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 -6/7 -5/7 -4/7 -1/2 -3/7 -5/12 -11/28 -3/8 -13/42 -3/10 -2/7 -31/112 -15/56 -1/4 -19/84 -5/28 -1/6 -2/13 -1/7 -1/8 0/1 1/7 1/6 2/11 1/5 3/14 2/9 3/13 1/4 3/11 2/7 3/10 1/3 5/14 2/5 5/12 3/7 1/2 5/9 4/7 9/14 2/3 29/42 5/7 11/14 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 -6/7 -1/1 -5/6 -1/2 1/0 -9/11 -1/1 0/1 -4/5 -1/1 1/0 -11/14 -1/1 -7/9 -1/1 -2/3 -17/22 -3/4 -1/2 -10/13 -3/5 -1/2 -3/4 -1/2 1/0 -11/15 2/1 1/0 -8/11 -2/1 -1/1 -5/7 -1/1 -12/17 -1/1 -2/3 -19/27 -1/1 -2/3 -7/10 -3/4 -1/2 -9/13 -1/2 0/1 -11/16 -1/2 1/0 -2/3 -1/1 0/1 -9/14 -1/1 -7/11 -1/1 -4/5 -12/19 -1/1 -2/3 -29/46 -3/4 -1/2 -17/27 -1/1 -2/3 -5/8 -3/4 -1/2 -13/21 -2/3 -8/13 -1/1 -1/2 -11/18 -5/8 -1/2 -3/5 -1/2 0/1 -10/17 -1/1 0/1 -17/29 -1/1 0/1 -7/12 -1/2 1/0 -4/7 -1/1 -9/16 -3/4 -1/2 -14/25 -1/1 -2/3 -19/34 -3/4 -1/2 -5/9 -1/1 -2/3 -6/11 -2/3 -3/5 -7/13 -6/11 -1/2 -1/2 -1/2 1/0 -5/11 -2/5 -1/3 -4/9 -1/3 0/1 -3/7 0/1 -8/19 -1/1 0/1 -13/31 -1/1 0/1 -5/12 -1/2 1/0 -17/41 -1/2 0/1 -29/70 0/1 -12/29 -1/1 0/1 -7/17 -1/1 0/1 -2/5 -1/1 -1/2 -11/28 -1/2 -9/23 -1/2 -4/9 -7/18 -1/2 -3/8 -5/13 -1/2 0/1 -13/34 -1/2 -3/8 -8/21 -1/3 -11/29 -1/3 0/1 -3/8 -1/2 -1/4 -10/27 -1/3 0/1 -7/19 -1/3 0/1 -4/11 -1/5 0/1 -5/14 0/1 -1/3 -1/1 0/1 -5/16 -1/2 1/0 -9/29 -1/1 0/1 -13/42 -1/1 -4/13 -1/1 -1/2 -3/10 -1/2 -1/4 -2/7 0/1 -5/18 1/2 1/0 -18/65 0/1 1/1 -31/112 1/2 1/0 -13/47 0/1 1/1 -8/29 0/1 1/1 -3/11 0/1 1/1 -7/26 5/2 1/0 -11/41 8/1 1/0 -15/56 1/0 -4/15 -3/1 1/0 -1/4 -1/2 1/0 -3/13 -1/2 -2/5 -5/22 -1/2 -1/4 -12/53 -1/3 0/1 -19/84 -1/2 -1/4 -7/31 -1/3 0/1 -2/9 -1/3 0/1 -3/14 0/1 -1/5 0/1 1/0 -2/11 -1/1 0/1 -5/28 -1/2 1/0 -3/17 -1/1 0/1 -1/6 -1/2 1/0 -2/13 -1/2 -1/3 -1/7 0/1 -1/8 -1/2 1/0 0/1 -1/1 0/1 1/7 0/1 1/6 -1/2 1/0 2/11 -1/1 0/1 1/5 -1/2 0/1 3/14 0/1 2/9 0/1 1/1 5/22 1/2 1/0 3/13 2/1 1/0 1/4 -1/2 1/0 4/15 -3/5 -1/2 3/11 -1/3 0/1 2/7 0/1 5/17 0/1 1/1 8/27 0/1 1/1 3/10 1/2 1/0 4/13 -1/1 1/0 5/16 -1/2 1/0 1/3 -1/1 0/1 5/14 0/1 4/11 0/1 1/3 7/19 0/1 1/1 17/46 1/2 1/0 10/27 0/1 1/1 3/8 1/2 1/0 8/21 1/1 5/13 0/1 1/0 7/18 3/2 1/0 2/5 -1/1 1/0 7/17 -1/1 0/1 12/29 -1/1 0/1 5/12 -1/2 1/0 3/7 0/1 7/16 1/2 1/0 11/25 0/1 1/1 15/34 1/2 1/0 4/9 0/1 1/1 5/11 1/1 2/1 6/13 5/1 1/0 1/2 -1/2 1/0 6/11 -3/1 -2/1 5/9 -2/1 -1/1 4/7 -1/1 11/19 -1/1 0/1 18/31 -1/1 0/1 7/12 -1/2 1/0 24/41 -1/1 1/0 41/70 -1/1 17/29 -1/1 0/1 10/17 -1/1 0/1 3/5 0/1 1/0 17/28 1/0 14/23 -5/1 1/0 11/18 -5/2 1/0 8/13 -1/1 1/0 21/34 -5/2 1/0 13/21 -2/1 18/29 -2/1 -1/1 5/8 -3/2 1/0 17/27 -2/1 -1/1 12/19 -2/1 -1/1 7/11 -4/3 -1/1 9/14 -1/1 2/3 -1/1 0/1 11/16 -1/2 1/0 20/29 -1/1 0/1 29/42 0/1 9/13 0/1 1/0 7/10 -3/2 1/0 5/7 -1/1 13/18 -3/4 -1/2 47/65 -1/1 -2/3 81/112 -3/4 -1/2 34/47 -1/1 -2/3 21/29 -1/1 -2/3 8/11 -1/1 -2/3 19/26 -7/12 -1/2 30/41 -9/17 -1/2 41/56 -1/2 11/15 -1/2 -2/5 3/4 -1/2 1/0 10/13 -3/1 1/0 17/22 -3/2 1/0 41/53 -2/1 -1/1 65/84 -3/2 1/0 24/31 -2/1 -1/1 7/9 -2/1 -1/1 11/14 -1/1 4/5 -1/1 -1/2 9/11 -1/1 0/1 23/28 -1/2 1/0 14/17 -1/1 0/1 5/6 -1/2 1/0 11/13 -2/1 1/0 6/7 -1/1 7/8 -1/2 1/0 1/1 -1/1 0/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(85,74,-224,-195) (-1/1,-6/7) -> (-8/21,-11/29) Hyperbolic Matrix(139,118,-364,-309) (-6/7,-5/6) -> (-13/34,-8/21) Hyperbolic Matrix(29,24,-168,-139) (-5/6,-9/11) -> (-3/17,-1/6) Hyperbolic Matrix(57,46,140,113) (-9/11,-4/5) -> (2/5,7/17) Hyperbolic Matrix(111,88,140,111) (-4/5,-11/14) -> (11/14,4/5) Hyperbolic Matrix(197,154,252,197) (-11/14,-7/9) -> (7/9,11/14) Hyperbolic Matrix(281,218,-504,-391) (-7/9,-17/22) -> (-19/34,-5/9) Hyperbolic Matrix(449,346,728,561) (-17/22,-10/13) -> (8/13,21/34) Hyperbolic Matrix(29,22,112,85) (-10/13,-3/4) -> (1/4,4/15) Hyperbolic Matrix(27,20,112,83) (-3/4,-11/15) -> (3/13,1/4) Hyperbolic Matrix(167,122,-308,-225) (-11/15,-8/11) -> (-6/11,-7/13) Hyperbolic Matrix(139,100,-196,-141) (-8/11,-5/7) -> (-5/7,-12/17) Parabolic Matrix(559,394,952,671) (-12/17,-19/27) -> (17/29,10/17) Hyperbolic Matrix(953,670,-1512,-1063) (-19/27,-7/10) -> (-29/46,-17/27) Hyperbolic Matrix(141,98,364,253) (-7/10,-9/13) -> (5/13,7/18) Hyperbolic Matrix(337,232,-812,-559) (-9/13,-11/16) -> (-5/12,-17/41) Hyperbolic Matrix(309,212,532,365) (-11/16,-2/3) -> (18/31,7/12) Hyperbolic Matrix(55,36,84,55) (-2/3,-9/14) -> (9/14,2/3) Hyperbolic Matrix(197,126,308,197) (-9/14,-7/11) -> (7/11,9/14) Hyperbolic Matrix(139,88,308,195) (-7/11,-12/19) -> (4/9,5/11) Hyperbolic Matrix(923,582,-3332,-2101) (-12/19,-29/46) -> (-5/18,-18/65) Hyperbolic Matrix(197,124,224,141) (-17/27,-5/8) -> (7/8,1/1) Hyperbolic Matrix(29,18,-224,-139) (-5/8,-13/21) -> (-1/7,-1/8) Hyperbolic Matrix(55,34,-364,-225) (-13/21,-8/13) -> (-2/13,-1/7) Hyperbolic Matrix(111,68,364,223) (-8/13,-11/18) -> (3/10,4/13) Hyperbolic Matrix(197,120,-504,-307) (-11/18,-3/5) -> (-9/23,-7/18) Hyperbolic Matrix(27,16,140,83) (-3/5,-10/17) -> (2/11,1/5) Hyperbolic Matrix(589,346,812,477) (-10/17,-17/29) -> (21/29,8/11) Hyperbolic