92,223) (-17/7,-29/12) -> (29/8,11/3) Hyperbolic Matrix(337,812,-232,-559) (-29/12,-12/5) -> (-16/11,-29/20) Hyperbolic Matrix(83,196,-36,-85) (-12/5,-7/3) -> (-7/3,-16/7) Parabolic Matrix(197,448,62,141) (-16/7,-25/11) -> (3/1,16/5) Hyperbolic Matrix(1037,2352,-802,-1819) (-25/11,-34/15) -> (-22/17,-53/41) Hyperbolic Matrix(139,308,88,195) (-9/4,-11/5) -> (11/7,19/12) Hyperbolic Matrix(197,420,-144,-307) (-13/6,-2/1) -> (-26/19,-41/30) Hyperbolic Matrix(167,308,-122,-225) (-2/1,-11/6) -> (-11/8,-26/19) Hyperbolic Matrix(169,308,62,113) (-11/6,-9/5) -> (19/7,11/4) Hyperbolic Matrix(111,196,-64,-113) (-9/5,-7/4) -> (-7/4,-19/11) Parabolic Matrix(503,868,-390,-673) (-19/11,-31/18) -> (-31/24,-9/7) Hyperbolic Matrix(309,532,212,365) (-31/18,-12/7) -> (16/11,3/2) Hyperbolic Matrix(1541,2632,476,813) (-41/24,-70/41) -> (42/13,13/4) Hyperbolic Matrix(1903,3248,590,1007) (-70/41,-29/17) -> (29/9,42/13) Hyperbolic Matrix(559,952,394,671) (-29/17,-17/10) -> (17/12,27/19) Hyperbolic Matrix(83,140,16,27) (-17/10,-5/3) -> (5/1,11/2) Hyperbolic Matrix(475,784,186,307) (-5/3,-28/17) -> (28/11,23/9) Hyperbolic Matrix(477,784,188,309) (-28/17,-23/14) -> (5/2,28/11) Hyperbolic Matrix(223,364,68,111) (-18/11,-13/8) -> (13/4,10/3) Hyperbolic Matrix(449,728,346,561) (-13/8,-34/21) -> (22/17,13/10) Hyperbolic Matrix(279,448,104,167) (-29/18,-8/5) -> (8/3,27/10) Hyperbolic Matrix(281,448,106,169) (-8/5,-27/17) -> (29/11,8/3) Hyperbolic Matrix(1147,1820,-830,-1317) (-27/17,-19/12) -> (-47/34,-29/21) Hyperbolic Matrix(195,308,88,139) (-19/12,-11/7) -> (11/5,9/4) Hyperbolic Matrix(197,308,126,197) (-11/7,-14/9) -> (14/9,11/7) Hyperbolic Matrix(55,84,36,55) (-14/9,-3/2) -> (3/2,14/9) Hyperbolic Matrix(307,448,172,251) (-3/2,-16/11) -> (16/9,25/14) Hyperbolic Matrix(2241,3248,928,1345) (-29/20,-42/29) -> (70/29,29/12) Hyperbolic Matrix(1819,2632,754,1091) (-42/29,-13/9) -> (41/17,70/29) Hyperbolic Matrix(253,364,98,141) (-13/9,-10/7) -> (18/7,13/5) Hyperbolic Matrix(139,196,-100,-141) (-10/7,-7/5) -> (-7/5,-18/13) Parabolic Matrix(9071,12544,2510,3471) (-65/47,-112/81) -> (112/31,47/13) Hyperbolic Matrix(9073,12544,2512,3473) (-112/81,-47/34) -> (65/18,112/31) Hyperbolic Matrix(589,812,346,477) (-29/21,-11/8) -> (17/10,29/17) Hyperbolic Matrix(2295,3136,614,839) (-41/30,-56/41) -> (56/15,15/4) Hyperbolic Matrix(2297,3136,616,841) (-56/41,-15/11) -> (41/11,56/15) Hyperbolic Matrix(83,112,20,27) (-15/11,-4/3) -> (4/1,13/3) Hyperbolic Matrix(85,112,22,29) (-4/3,-13/10) -> (15/4,4/1) Hyperbolic Matrix(281,364,44,57) (-13/10,-22/17) -> (6/1,13/2) Hyperbolic