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 -1/1 -1/3 -6/1 -1/2 1/0 -11/2 -1/2 -1/3 -5/1 -1/2 0/1 -14/3 0/1 -9/2 0/1 1/0 -22/5 -1/2 1/0 -13/3 -1/1 0/1 -4/1 -1/2 -15/4 -1/3 0/1 -11/3 -1/4 0/1 -7/2 0/1 -17/5 0/1 1/4 -27/8 1/3 2/5 -10/3 1/2 1/0 -13/4 -1/1 0/1 -16/5 1/2 -3/1 0/1 1/0 -14/5 1/0 -11/4 -1/1 1/0 -19/7 1/1 1/0 -46/17 -5/2 1/0 -27/10 -1/1 0/1 -8/3 1/0 -21/8 -2/1 0/1 -13/5 -2/1 -1/1 -18/7 -1/2 1/0 -5/2 -1/1 1/0 -17/7 -2/1 1/0 -29/12 -4/3 -1/1 -12/5 1/0 -7/3 -1/1 -16/7 -3/4 -25/11 -2/3 -1/2 -34/15 -3/4 -1/2 -9/4 -2/3 -1/2 -11/5 -1/2 0/1 -13/6 -1/1 0/1 -2/1 -1/2 1/0 -11/6 -1/1 -1/2 -9/5 -1/2 -1/3 -7/4 0/1 -19/11 1/2 1/1 -31/18 1/2 1/1 -12/7 1/0 -41/24 -1/3 0/1 -70/41 0/1 -29/17 0/1 1/3 -17/10 1/1 1/0 -5/3 0/1 1/0 -28/17 1/0 -23/14 -1/1 1/0 -18/11 -1/2 1/0 -13/8 0/1 1/1 -34/21 1/2 1/0 -21/13 -1/1 1/1 -29/18 0/1 1/1 -8/5 1/0 -27/17 -1/1 0/1 -19/12 -2/1 1/0 -11/7 0/1 1/0 -14/9 1/0 -3/2 -1/1 1/0 -16/11 -3/2 -29/20 -6/5 -1/1 -42/29 -1/1 -13/9 -1/1 0/1 -10/7 -3/2 1/0 -7/5 -1/1 -18/13 -7/8 -5/6 -65/47 -21/25 -5/6 -112/81 -5/6 -47/34 -5/6 -14/17 -29/21 -9/11 -4/5 -11/8 -1/1 -3/4 -26/19 -3/4 -1/2 -41/30 -1/1 -2/3 -56/41 -3/4 -1/2 -15/11 -1/1 -2/3 -4/3 -1/2 -13/10 -1/1 0/1 -22/17 -1/2 1/0 -53/41 0/1 1/0 -84/65 1/0 -31/24 -1/1 1/0 -9/7 -1/1 1/0 -14/11 -1/1 -5/4 -1/1 -1/2 -11/9 -2/3 -1/2 -28/23 -1/2 -17/14 -1/2 -1/3 -6/5 -1/2 1/0 -13/11 -1/1 0/1 -7/6 -2/3 0/1 -8/7 -1/2 -1/1 -1/1 0/1 0/1 -1/2 1/0 1/1 -1/1 0/1 7/6 -2/1 0/1 6/5 -1/2 1/0 11/9 -2/1 1/0 5/4 -1/1 1/0 14/11 -1/1 9/7 -1/1 -1/2 22/17 -1/2 1/0 13/10 -1/1 0/1 4/3 1/0 15/11 -2/1 -1/1 11/8 -3/2 -1/1 7/5 -1/1 17/12 -1/1 -5/6 27/19 -4/5 -7/9 10/7 -3/4 -1/2 13/9 -1/1 0/1 16/11 -3/4 3/2 -1/1 -1/2 14/9 -1/2 11/7 -1/2 0/1 19/12 -2/3 -1/2 46/29 -1/2 -3/8 27/17 -1/1 0/1 8/5 -1/2 21/13 -1/1 -1/3 13/8 -1/3 0/1 18/11 -1/2 1/0 5/3 -1/2 0/1 17/10 -1/2 -1/3 29/17 -1/5 0/1 12/7 -1/2 7/4 0/1 16/9 1/2 25/14 1/1 1/0 34/19 1/2 1/0 9/5 1/1 1/0 11/6 -1/1 1/0 13/7 -1/1 0/1 2/1 -1/2 1/0 11/5 0/1 1/0 9/4 -2/1 1/0 7/3 -1/1 19/8 -3/4 -2/3 31/13 -3/4 -2/3 12/5 -1/2 41/17 -2/1 -1/1 70/29 -1/1 29/12 -1/1 -4/5 17/7 -2/3 -1/2 5/2 -1/1 -1/2 28/11 -1/2 23/9 -1/2 0/1 18/7 -1/2 1/0 13/5 -1/1 -2/3 34/13 -3/4 -1/2 21/8 -2/3 0/1 29/11 -1/1 -2/3 8/3 -1/2 27/10 -1/1 0/1 19/7 -1/2 -1/3 11/4 -1/1 -1/2 14/5 -1/2 3/1 -1/2 0/1 16/5 -1/4 29/9 -1/7 0/1 42/13 0/1 13/4 -1/1 0/1 10/3 -1/2 -1/4 7/2 0/1 18/5 1/6 1/4 65/18 4/17 1/4 112/31 1/4 47/13 1/4 3/11 29/8 2/7 1/3 11/3 0/1 1/2 26/7 1/2 1/0 41/11 0/1 1/1 56/15 1/2 1/0 15/4 0/1 1/1 4/1 1/0 13/3 -1/1 0/1 22/5 -1/2 1/0 53/12 -1/1 -1/2 84/19 -1/2 31/7 -1/2 0/1 9/2 -1/2 0/1 14/3 0/1 5/1 0/1 1/0 11/2 1/1 1/0 28/5 1/0 17/3 -2/1 1/0 6/1 -1/2 1/0 13/2 -1/1 0/1 7/1 -1/1 1/1 8/1 1/0 1/0 -1/1 0/1 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(1,0,2,1) Matrix(55,364,-34,-225) -> Matrix(1,0,2,1) Matrix(29,168,-24,-139) -> Matrix(1,0,0,1) Matrix(27,140,16,83) -> Matrix(1,0,0,1) Matrix(29,140,6,29) -> Matrix(1,0,2,1) Matrix(55,252,12,55) -> Matrix(1,0,-2,1) Matrix(113,504,-50,-223) -> Matrix(1,2,-2,-3) Matrix(167,728,64,279) -> Matrix(1,2,-2,-3) Matrix(27,112,20,83) -> Matrix(3,2,-2,-1) Matrix(29,112,22,85) -> Matrix(1,0,2,1) Matrix(83,308,-38,-141) -> Matrix(1,0,2,1) Matrix(55,196,-16,-57) -> Matrix(1,0,8,1) Matrix(281,952,116,393) -> Matrix(7,-2,-10,3) Matrix(449,1512,-166,-559) -> Matrix(5,-2,-2,1) Matrix(111,364,68,223) -> Matrix(1,0,-2,1) Matrix(253,812,-148,-475) -> Matrix(1,0,-2,1) Matrix(167,532,70,223) -> Matrix(3,-2,-4,3) Matrix(29,84,10,29) -> Matrix(1,0,-2,1) Matrix(111,308,40,111) -> Matrix(1,2,-2,-3) Matrix(113,308,62,169) -> Matrix(1,0,0,1) Matrix(1231,3332,-890,-2409) -> Matrix(5,16,-6,-19) Matrix(83,224,10,27) -> Matrix(1,0,0,1) Matrix(85,224,-74,-195) -> Matrix(1,2,-2,-3) Matrix(139,364,-118,-309) -> Matrix(1,2,-2,-3) Matrix(141,364,98,253) -> Matrix(1,2,-2,-3) Matrix(197,504,-120,-307) -> Matrix(1,0,0,1) Matrix(57,140,46,113) -> Matrix(1,0,0,1) Matrix(335,812,92,223) -> Matrix(1,2,2,5) Matrix(337,812,-232,-559) -> Matrix(3,2,-2,-1) Matrix(83,196,-36,-85) -> Matrix(3,4,-4,-5) Matrix(197,448,62,141) -> Matrix(3,2,-8,-5) Matrix(1037,2352,-802,-1819) -> Matrix(3,2,-2,-1) Matrix(139,308,88,195) -> Matrix(1,0,0,1) Matrix(197,420,-144,-307) -> Matrix(1,2,-2,-3) Matrix(167,308,-122,-225) -> Matrix(1,2,-2,-3) Matrix(169,308,62,113) -> Matrix(1,0,0,1) Matrix(111,196,-64,-113) -> Matrix(1,0,4,1) Matrix(503,868,-390,-673) -> Matrix(1,0,-2,1) Matrix(309,532,212,365) -> Matrix(3,-2,-4,3) Matrix(1541,2632,476,813) -> Matrix(1,0,2,1) Matrix(1903,3248,590,1007) -> Matrix(1,0,-10,1) Matrix(559,952,394,671) -> Matrix(5,-4,-6,5) Matrix(83,140,16,27) -> Matrix(1,0,0,1) Matrix(475,784,186,307) -> Matrix(1,0,-2,1) Matrix(477,784,188,309