Matrix(253,148,-812,-475) (-17/29,-7/12) -> (-5/16,-9/29) Hyperbolic Matrix(111,64,-196,-113) (-7/12,-4/7) -> (-4/7,-9/16) Parabolic Matrix(307,172,448,251) (-9/16,-14/25) -> (2/3,11/16) Hyperbolic Matrix(533,298,-2352,-1315) (-14/25,-19/34) -> (-5/22,-12/53) Hyperbolic Matrix(113,62,308,169) (-5/9,-6/11) -> (4/11,7/19) Hyperbolic Matrix(113,60,-420,-223) (-7/13,-1/2) -> (-7/26,-11/41) Hyperbolic Matrix(83,38,-308,-141) (-1/2,-5/11) -> (-3/11,-7/26) Hyperbolic Matrix(195,88,308,139) (-5/11,-4/9) -> (12/19,7/11) Hyperbolic Matrix(83,36,-196,-85) (-4/9,-3/7) -> (-3/7,-8/19) Parabolic Matrix(195,82,-868,-365) (-8/19,-13/31) -> (-7/31,-2/9) Hyperbolic Matrix(167,70,532,223) (-13/31,-5/12) -> (5/16,1/3) Hyperbolic Matrix(1819,754,2632,1091) (-17/41,-29/70) -> (29/42,9/13) Hyperbolic Matrix(2241,928,3248,1345) (-29/70,-12/29) -> (20/29,29/42) Hyperbolic Matrix(281,116,952,393) (-12/29,-7/17) -> (5/17,8/27) Hyperbolic Matrix(113,46,140,57) (-7/17,-2/5) -> (4/5,9/11) Hyperbolic Matrix(477,188,784,309) (-2/5,-11/28) -> (17/28,14/23) Hyperbolic Matrix(475,186,784,307) (-11/28,-9/23) -> (3/5,17/28) Hyperbolic Matrix(253,98,364,141) (-7/18,-5/13) -> (9/13,7/10) Hyperbolic Matrix(167,64,728,279) (-5/13,-13/34) -> (5/22,3/13) Hyperbolic Matrix(281,106,448,169) (-11/29,-3/8) -> (5/8,17/27) Hyperbolic Matrix(279,104,448,167) (-3/8,-10/27) -> (18/29,5/8) Hyperbolic Matrix(503,186,-1820,-673) (-10/27,-7/19) -> (-13/47,-8/29) Hyperbolic Matrix(169,62,308,113) (-7/19,-4/11) -> (6/11,5/9) Hyperbolic Matrix(111,40,308,111) (-4/11,-5/14) -> (5/14,4/11) Hyperbolic Matrix(29,10,84,29) (-5/14,-1/3) -> (1/3,5/14) Hyperbolic Matrix(197,62,448,141) (-1/3,-5/16) -> (7/16,11/25) Hyperbolic Matrix(1903,590,3248,1007) (-9/29,-13/42) -> (41/70,17/29) Hyperbolic Matrix(1541,476,2632,813) (-13/42,-4/13) -> (24/41,41/70) Hyperbolic Matrix(223,68,364,111) (-4/13,-3/10) -> (11/18,8/13) Hyperbolic Matrix(55,16,-196,-57) (-3/10,-2/7) -> (-2/7,-5/18) Parabolic Matrix(9073,2512,12544,3473) (-18/65,-31/112) -> (81/112,34/47) Hyperbolic Matrix(9071,2510,12544,3471) (-31/112,-13/47) -> (47/65,81/112) Hyperbolic Matrix(335,92,812,223) (-8/29,-3/11) -> (7/17,12/29) Hyperbolic Matrix(2297,616,3136,841) (-11/41,-15/56) -> (41/56,11/15) Hyperbolic Matrix(2295,614,3136,839) (-15/56,-4/15) -> (30/41,41/56) Hyperbolic Matrix(85,22,112,29) (-4/15,-1/4) -> (3/4,10/13) Hyperbolic Matrix(83,20,112,27) (-1/4,-3/13) -> (11/15,3/4) Hyperbolic Matrix(307,70,364,83) (-3/13,-5/22) -> (5/6,11/13) Hyperbolic