Matrix(5459,7056,1234,1595) (-53/41,-84/65) -> (84/19,31/7) Hyperbolic Matrix(5461,7056,1236,1597) (-84/65,-31/24) -> (53/12,84/19) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(113,140,46,57) (-5/4,-11/9) -> (17/7,5/2) Hyperbolic Matrix(643,784,114,139) (-11/9,-28/23) -> (28/5,17/3) Hyperbolic Matrix(645,784,116,141) (-28/23,-17/14) -> (11/2,28/5) Hyperbolic Matrix(307,364,70,83) (-6/5,-13/11) -> (13/3,22/5) Hyperbolic Matrix(197,224,124,141) (-8/7,-1/1) -> (27/17,8/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(195,-224,74,-85) (1/1,7/6) -> (21/8,29/11) Hyperbolic Matrix(309,-364,118,-139) (7/6,6/5) -> (34/13,21/8) Hyperbolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(391,-504,218,-281) (9/7,22/17) -> (34/19,9/5) Hyperbolic Matrix(225,-308,122,-167) (15/11,11/8) -> (11/6,13/7) Hyperbolic Matrix(141,-196,100,-139) (11/8,7/5) -> (7/5,17/12) Parabolic Matrix(1063,-1512,670,-953) (27/19,10/7) -> (46/29,27/17) Hyperbolic Matrix(559,-812,232,-337) (13/9,16/11) -> (12/5,41/17) Hyperbolic Matrix(2101,-3332,582,-923) (19/12,46/29) -> (18/5,65/18) Hyperbolic Matrix(139,-224,18,-29) (8/5,21/13) -> (7/1,8/1) Hyperbolic Matrix(225,-364,34,-55) (21/13,13/8) -> (13/2,7/1) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(475,-812,148,-253) (29/17,12/7) -> (16/5,29/9) Hyperbolic Matrix(113,-196,64,-111) (12/7,7/4) -> (7/4,16/9) Parabolic Matrix(1315,-2352,298,-533) (25/14,34/19) -> (22/5,53/12) Hyperbolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(141,-308,38,-83) (2/1,11/5) -> (11/3,26/7) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(365,-868,82,-195) (19/8,31/13) -> (31/7,9/2) Hyperbolic Matrix(673,-1820,186,-503) (27/10,19/7) -> (47/13,29/8) Hyperbolic Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(29,224,-18,-139) -> Matrix(7,4,-58,-33) Matrix(55,364,-34,-225) -> Matrix(13,6,-102,-47) Matrix(29,168,-24,-139) -> Matrix(19,8,-202,-85) Matrix(27,140,16,83) -> Matrix(5,2,32,13) Matrix(29,140,6,29) -> Matrix(17,6,48,17) Matrix(55,252,12,55) -> Matrix(31,10,96,31) Matrix(113,504,-50,-223) -> Matrix(13,4,-88,-27) Matrix(167,728,64,279) -> Matrix(13,4,68,21) Matrix(27,112,20,83) -> Matrix(13,4,120,37) Matrix(29,112,22,85) -> Matrix(15,4,146,39) Matrix(83,308,-38,-141) -> Matrix(7,2,-46,-13) Matrix(55,196,-16,-57) -> Matrix(23,6,-96,-25) Matrix(281,952,116,393) -> Matrix(17,4,72,17) Matrix(449,1512,-166,-559) -> Matrix(53,12,-296,-67) Matrix(111,364,68,223) -> Matrix(7,2,52,15) Matrix(253,812,-148,-475) -> Matrix(25,6,-196,-47) Matrix(167,532,70,223) -> Matrix(9,2,58,13) Matrix(29,84,10,29) -> Matrix(9,2,40,9) Matrix(111,308,40,111) -> Matrix(31,6,160,31) Matrix(113,