) -> Matrix(1,2,-2,-3) Matrix(223,364,68,111) -> Matrix(1,0,-2,1) Matrix(449,728,346,561) -> Matrix(1,0,-2,1) Matrix(279,448,104,167) -> Matrix(1,0,-2,1) Matrix(281,448,106,169) -> Matrix(1,2,-2,-3) Matrix(1147,1820,-830,-1317) -> Matrix(5,-4,-6,5) Matrix(195,308,88,139) -> Matrix(1,0,0,1) Matrix(197,308,126,197) -> Matrix(1,0,-2,1) Matrix(55,84,36,55) -> Matrix(1,2,-2,-3) Matrix(307,448,172,251) -> Matrix(1,2,0,1) Matrix(2241,3248,928,1345) -> Matrix(9,10,-10,-11) Matrix(1819,2632,754,1091) -> Matrix(3,2,-2,-1) Matrix(253,364,98,141) -> Matrix(1,2,-2,-3) Matrix(139,196,-100,-141) -> Matrix(7,8,-8,-9) Matrix(9071,12544,2510,3471) -> Matrix(43,36,166,139) Matrix(9073,12544,2512,3473) -> Matrix(41,34,170,141) Matrix(589,812,346,477) -> Matrix(5,4,-14,-11) Matrix(2295,3136,614,839) -> Matrix(3,2,4,3) Matrix(2297,3136,616,841) -> Matrix(3,2,4,3) Matrix(83,112,20,27) -> Matrix(3,2,-2,-1) Matrix(85,112,22,29) -> Matrix(1,0,2,1) Matrix(281,364,44,57) -> Matrix(1,0,0,1) Matrix(5459,7056,1234,1595) -> Matrix(1,0,-2,1) Matrix(5461,7056,1236,1597) -> Matrix(1,2,-2,-3) Matrix(197,252,154,197) -> Matrix(1,2,-2,-3) Matrix(111,140,88,111) -> Matrix(3,2,-2,-1) Matrix(113,140,46,57) -> Matrix(1,0,0,1) Matrix(643,784,114,139) -> Matrix(7,4,-2,-1) Matrix(645,784,116,141) -> Matrix(5,2,2,1) Matrix(307,364,70,83) -> Matrix(1,0,0,1) Matrix(197,224,124,141) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(195,-224,74,-85) -> Matrix(1,2,-2,-3) Matrix(309,-364,118,-139) -> Matrix(1,2,-2,-3) Matrix(139,-168,24,-29) -> Matrix(1,0,0,1) Matrix(391,-504,218,-281) -> Matrix(1,0,2,1) Matrix(225,-308,122,-167) -> Matrix(1,2,-2,-3) Matrix(141,-196,100,-139) -> Matrix(7,8,-8,-9) Matrix(1063,-1512,670,-953) -> Matrix(5,4,-14,-11) Matrix(559,-812,232,-337) -> Matrix(3,2,-2,-1) Matrix(2101,-3332,582,-923) -> Matrix(5,2,22,9) Matrix(139,-224,18,-29) -> Matrix(1,0,2,1) Matrix(225,-364,34,-55) -> Matrix(1,0,2,1) Matrix(307,-504,120,-197) -> Matrix(1,0,0,1) Matrix(475,-812,148,-253) -> Matrix(1,0,-2,1) Matrix(113,-196,64,-111) -> Matrix(1,0,4,1) Matrix(1315,-2352,298,-533) -> Matrix(1,0,-2,1) Matrix(223,-420,60,-113) -> Matrix(1,0,2,1) Matrix(141,-308,38,-83) -> Matrix(1,0,2,1) Matrix(85,-196,36,-83) -> Matrix(3,4,-4,-5) Matrix(365,-868,82,-195) -> Matrix(3,2,-2,-1) Matrix(673,-1820,186,-503) -> Matrix(3,2,10,7) Matrix(57,-196,16,-55) -> Matrix(1,0,8,1) 