Matrix(5461,1236,7056,1597) (-12/53,-19/84) -> (65/84,24/31) Hyperbolic Matrix(5459,1234,7056,1595) (-19/84,-7/31) -> (41/53,65/84) Hyperbolic Matrix(55,12,252,55) (-2/9,-3/14) -> (3/14,2/9) Hyperbolic Matrix(29,6,140,29) (-3/14,-1/5) -> (1/5,3/14) Hyperbolic Matrix(83,16,140,27) (-1/5,-2/11) -> (10/17,3/5) Hyperbolic Matrix(645,116,784,141) (-2/11,-5/28) -> (23/28,14/17) Hyperbolic Matrix(643,114,784,139) (-5/28,-3/17) -> (9/11,23/28) Hyperbolic Matrix(281,44,364,57) (-1/6,-2/13) -> (10/13,17/22) Hyperbolic Matrix(83,10,224,27) (-1/8,0/1) -> (10/27,3/8) Hyperbolic Matrix(139,-18,224,-29) (0/1,1/7) -> (13/21,18/29) Hyperbolic Matrix(225,-34,364,-55) (1/7,1/6) -> (21/34,13/21) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(223,-50,504,-113) (2/9,5/22) -> (15/34,4/9) Hyperbolic Matrix(141,-38,308,-83) (4/15,3/11) -> (5/11,6/13) Hyperbolic Matrix(57,-16,196,-55) (3/11,2/7) -> (2/7,5/17) Parabolic Matrix(559,-166,1512,-449) (8/27,3/10) -> (17/46,10/27) Hyperbolic Matrix(475,-148,812,-253) (4/13,5/16) -> (7/12,24/41) Hyperbolic Matrix(2409,-890,3332,-1231) (7/19,17/46) -> (13/18,47/65) Hyperbolic Matrix(195,-74,224,-85) (3/8,8/21) -> (6/7,7/8) Hyperbolic Matrix(309,-118,364,-139) (8/21,5/13) -> (11/13,6/7) Hyperbolic Matrix(307,-120,504,-197) (7/18,2/5) -> (14/23,11/18) Hyperbolic Matrix(559,-232,812,-337) (12/29,5/12) -> (11/16,20/29) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(1819,-802,2352,-1037) (11/25,15/34) -> (17/22,41/53) Hyperbolic Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(225,-122,308,-167) (1/2,6/11) -> (8/11,19/26) Hyperbolic Matrix(113,-64,196,-111) (5/9,4/7) -> (4/7,11/19) Parabolic Matrix(673,-390,868,-503) (11/19,18/31) -> (24/31,7/9) Hyperbolic Matrix(1317,-830,1820,-1147) (17/27,12/19) -> (34/47,21/29) Hyperbolic Matrix(141,-100,196,-139) (7/10,5/7) -> (5/7,13/18) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(85,74,-224,-195) -> Matrix(1,0,-2,1) Matrix(139,118,-364,-309) -> Matrix(3,2,-8,-5) Matrix(29,24,-168,-139) -> Matrix(1,0,0,1) Matrix(57,46,140,113) -> Matrix(1,0,0,1) Matrix(111,88,140,111) -> Matrix(1,2,-2,-3) Matrix(197,154,252,197) -> Matrix(5,4,-4,-3) Matrix(281,218,-504,-391) -> Matrix(1,0,0,1) Matrix(449,346,728,561) -> Matrix(7,4,-2,-1) Matrix(29,22,112,85) -> Matrix(1,0,0,1) Matrix(27,20,112,83) -> Matrix(1,0,0,1) Matrix(167,122,-308,-225) -> Matrix(1,4,-2,-7) Matrix(139,100,-196,-141) -> Matrix(3,4,-4,-5) Matrix(559,394