308,62,169) -> Matrix(11,2,82,15) Matrix(1231,3332,-890,-2409) -> Matrix(167,30,-1542,-277) Matrix(83,224,10,27) -> Matrix(23,4,40,7) Matrix(85,224,-74,-195) -> Matrix(19,4,-214,-45) Matrix(139,364,-118,-309) -> Matrix(31,6,-336,-65) Matrix(141,364,98,253) -> Matrix(11,2,104,19) Matrix(197,504,-120,-307) -> Matrix(43,8,-328,-61) Matrix(57,140,46,113) -> Matrix(13,2,136,21) Matrix(335,812,92,223) -> Matrix(7,2,24,7) Matrix(337,812,-232,-559) -> Matrix(31,6,-274,-53) Matrix(83,196,-36,-85) -> Matrix(11,2,-72,-13) Matrix(197,448,62,141) -> Matrix(25,4,106,17) Matrix(1037,2352,-802,-1819) -> Matrix(27,4,-250,-37) Matrix(139,308,88,195) -> Matrix(11,2,82,15) Matrix(197,420,-144,-307) -> Matrix(13,2,-124,-19) Matrix(167,308,-122,-225) -> Matrix(13,2,-124,-19) Matrix(169,308,62,113) -> Matrix(15,2,82,11) Matrix(111,196,-64,-113) -> Matrix(13,2,-98,-15) Matrix(503,868,-390,-673) -> Matrix(17,2,-162,-19) Matrix(309,532,212,365) -> Matrix(13,2,110,17) Matrix(1541,2632,476,813) -> Matrix(63,8,244,31) Matrix(1903,3248,590,1007) -> Matrix(129,16,524,65) Matrix(559,952,394,671) -> Matrix(35,4,306,35) Matrix(83,140,16,27) -> Matrix(13,2,32,5) Matrix(475,784,186,307) -> Matrix(89,12,482,65) Matrix(477,784,188,309) -> Matrix(91,12,508,67) Matrix(223,364,68,111) -> Matrix(15,2,52,7) Matrix(449,728,346,561) -> Matrix(31,4,302,39) Matrix(279,448,104,167) -> Matrix(1,0,14,1) Matrix(281,448,106,169) -> Matrix(1,0,12,1) Matrix(1147,1820,-830,-1317) -> Matrix(133,18,-1234,-167) Matrix(195,308,88,139) -> Matrix(15,2,82,11) Matrix(197,308,126,197) -> Matrix(47,6,368,47) Matrix(55,84,36,55) -> Matrix(17,2,144,17) Matrix(307,448,172,251) -> Matrix(35,4,236,27) Matrix(2241,3248,928,1345) -> Matrix(143,16,706,79) Matrix(1819,2632,754,1091) -> Matrix(73,8,374,41) Matrix(253,364,98,141) -> Matrix(19,2,104,11) Matrix(139,196,-100,-141) -> Matrix(53,6,-486,-55) Matrix(9071,12544,2510,3471) -> Matrix(1627,176,6092,659) Matrix(9073,12544,2512,3473) -> Matrix(1629,176,6118,661) Matrix(589,812,346,477) -> Matrix(19,2,180,19) Matrix(2295,3136,614,839) -> Matrix(1,0,14,1) Matrix(2297,3136,616,841) -> Matrix(1,0,12,1) Matrix(83,112,20,27) -> Matrix(37,4,120,13) Matrix(85,112,22,29) -> Matrix(39,4,146,15) Matrix(281,364,44,57) -> Matrix(21,2,52,5) Matrix(5459,7056,1234,1595) -> Matrix(189,20,652,69) Matrix(5461,7056,1236,1597) -> Matrix(191,20,678,71) Matrix(197,252,154,197) -> Matrix(99,10,980,99) Matrix(111,140,88,111) -> Matrix(61,6,620,61) Matrix(113,140,46,57) -> Matrix(21,2,136,13) Matrix(643,784,114,139) -> Matrix(377,36,932,89) Matrix(645,784,116,141) -> Matrix(379,36,958,91) Matrix(307,364,70