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): 20 Degree of the the map X: 20 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: 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 -1/2 1/0 1/1 -1/1 0/1 7/6 -2/1 0/1 6/5 -1/2 1/0 11/9 -2/1 1/0 5/4 -1/1 1/0 14/11 -1/1 9/7 -1/1 -1/2 22/17 -1/2 1/0 13/10 -1/1 0/1 4/3 1/0 15/11 -2/1 -1/1 11/8 -3/2 -1/1 7/5 -1/1 17/12 -1/1 -5/6 27/19 -4/5 -7/9 10/7 -3/4 -1/2 13/9 -1/1 0/1 16/11 -3/4 3/2 -1/1 -1/2 14/9 -1/2 11/7 -1/2 0/1 19/12 -2/3 -1/2 46/29 -1/2 -3/8 27/17 -1/1 0/1 8/5 -1/2 21/13 -1/1 -1/3 13/8 -1/3 0/1 18/11 -1/2 1/0 5/3 -1/2 0/1 17/10 -1/2 -1/3 29/17 -1/5 0/1 12/7 -1/2 7/4 0/1 16/9 1/2 25/14 1/1 1/0 34/19 1/2 1/0 9/5 1/1 1/0 11/6 -1/1 1/0 13/7 -1/1 0/1 2/1 -1/2 1/0 11/5 0/1 1/0 9/4 -2/1 1/0 7/3 -1/1 19/8 -3/4 -2/3 31/13 -3/4 -2/3 12/5 -1/2 41/17 -2/1 -1/1 70/29 -1/1 29/12 -1/1 -4/5 17/7 -2/3 -1/2 5/2 -1/1 -1/2 28/11 -1/2 23/9 -1/2 0/1 18/7 -1/2 1/0 13/5 -1/1 -2/3 34/13 -3/4 -1/2 21/8 -2/3 0/1 29/11 -1/1 -2/3 8/3 -1/2 27/10 -1/1 0/1 19/7 -1/2 -1/3 11/4 -1/1 -1/2 14/5 -1/2 3/1 -1/2 0/1 16/5 -1/4 29/9 -1/7 0/1 42/13 0/1 13/4 -1/1 0/1 10/3 -1/2 -1/4 7/2 0/1 18/5 1/6 1/4 65/18 4/17 1/4 112/31 1/4 47/13 1/4 3/11 29/8 2/7 1/3 11/3 0/1 1/2 26/7 1/2 1/0 41/11 0/1 1/1 56/15 1/2 1/0 15/4 0/1 1/1 4/1 1/0 13/3 -1/1 0/1 22/5 -1/2 1/0 53/12 -1/1 -1/2 84/19 -1/2 31/7 -1/2 0/1 9/2 -1/2 0/1 14/3 0/1 5/1 0/1 1/0 11/2 1/1 1/0 28/5 1/0 17/3 -2/1 1/0 6/1 -1/2 1/0 13/2 -1/1 0/1 7/1 -1/1 1/1 8/1 1/0 1/0 -1/1 0/1 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(1,1,-2,-1) Matrix(195,-224,74,-85) -> Matrix(1,2,-2,-3) Matrix(309,-364,118,-139) -> Matrix(1,2,-2,-3) Matrix(139,-168,24,-29) -> Matrix(1,0,0,1) Matrix(113,-140,67,-83) -> Matrix(1,1,-2,-1) Matrix(111,-140,23,-29) -> Matrix(1,1,0,1) Matrix(197,-252,43,-55) -> Matrix(1,1,-4,-3) Matrix(391,-504,218,-281) -> Matrix(1,0,2,1) Matrix(561,-728,215,-279) -> Matrix(3,1,-4,-1) Matrix(85,-112,63,-83) -> Matrix(1,-1,0,1) Matrix(225,-308,122,-167) -> Matrix(1,2,-2,-3) Matrix(141,-196,100,-139) -> Matrix(7,8,-8,-9) Matrix(671,-952,277,-393) -> Matrix(11,9,-16,-13) Matrix(1063,-1512,670,-953) -> Matrix(5,4,-14,-11) Matrix(253,-364,155,-223) -> Matrix(1,1,-4,-3) Matrix(559,-812,232,-337) -> Matrix(3,2,-2,-1) Matrix(365,-532,153,-223) -> Matrix(7,5,-10,-7) Matrix(55,-84,19,-29) -> Matrix(1,1,-4,-3) Matrix(197,-308,71,-111) -> Matrix(3,1,-4,-1) Matrix(195,-308,107,-169) -> Matrix(1,1,-2,-1) Matrix(2101,-3332,582,-923) -> Matrix(5,2,22,9) Matrix(141,-224,17,-27) -> Matrix(1,1,-2,-1) Matrix(139,-224,18,-29) -> Matrix(1,0,2,1) Matrix(225,-364,34,-55) -> Matrix(1,0,2,1) Matrix(307,-504,120,-197) -> Matrix(1,0,0,1) Matrix(477,-812,131,-223) -> Matrix(3,1,8,3) Matrix(475,-812,148,-253) -> Matrix(1,0,-2,1) Matrix(113,-196,64,-111) -> Matrix(1,0,4,1) Matrix(251,-448,79,-141) -> Matrix(1,-1,-2,3) Matrix(1315,-2352,298,-533) -> Matrix(1,0,-2,1) Matrix(223,-420,60,-113) -> Matrix(1,0,2,1) Matrix(141,-308,38,-83) -> Matrix(1,0,2,1) Matrix(139,-308,51,-113) -> Matrix(1,1,-2,-1) Matrix(85,-196,36,-83) -> Matrix(3,4,-4,-5) Matrix(365,-868,82,-195) -> Matrix(3,2,-2,-1) Matrix(1091,-2632,337,-813) -> Matrix(1,1,0,1) Matrix(1345,-3248,417,-1007) -> Matrix(1,1,-12,-11) Matrix(57,-140,11,-27) -> Matrix(1,1,-2,-1) Matrix(309,-784,121,-307) -> Matrix(1,1,-4,-3) Matrix(141,-364,43,-111) -> Matrix(1,1,-4,-3) Matrix(169,-448,63,-167) -> Matrix(1,1,-4,-3) Matrix(673,-1820,186,-503) -> Matrix(3,2,10,7) Matrix(57,-196,16,-55) -> Matrix(1,0,8,1) Matrix(3473,-12544,961,-3471) -> Matrix(29,-7,112,-27) Matrix(841,-3136,225,-839) -> Matrix(1,-1,2,-1) Matrix(29,-112,7,-27) -> Matrix(1,-1,0,1) Matrix(83,-364,13,-57) -> Matrix(1,1,-2,-1) Matrix(1597,-7056,361,-1595) -> Matrix(1,1,-4,-3) Matrix(141,-784,25,-139) -> 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): 20 ----------------------------------------------------------------------- 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 (-1/1,0/1).(-1/2,1/0) 0 1 2/1 (-1/2,1/0) 0 7 11/5 (0/1,1/0) 0 14 9/4 (-2/1,1/0) 0 14 7/3 -1/1 2 2 19/8 (-3/4,-2/3) 0 14 31/13 (-3/4,-2/3) 0 14 12/5 -1/2 1 7 29/12 (-1/1,-4/5) 0 14 17/7 (-2/3,-1/2) 0 14 5/2 (-1/1,-1/2) 0 14 28/11 -1/2 1 1 18/7 (-1/2,1/0) 0 7 13/5 (-1/1,-2/3) 0 14 21/8 0 2 8/3 -1/2 1 7 27/10 (-1/1,0/1) 0 14 19/7 (-1/2,-1/3) 0 14 11/4 (-1/1,-1/2) 0 14 14/5 -1/2 1 1 3/1 (-1/2,0/1) 0 14 16/5 -1/4 1 7 29/9 (-1/7,0/1) 0 14 42/13 0/1 6 1 13/4 (-1/1,0/1) 0 14 10/3 (-1/2,-1/4) 0 7 7/2 0/1 4 2 18/5 (1/6,1/4) 0 7 112/31 1/4 7 1 47/13 (1/4,3/11) 0 14 29/8 (2/7,1/3) 0 14 11/3 (0/1,1/2) 0 14 26/7 (1/2,1/0) 0 7 56/15 (0/1,1/1).