,952,671) -> Matrix(3,2,-2,-1) Matrix(953,670,-1512,-1063) -> Matrix(1,0,0,1) Matrix(141,98,364,253) -> Matrix(1,0,2,1) Matrix(337,232,-812,-559) -> Matrix(1,0,0,1) Matrix(309,212,532,365) -> Matrix(1,0,0,1) Matrix(55,36,84,55) -> Matrix(1,0,0,1) Matrix(197,126,308,197) -> Matrix(9,8,-8,-7) Matrix(139,88,308,195) -> Matrix(3,2,4,3) Matrix(923,582,-3332,-2101) -> Matrix(3,2,4,3) Matrix(197,124,224,141) -> Matrix(3,2,-2,-1) Matrix(29,18,-224,-139) -> Matrix(3,2,-2,-1) Matrix(55,34,-364,-225) -> Matrix(3,2,-8,-5) Matrix(111,68,364,223) -> Matrix(3,2,-2,-1) Matrix(197,120,-504,-307) -> Matrix(7,4,-16,-9) Matrix(27,16,140,83) -> Matrix(1,0,0,1) Matrix(589,346,812,477) -> Matrix(1,2,-2,-3) Matrix(253,148,-812,-475) -> Matrix(1,0,0,1) Matrix(111,64,-196,-113) -> Matrix(1,2,-2,-3) Matrix(307,172,448,251) -> Matrix(3,2,-2,-1) Matrix(533,298,-2352,-1315) -> Matrix(3,2,-8,-5) Matrix(113,62,308,169) -> Matrix(3,2,4,3) Matrix(113,60,-420,-223) -> Matrix(5,2,2,1) Matrix(83,38,-308,-141) -> Matrix(5,2,2,1) Matrix(195,88,308,139) -> Matrix(7,2,-4,-1) Matrix(83,36,-196,-85) -> Matrix(1,0,2,1) Matrix(195,82,-868,-365) -> Matrix(1,0,-2,1) Matrix(167,70,532,223) -> Matrix(1,0,0,1) Matrix(1819,754,2632,1091) -> Matrix(1,0,2,1) Matrix(2241,928,3248,1345) -> Matrix(1,0,0,1) Matrix(281,116,952,393) -> Matrix(1,0,2,1) Matrix(113,46,140,57) -> Matrix(1,0,0,1) Matrix(477,188,784,309) -> Matrix(11,6,-2,-1) Matrix(475,186,784,307) -> Matrix(9,4,2,1) Matrix(253,98,364,141) -> Matrix(1,0,2,1) Matrix(167,64,728,279) -> Matrix(5,2,2,1) Matrix(281,106,448,169) -> Matrix(7,2,-4,-1) Matrix(279,104,448,167) -> Matrix(7,2,-4,-1) Matrix(503,186,-1820,-673) -> Matrix(1,0,4,1) Matrix(169,62,308,113) -> Matrix(7,2,-4,-1) Matrix(111,40,308,111) -> Matrix(1,0,8,1) Matrix(29,10,84,29) -> Matrix(1,0,0,1) Matrix(197,62,448,141) -> Matrix(1,0,2,1) Matrix(1903,590,3248,1007) -> Matrix(1,0,0,1) Matrix(1541,476,2632,813) -> Matrix(3,2,-2,-1) Matrix(223,68,364,111) -> Matrix(3,2,-2,-1) Matrix(55,16,-196,-57) -> Matrix(1,0,4,1) Matrix(9073,2512,12544,3473) -> Matrix(3,-2,-4,3) Matrix(9071,2510,12544,3471) -> Matrix(3,-2,-4,3) Matrix(335,92,812,223) -> Matrix(1,0,-2,1) Matrix(2297,616,3136,841) -> Matrix(1,-10,-2,21) Matrix(2295,614,3136,839) -> Matrix(1,12,-2,-23) Matrix(85,22,112,29) -> Matrix(1,0,0,1) Matrix(83,20,112,27) -> Matrix(1,0,0,1) Matrix(307,70,364,83) -> Matrix(1,0,2,1) Matrix(5461,1236,7056,1597) -> Matrix(7,2,-4,-1) Matrix(5459,1234,7056,1595