,83) -> Matrix(21,2,52,5) Matrix(197,224,124,141) -> Matrix(45,4,326,29) Matrix(1,0,2,1) -> Matrix(1,0,26,1) Matrix(195,-224,74,-85) -> Matrix(45,-4,214,-19) Matrix(309,-364,118,-139) -> Matrix(65,-6,336,-31) Matrix(139,-168,24,-29) -> Matrix(85,-8,202,-19) Matrix(391,-504,218,-281) -> Matrix(39,-4,244,-25) Matrix(225,-308,122,-167) -> Matrix(19,-2,124,-13) Matrix(141,-196,100,-139) -> Matrix(55,-6,486,-53) Matrix(1063,-1512,670,-953) -> Matrix(103,-12,764,-89) Matrix(559,-812,232,-337) -> Matrix(53,-6,274,-31) Matrix(2101,-3332,582,-923) -> Matrix(223,-30,840,-113) Matrix(139,-224,18,-29) -> Matrix(33,-4,58,-7) Matrix(225,-364,34,-55) -> Matrix(47,-6,102,-13) Matrix(307,-504,120,-197) -> Matrix(61,-8,328,-43) Matrix(475,-812,148,-253) -> Matrix(47,-6,196,-25) Matrix(113,-196,64,-111) -> Matrix(15,-2,98,-13) Matrix(1315,-2352,298,-533) -> Matrix(25,-4,94,-15) Matrix(223,-420,60,-113) -> Matrix(13,-2,46,-7) Matrix(141,-308,38,-83) -> Matrix(13,-2,46,-7) Matrix(85,-196,36,-83) -> Matrix(13,-2,72,-11) Matrix(365,-868,82,-195) -> Matrix(9,-2,32,-7) Matrix(673,-1820,186,-503) -> Matrix(101,-18,376,-67) Matrix(57,-196,16,-55) -> Matrix(25,-6,96,-23) 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): 48 Degree of the the map X: 48 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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 7/6 4/3 7/5 7/4 2/1 7/3 12/5 28/11 8/3 14/5 3/1 16/5 42/13 10/3 7/2 112/31 56/15 4/1 84/19 9/2 14/3 5/1 11/2 28/5 6/1 13/2 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/13 7/6 1/11 6/5 4/43 2/21 11/9 7/73 5/4 1/10 14/11 1/10 9/7 5/49 22/17 0/1 2/19 13/10 1/10 4/3 2/19 15/11 1/9 11/8 1/10 7/5 1/9 17/12 5/44 27/19 11/95 10/7 0/1 2/17 13/9 1/9 16/11 4/35 3/2 1/8 14/9 1/8 11/7 3/23 19/12 5/38 46/29 2/15 12/89 27/17 7/51 8/5 0/1 21/13 1/8 13/8 1/8 18/11 4/31 2/15 5/3 1/7 17/10 1/10 29/17 5/41 12/7 2/15 7/4 1/7 16/9 0/1 25/14 3/20 34/19 2/13 4/25 9/5 1/5 11/6 1/6 13/7 1/7 2/1 0/1 2/13 11/5 1/7 9/4 1/8 7/3 1/6 19/8 3/16 31/13 1/7 12/5 2/11 41/17 1/5 70/29 1/5 29/12 5/24 17/7 1/3 5/2 1/6 28/11 2/11 23/9 5/27 18/7 2/11 4/21 13/5 1/5 34/13 2/11 4/21 21/8 1/5 29/11 7/33 8/3 0/1 27/10 7/40 19/7 5/27 11/4 3/16 14/5 1/5 3/1 1/5 16/5 4/17 29/9 11/45 42/13 1/4 13/4 1/4 10/3 0/1 2/9 7/2 1/4 18/5 6/23 4/15 65/18 25/94 112/31 4/15 47/13 19/71 29/8 13/48 11/3 1/3 26/7 0/1 2/7 41/11 1/3 56/15 0/1 15/4 1/4 4/1 2/7 13/3 1/3 22/5 0/1 2/7 53/12 5/18 84/19 2/7 31/7 5/17 9/2 5/16 14/3 1/3 5/1 1/3 11/2 7/18 28/5 2/5 17/3 11/27 6/1 2/5 4/9 13/2 1/2 7/1 1/2 8/1 4/7 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(195,-224,74,-85) (1/1,7/6) -> (21/8,29/11) Hyperbolic