(1/2,1/0) 0 1 15/4 (0/1,1/1) 0 14 4/1 1/0 1 7 13/3 (-1/1,0/1) 0 14 22/5 (-1/2,1/0) 0 7 84/19 -1/2 1 1 31/7 (-1/2,0/1) 0 14 9/2 (-1/2,0/1) 0 14 14/3 0/1 2 1 5/1 (0/1,1/0) 0 14 11/2 (1/1,1/0) 0 14 28/5 1/0 3 1 6/1 (-1/2,1/0) 0 7 13/2 (-1/1,0/1) 0 14 7/1 0 2 8/1 1/0 1 7 1/0 (-1/1,0/1) 0 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,1,-1) -> Matrix(1,1,0,-1) (0/1,2/1) -> (-1/2,1/0) Matrix(141,-308,38,-83) -> Matrix(1,0,2,1) 0/1 Matrix(139,-308,51,-113) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(85,-196,36,-83) -> Matrix(3,4,-4,-5) -1/1 Matrix(365,-868,82,-195) -> Matrix(3,2,-2,-1) -1/1 Matrix(223,-532,70,-167) -> Matrix(3,2,-10,-7) Matrix(337,-812,105,-253) -> Matrix(1,1,-2,-3) Matrix(335,-812,92,-223) -> Matrix(3,2,8,5) Matrix(57,-140,11,-27) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(111,-280,44,-111) -> Matrix(3,2,-4,-3) (5/2,28/11) -> (-1/1,-1/2) Matrix(197,-504,77,-197) -> Matrix(1,1,0,-1) (28/11,18/7) -> (-1/2,1/0) Matrix(141,-364,43,-111) -> Matrix(1,1,-4,-3) -1/2 Matrix(139,-364,21,-55) -> Matrix(1,1,2,1) Matrix(85,-224,11,-29) -> Matrix(1,1,2,1) Matrix(83,-224,10,-27) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(673,-1820,186,-503) -> Matrix(3,2,10,7) Matrix(111,-308,40,-111) -> Matrix(3,2,-4,-3) (11/4,14/5) -> (-1/1,-1/2) Matrix(29,-84,10,-29) -> Matrix(-1,0,4,1) (14/5,3/1) -> (-1/2,0/1) Matrix(755,-2436,234,-755) -> Matrix(-1,0,14,1) (29/9,42/13) -> (-1/7,0/1) Matrix(337,-1092,104,-337) -> Matrix(-1,0,2,1) (42/13,13/4) -> (-1/1,0/1) Matrix(57,-196,16,-55) -> Matrix(1,0,8,1) 0/1 Matrix(559,-2016,155,-559) -> Matrix(5,-1,24,-5) (18/5,112/31) -> (1/6,1/4) Matrix(2913,-10528,806,-2913) -> Matrix(23,-6,88,-23) (112/31,47/13) -> (1/4,3/11) Matrix(391,-1456,105,-391) -> Matrix(-1,1,0,1) (26/7,56/15) -> (1/2,1/0) Matrix(449,-1680,120,-449) -> Matrix(1,0,2,-1) (56/15,15/4) -> (0/1,1/1) Matrix(29,-112,7,-27) -> Matrix(1,-1,0,1) 1/0 Matrix(83,-364,13,-57) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(419,-1848,95,-419) -> Matrix(1,1,0,-1) (22/5,84/19) -> (-1/2,1/0) Matrix(1177,-5208,266,-1177) -> Matrix(-1,0,4,1) (84/19,31/7) -> (-1/2,0/1) Matrix(55,-252,12,-55) -> Matrix(-1,0,4,1) (9/2,14/3) -> (-1/2,0/1) Matrix(29,-140,6,-29) -> Matrix(1,0,0,-1) (14/3,5/1) -> (0/1,1/0) Matrix(111,-616,20,-111) -> Matrix(-1,2,0,1) (11/2,28/5) -> (1/1,1/0) Matrix(29,-168,5,-29) -> Matrix(1,1,0,-1) (28/5,6/1) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.