) -> Matrix(7,2,-4,-1) Matrix(55,12,252,55) -> Matrix(1,0,4,1) Matrix(29,6,140,29) -> Matrix(1,0,-2,1) Matrix(83,16,140,27) -> Matrix(1,0,0,1) Matrix(645,116,784,141) -> Matrix(1,0,0,1) Matrix(643,114,784,139) -> Matrix(1,0,0,1) Matrix(281,44,364,57) -> Matrix(3,2,-2,-1) Matrix(83,10,224,27) -> Matrix(1,0,2,1) Matrix(139,-18,224,-29) -> Matrix(3,2,-2,-1) Matrix(225,-34,364,-55) -> Matrix(1,-2,0,1) Matrix(139,-24,168,-29) -> Matrix(1,0,0,1) Matrix(223,-50,504,-113) -> Matrix(1,0,0,1) Matrix(141,-38,308,-83) -> Matrix(5,2,2,1) Matrix(57,-16,196,-55) -> Matrix(1,0,4,1) Matrix(559,-166,1512,-449) -> Matrix(1,0,0,1) Matrix(475,-148,812,-253) -> Matrix(1,0,0,1) Matrix(2409,-890,3332,-1231) -> Matrix(3,-2,-4,3) Matrix(195,-74,224,-85) -> Matrix(1,0,-2,1) Matrix(309,-118,364,-139) -> Matrix(1,-2,0,1) Matrix(307,-120,504,-197) -> Matrix(1,-4,0,1) Matrix(559,-232,812,-337) -> Matrix(1,0,0,1) Matrix(85,-36,196,-83) -> Matrix(1,0,2,1) Matrix(1819,-802,2352,-1037) -> Matrix(1,-2,0,1) Matrix(307,-144,420,-197) -> Matrix(1,4,-2,-7) Matrix(225,-122,308,-167) -> Matrix(1,4,-2,-7) Matrix(113,-64,196,-111) -> Matrix(1,2,-2,-3) Matrix(673,-390,868,-503) -> Matrix(3,2,-2,-1) Matrix(1317,-830,1820,-1147) -> Matrix(3,4,-4,-5) Matrix(141,-100,196,-139) -> Matrix(3,4,-4,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 Degree of the the map X: 16 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 1/7 1/6 2/11 1/5 3/14 2/9 3/13 1/4 2/7 3/10 1/3 5/14 3/8 5/12 3/7 1/2 4/7 7/12 17/28 5/8 29/42 5/7 81/112 41/56 3/4 65/84 23/28 6/7 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 0/1 1/7 0/1 1/6 -1/2 1/0 2/11 -1/1 0/1 1/5 -1/2 0/1 3/14 0/1 2/9 0/1 1/1 5/22 1/2 1/0 3/13 2/1 1/0 1/4 -1/2 1/0 4/15 -3/5 -1/2 3/11 -1/3 0/1 2/7 0/1 5/17 0/1 1/1 8/27 0/1 1/1 3/10 1/2 1/0 4/13 -1/1 1/0 5/16 -1/2 1/0 1/3 -1/1 0/1 5/14 0/1 4/11 0/1 1/3 7/19 0/1 1/1 17/46 1/2 1/0 10/27 0/1 1/1 3/8 1/2 1/0 8/21 1/1 5/13 0/1 1/0 7/18 3/2 1/0 2/5 -1/1 1/0 7/17 -1/1 0/1 12/29 -1/1 0/1 5/12 -1/2 1/0 3/7 0/1 7/16 1/2 1/0 11/25 0/1 1/1 15/34 1/2 1/0 4/9 0/1 1/1 5/11 1/1 2/1 6/13 5/1 1/0 1/2 -1/2 1/0 6/11 -3/1 -2/1 5/9 -2/1 -1/1 4/7 -1/1 11/19 -1/1 0/1 18/31 -1/1 0/1 7/12 -1/2 1/0 24/41 -1/1 1/0 41/70 -1/1 17/29 -1/1 0/1 10/17 -1/1 0/1 3/5 0/1 1/0 17/28 1/0 14/23 -5/1 1/0 11/18 -5/2 1/0 8/13 -1/1 1/0 21/34 -5/2 1/0 13/21 -2/1 18/29 -2/1 -1/1 5/8 -3/2 1/0 17/27 -2/1 -1/1 12/19 -2/1 -1/1 7/11 -4/3 -1/1 9/14 -1/1 2/3 -1/1 0/1 11/16 -1/2 1/0 20/29 -1/1 0/1 29/42 