Matrix(309,-364,118,-139) (7/6,6/5) -> (34/13,21/8) Hyperbolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(113,-140,67,-83) (11/9,5/4) -> (5/3,17/10) Hyperbolic Matrix(111,-140,23,-29) (5/4,14/11) -> (14/3,5/1) Hyperbolic Matrix(197,-252,43,-55) (14/11,9/7) -> (9/2,14/3) Hyperbolic Matrix(391,-504,218,-281) (9/7,22/17) -> (34/19,9/5) Hyperbolic Matrix(561,-728,215,-279) (22/17,13/10) -> (13/5,34/13) Hyperbolic Matrix(85,-112,63,-83) (13/10,4/3) -> (4/3,15/11) Parabolic Matrix(225,-308,122,-167) (15/11,11/8) -> (11/6,13/7) Hyperbolic Matrix(141,-196,100,-139) (11/8,7/5) -> (7/5,17/12) Parabolic Matrix(671,-952,277,-393) (17/12,27/19) -> (29/12,17/7) Hyperbolic Matrix(1063,-1512,670,-953) (27/19,10/7) -> (46/29,27/17) Hyperbolic Matrix(253,-364,155,-223) (10/7,13/9) -> (13/8,18/11) Hyperbolic Matrix(559,-812,232,-337) (13/9,16/11) -> (12/5,41/17) Hyperbolic Matrix(365,-532,153,-223) (16/11,3/2) -> (31/13,12/5) Hyperbolic Matrix(55,-84,19,-29) (3/2,14/9) -> (14/5,3/1) Hyperbolic Matrix(197,-308,71,-111) (14/9,11/7) -> (11/4,14/5) Hyperbolic Matrix(195,-308,107,-169) (11/7,19/12) -> (9/5,11/6) Hyperbolic Matrix(2101,-3332,582,-923) (19/12,46/29) -> (18/5,65/18) Hyperbolic Matrix(141,-224,17,-27) (27/17,8/5) -> (8/1,1/0) Hyperbolic Matrix(139,-224,18,-29) (8/5,21/13) -> (7/1,8/1) Hyperbolic Matrix(225,-364,34,-55) (21/13,13/8) -> (13/2,7/1) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(477,-812,131,-223) (17/10,29/17) -> (29/8,11/3) Hyperbolic Matrix(475,-812,148,-253) (29/17,12/7) -> (16/5,29/9) Hyperbolic Matrix(113,-196,64,-111) (12/7,7/4) -> (7/4,16/9) Parabolic Matrix(251,-448,79,-141) (16/9,25/14) -> (3/1,16/5) Hyperbolic Matrix(1315,-2352,298,-533) (25/14,34/19) -> (22/5,53/12) Hyperbolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(141,-308,38,-83) (2/1,11/5) -> (11/3,26/7) Hyperbolic Matrix(139,-308,51,-113) (11/5,9/4) -> (19/7,11/4) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(365,-868,82,-195) (19/8,31/13) -> (31/7,9/2) Hyperbolic Matrix(1091,-2632,337,-813) (41/17,70/29) -> (42/13,13/4) Hyperbolic Matrix(1345,-3248,417,-1007) (70/29,29/12) -> (29/9,42/13) Hyperbolic Matrix(57,-140,11,-27) (17/7,5/2) -> (5/1,11/2) Hyperbolic Matrix(309,-784,121,-307) (5/2,28/11) -> (28/11,23/9) Parabolic Matrix(141,-364,43,-111) (18/7,13/5) -> (13/4,10/3) Hyperbolic Matrix(169,-448,63,-167) (29/11,8/3) -> (8/3,27/10) Parabolic Matrix(673,-1820,186,-503) (27/10,19/7) -> (47/13,29/8) Hyperbolic Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic Matrix(3473,-12544,961,-3471) (65/18,112/31) -> (112/31,47/13) Parabolic Matrix(841,-3136,225,-839) (41/11,56/15) -> (56/15,15/4) Parabolic Matrix(29,-112,7,-27) (15/4,4/1) -> (4/1,13/3) Parabolic Matrix(83,-364,13,-57) (13/3,22/5) -> (6/1,13/2) Hyperbolic Matrix(1597,-7056,361,-1595) (53/12,84/19) -> (84/19,31/7) Parabolic Matrix(141,-784,25,-139) (11/2,28/5) -> (28/5,17/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,13,1) Matrix(195,-224,74,-85) -> Matrix(45,-4,214,-19) Matrix(309,-364,118,-139) -> Matrix(65,-6,336,-31) Matrix(139,-168,24,-29) -> Matrix(85,-8,202,-19) Matrix(113,-140,67,-83) -> Matrix(21,-2,137,-13) Matrix(111,-140,23,-29) -> Matrix(61,-6,173,-17) Matrix(197,-252,43,-55) -> Matrix(99,-10,307,-31) Matrix(391,-504,218,-281) -> Matrix(39,-4,244,-25) Matrix(561,-728,215,-279) -> Matrix(39,-4,205,-21) Matrix(85,-112,63,-83) -> Matrix(39,-4,361,-37) Matrix(225,-308,122,-167) -> Matrix(19,-2,124,-13) Matrix(141,-196,100,-139) -> Matrix(55,-6,486,-53) Matrix(671,-952,277,-393) -> Matrix(35,-4,149,-17) Matrix(1063,-1512,670,-953) -> Matrix(103,-12,764,-89) Matrix(253,-364,155,-223) -> Matrix(19,-2,143,-15) Matrix(559,-812,232,-337) -> Matrix(53,-6,274,-31) Matrix(365,-532,153,-223) -> Matrix(17,-2,111,-13) Matrix(55,-84,19,-29) -> Matrix(17,-2,77,-9) Matrix(197,-308,71,-111) -> Matrix(47,-6,243,-31) Matrix(195,-308,107,-169) -> Matrix(15,-2,113,-15) Matrix(2101,-3332,582,-923) -> Matrix(223,-30,840,-113) Matrix(141,-224,17,-27) -> Matrix(29,-4,51,-7) Matrix(139,-224,18,-29) -> Matrix(33,-4,58,-7) Matrix(225,-364,34,-55) -> Matrix(47,-6,102,-13) Matrix(307,-504,120,-197) -> Matrix(61,-8,328,-43) Matrix(477,-812,131,-223) -> Matrix(19,-2,67,-7) Matrix(475,-812,148,-253) -> Matrix(47,-6,196,-25) Matrix(113,-196,64,-111) -> Matrix(15,-2,98,-13) Matrix(251,-448,79,-141) -> Matrix(27,-4,115,-17) Matrix(1315,-2352,298,-533) -> Matrix(25,-4,94,-15) Matrix(223,-420,60,-113) -> Matrix(13,-2,46,-7) Matrix(141,-308,38,-83) -> Matrix(13,-2,46,-7) Matrix(139,-308,51,-113) -> Matrix(11,-2,61,-11) Matrix(85,-196,36,-83) -> Matrix(13,-2,72,-11) Matrix(365,-868,82,-195) -> Matrix(9,-2,32,-7) Matrix(1091,-2632,337,-813) -> Matrix(41,-8,159,-31) Matrix(1345,-3248,417,-1007) -> Matrix(79,-16,321,-65) Matrix(57,-140,11,-27) -> Matrix(13,-2,33,-5) Matrix(309,-784,121,-307) -> Matrix(67,-12,363,-65) Matrix(141,-364,43,-111) -> Matrix(11,-2,39,-7) Matrix(169,-448,63,-167) -> Matrix(1,0,1,1) Matrix(673,-1820,186,-503) -> Matrix(101,-18,376,-67) Matrix(57,-196,16,-55) -> Matrix(25,-6,96,-23) Matrix(3473,-12544,961,-3471) -> Matrix(661,-176,2475,-659) Matrix(841,-3136,225,-839) -> Matrix(1,0,1,1) Matrix(29,-112,7,-27) -> Matrix(15,-4,49,-13) Matrix(83,-364,13,-57) -> Matrix(5,-2,13,-5) Matrix(1597,-7056,361,-1595) -> Matrix(71,-20,245,-69) Matrix(141,-784,25,-139) -> Matrix(91,-36,225,-89) 