0/1 9/13 0/1 1/0 7/10 -3/2 1/0 5/7 -1/1 13/18 -3/4 -1/2 47/65 -1/1 -2/3 81/112 -3/4 -1/2 34/47 -1/1 -2/3 21/29 -1/1 -2/3 8/11 -1/1 -2/3 19/26 -7/12 -1/2 30/41 -9/17 -1/2 41/56 -1/2 11/15 -1/2 -2/5 3/4 -1/2 1/0 10/13 -3/1 1/0 17/22 -3/2 1/0 41/53 -2/1 -1/1 65/84 -3/2 1/0 24/31 -2/1 -1/1 7/9 -2/1 -1/1 11/14 -1/1 4/5 -1/1 -1/2 9/11 -1/1 0/1 23/28 -1/2 1/0 14/17 -1/1 0/1 5/6 -1/2 1/0 11/13 -2/1 1/0 6/7 -1/1 7/8 -1/2 1/0 1/1 -1/1 0/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(139,-18,224,-29) (0/1,1/7) -> (13/21,18/29) Hyperbolic Matrix(225,-34,364,-55) (1/7,1/6) -> (21/34,13/21) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(57,-11,140,-27) (2/11,1/5) -> (2/5,7/17) Hyperbolic Matrix(111,-23,140,-29) (1/5,3/14) -> (11/14,4/5) Hyperbolic Matrix(197,-43,252,-55) (3/14,2/9) -> (7/9,11/14) Hyperbolic Matrix(223,-50,504,-113) (2/9,5/22) -> (15/34,4/9) Hyperbolic Matrix(449,-103,728,-167) (5/22,3/13) -> (8/13,21/34) Hyperbolic Matrix(29,-7,112,-27) (3/13,1/4) -> (1/4,4/15) Parabolic Matrix(141,-38,308,-83) (4/15,3/11) -> (5/11,6/13) Hyperbolic Matrix(57,-16,196,-55) (3/11,2/7) -> (2/7,5/17) Parabolic Matrix(559,-165,952,-281) (5/17,8/27) -> (17/29,10/17) Hyperbolic Matrix(559,-166,1512,-449) (8/27,3/10) -> (17/46,10/27) Hyperbolic Matrix(141,-43,364,-111) (3/10,4/13) -> (5/13,7/18) Hyperbolic Matrix(475,-148,812,-253) (4/13,5/16) -> (7/12,24/41) Hyperbolic Matrix(309,-97,532,-167) (5/16,1/3) -> (18/31,7/12) Hyperbolic Matrix(55,-19,84,-29) (1/3,5/14) -> (9/14,2/3) Hyperbolic Matrix(197,-71,308,-111) (5/14,4/11) -> (7/11,9/14) Hyperbolic Matrix(139,-51,308,-113) (4/11,7/19) -> (4/9,5/11) Hyperbolic Matrix(2409,-890,3332,-1231) (7/19,17/46) -> (13/18,47/65) Hyperbolic Matrix(197,-73,224,-83) (10/27,3/8) -> (7/8,1/1) Hyperbolic Matrix(195,-74,224,-85) (3/8,8/21) -> (6/7,7/8) Hyperbolic Matrix(309,-118,364,-139) (8/21,5/13) -> (11/13,6/7) Hyperbolic Matrix(307,-120,504,-197) (7/18,2/5) -> (14/23,11/18) Hyperbolic Matrix(589,-243,812,-335) (7/17,12/29) -> (21/29,8/11) Hyperbolic Matrix(559,-232,812,-337) (12/29,5/12) -> (11/16,20/29) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(307,-135,448,-197) (7/16,11/25) -> (2/3,11/16) Hyperbolic Matrix(1819,-802,2352,-1037) (11/25,15/34) -> (17/22,41/53) Hyperbolic Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(225,-122,308,-167) (1/2,6/11) -> (8/11,19/26) Hyperbolic Matrix(195,-107,308,-169) (6/11,5/9) -> (12/19,7/11) Hyperbolic