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): 48 ----------------------------------------------------------------------- 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 13 1 2/1 (0/1,2/13) 0 7 11/5 1/7 1 14 9/4 1/8 1 14 7/3 1/6 1 2 19/8 3/16 1 14 31/13 1/7 1 14 12/5 2/11 1 7 29/12 5/24 1 14 17/7 1/3 1 14 5/2 1/6 1 14 28/11 2/11 3 1 18/7 (2/11,4/21) 0 7 13/5 1/5 1 14 21/8 1/5 5 2 8/3 0/1 1 7 27/10 7/40 1 14 19/7 5/27 1 14 11/4 3/16 1 14 14/5 1/5 4 1 3/1 1/5 1 14 16/5 4/17 1 7 29/9 11/45 1 14 42/13 1/4 12 1 13/4 1/4 1 14 10/3 (0/1,2/9) 0 7 7/2 1/4 3 2 18/5 (6/23,4/15) 0 7 112/31 4/15 11 1 47/13 19/71 1 14 29/8 13/48 1 14 11/3 1/3 1 14 26/7 (0/1,2/7) 0 7 56/15 0/1 1 1 15/4 1/4 1 14 4/1 2/7 1 7 13/3 1/3 1 14 22/5 (0/1,2/7) 0 7 84/19 2/7 5 1 31/7 5/17 1 14 9/2 5/16 1 14 14/3 1/3 8 1 5/1 1/3 1 14 11/2 7/18 1 14 28/5 2/5 9 1 6/1 (2/5,4/9) 0 7 13/2 1/2 1 14 7/1 1/2 5 2 8/1 4/7 1 7 1/0 1/0 1 14 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(141,-308,38,-83) (2/1,11/5) -> (11/3,26/7) Hyperbolic Matrix(139,-308,51,-113) (11/5,9/4) -> (19/7,11/4) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(365,-868,82,-195) (19/8,31/13) -> (31/7,9/2) Hyperbolic Matrix(223,-532,70,-167) (31/13,12/5) -> (3/1,16/5) Glide Reflection Matrix(337,-812,105,-253) (12/5,29/12) -> (16/5,29/9) Glide Reflection Matrix(335,-812,92,-223) (29/12,17/7) -> (29/8,11/3) Glide Reflection Matrix(57,-140,11,-27) (17/7,5/2) -> (5/1,11/2) Hyperbolic Matrix(111,-280,44,-111) (5/2,28/11) -> (5/2,28/11) Reflection Matrix(197,-504,77,-197) (28/11,18/7) -> (28/11,18/7) Reflection Matrix(141,-364,43,-111) (18/7,13/5) -> (13/4,10/3) Hyperbolic Matrix(139,-364,21,-55) (13/5,21/8) -> (13/2,7/1) Glide Reflection Matrix(85,-224,11,-29) (21/8,8/3) -> (7/1,8/1) Glide Reflection Matrix(83,-224,10,-27) (8/3,27/10) -> (8/1,1/0) Glide Reflection Matrix(673,-1820,186,-503) (27/10,19/7) -> (47/13,29/8) Hyperbolic Matrix(111,-308,40,-111) (11/4,14/5) -> (11/4,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(755,-2436,234,-755) (29/9,42/13) -> (29/9,42/13) Reflection Matrix(337,-1092,104,-337) (42/13,13/4) -> (42/13,13/4) Reflection Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic Matrix(559,-2016,155,-559) (18/5,112/31) -> (18/5,112/31) Reflection Matrix(2913,-10528,806,-2913) (112/31,47/13) -> (112/31,47/13) Reflection Matrix(391,-1456,105,-391) (26/7,56/15) -> (26/7,56/15) Reflection Matrix(449,-1680,120,-449) (56/15,15/4) -> (56/15,15/4) Reflection Matrix(29,-112,7,-27) (15/4,4/1) -> (4/1,13/3) Parabolic Matrix(83,-364,13,-57) (13/3,22/5) -> (6/1,13/2) Hyperbolic Matrix(419,-1848,95,-419) (22/5,84/19) -> (22/5,84/19) Reflection Matrix(1177,-5208,266,-1177) (84/19,31/7) -> (84/19,31/7) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(111,-616,20,-111) (11/2,28/5) -> (11/2,28/5) Reflection Matrix(29,-168,5,-29) (28/5,6/1) -> (28/5,6/1) 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(1,0,13,-1) (0/1,2/1) -> (0/1,2/13) Matrix(141,-308,38,-83) -> Matrix(13,-2,46,-7) Matrix(139,-308,51,-113) -> Matrix(11,-2,61,-11) (0/1,2/11).(1/6,1/5) Matrix(85,-196,36,-83) -> Matrix(13,-2,72,-11) 1/6 Matrix(365,-868,82,-195) -> Matrix(9,-2,32,-7) 1/4 Matrix(223,-532,70,-167) -> Matrix(13,-2,58,-9) Matrix(337,-812,105,-253) -> Matrix(31,-6,129,-25) Matrix(335,-812,92,-223) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(57,-140,11,-27) -> Matrix(13,-2,33,-5) Matrix(111,-280,44,-111) -> Matrix(23,-4,132,-23) (5/2,28/11) -> (1/6,2/11) Matrix(197,-504,77,-197) -> Matrix(43,-8,231,-43) (28/11,18/7) -> (2/11,4/21) Matrix(141,-364,43,-111) -> Matrix(11,-2,39,-7) Matrix(139,-364,21,-55) -> Matrix(31,-6,67,-13) Matrix(85,-224,11,-29) -> Matrix(19,-4,33,-7) Matrix(83,-224,10,-27) -> Matrix(23,-4,40,-7) Matrix(673,-1820,186,-503) -> Matrix(101,-18,376,-67) Matrix(111,-308,40,-111) -> Matrix(31,-6,160,-31) (11/4,14/5) -> (3/16,1/5) Matrix(29,-84,10,-29) -> Matrix(9,-2,40,-9) (14/5,3/1) -> (1/5,1/4) Matrix(755,-2436,234,-755) -> Matrix(89,-22,360,-89) (29/9,42/13) -> (11/45,1/4) Matrix(337,-1092,104,-337) -> Matrix(7,-2,24,-7) (42/13,13/4) -> (1/4,1/3) Matrix(57,-196,16,-55) -> Matrix(25,-6,96,-23) 1/4 Matrix(559,-2016,155,-559) -> Matrix(91,-24,345,-91) (18/5,112/31) -> (6/23,4/15) Matrix(2913,-10528,806,-2913) -> Matrix(569,-152,2130,-569) (112/31,47/13) -> (4/15,19/71) Matrix(391,-1456,105,-391) -> Matrix(1,0,7,-1) (26/7,56/15) -> (0/1,2/7) Matrix(449,-1680,120,-449) -> Matrix(1,0,8,-1) (56/15,15/4) -> (0/1,1/4) Matrix(29,-112,7,-27) -> Matrix(15,-4,49,-13) 2/7 Matrix(83,-364,13,-57) -> Matrix(5,-2,13,-5) (0/1,2/5).(1/3,1/2) Matrix(419,-1848,95,-419) -> Matrix(1,0,7,-1) (22/5,84/19) -> (0/1,2/7) Matrix(1177,-5208,266,-1177) -> Matrix(69,-20,238,-69) (84/19,31/7) -> (2/7,5/17) Matrix(55,-252,12,-55) -> Matrix(31,-10,96,-31) (9/2,14/3) -> (5/16,1/3) Matrix(29,-140,6,-29) -> Matrix(17,-6,48,-17) (14/3,5/1) -> (1/3,3/8) Matrix(111,-616,20,-111) -> Matrix(71,-28,180,-71) (11/2,28/5) -> (7/18,2/5) Matrix(29,-168,5,-29) -> Matrix(19,-8,45,-19) (28/5,6/1) -> (2/5,4/9) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.