Matrix(113,-64,196,-111) (5/9,4/7) -> (4/7,11/19) Parabolic Matrix(673,-390,868,-503) (11/19,18/31) -> (24/31,7/9) Hyperbolic Matrix(1819,-1065,2632,-1541) (24/41,41/70) -> (29/42,9/13) Hyperbolic Matrix(2241,-1313,3248,-1903) (41/70,17/29) -> (20/29,29/42) Hyperbolic Matrix(113,-67,140,-83) (10/17,3/5) -> (4/5,9/11) Hyperbolic Matrix(477,-289,784,-475) (3/5,17/28) -> (17/28,14/23) Parabolic Matrix(253,-155,364,-223) (11/18,8/13) -> (9/13,7/10) Hyperbolic Matrix(281,-175,448,-279) (18/29,5/8) -> (5/8,17/27) Parabolic Matrix(1317,-830,1820,-1147) (17/27,12/19) -> (34/47,21/29) Hyperbolic Matrix(141,-100,196,-139) (7/10,5/7) -> (5/7,13/18) Parabolic Matrix(9073,-6561,12544,-9071) (47/65,81/112) -> (81/112,34/47) Parabolic Matrix(2297,-1681,3136,-2295) (30/41,41/56) -> (41/56,11/15) Parabolic Matrix(85,-63,112,-83) (11/15,3/4) -> (3/4,10/13) Parabolic Matrix(307,-237,364,-281) (10/13,17/22) -> (5/6,11/13) Hyperbolic Matrix(5461,-4225,7056,-5459) (41/53,65/84) -> (65/84,24/31) Parabolic Matrix(645,-529,784,-643) (9/11,23/28) -> (23/28,14/17) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,1,-2,-1) Matrix(139,-18,224,-29) -> Matrix(3,2,-2,-1) Matrix(225,-34,364,-55) -> Matrix(1,-2,0,1) Matrix(139,-24,168,-29) -> Matrix(1,0,0,1) Matrix(57,-11,140,-27) -> Matrix(1,1,-2,-1) Matrix(111,-23,140,-29) -> Matrix(3,1,-4,-1) Matrix(197,-43,252,-55) -> Matrix(3,-1,-2,1) Matrix(223,-50,504,-113) -> Matrix(1,0,0,1) Matrix(449,-103,728,-167) -> Matrix(1,-3,0,1) Matrix(29,-7,112,-27) -> Matrix(1,1,-2,-1) Matrix(141,-38,308,-83) -> Matrix(5,2,2,1) Matrix(57,-16,196,-55) -> Matrix(1,0,4,1) Matrix(559,-165,952,-281) -> Matrix(1,-1,0,1) Matrix(559,-166,1512,-449) -> Matrix(1,0,0,1) Matrix(141,-43,364,-111) -> Matrix(1,1,0,1) Matrix(475,-148,812,-253) -> Matrix(1,0,0,1) Matrix(309,-97,532,-167) -> Matrix(1,1,-2,-1) Matrix(55,-19,84,-29) -> Matrix(1,1,-2,-1) Matrix(197,-71,308,-111) -> Matrix(7,-1,-6,1) Matrix(139,-51,308,-113) -> Matrix(1,-1,2,-1) Matrix(2409,-890,3332,-1231) -> Matrix(3,-2,-4,3) Matrix(197,-73,224,-83) -> Matrix(1,-1,0,1) Matrix(195,-74,224,-85) -> Matrix(1,0,-2,1) Matrix(309,-118,364,-139) -> Matrix(1,-2,0,1) Matrix(307,-120,504,-197) -> Matrix(1,-4,0,1) Matrix(589,-243,812,-335) -> Matrix(3,1,-4,-1) Matrix(559,-232,812,-337) -> Matrix(1,0,0,1) Matrix(85,-36,196,-83) -> Matrix(1,0,2,1) Matrix(307,-135,448,-197) -> Matrix(1,-1,0,1) Matrix(1819,-802,2352,-1037) -> Matrix(1,-2,0,1) Matrix(307,-144,420,-197) -> Matrix(1,4,-2,-7) Matrix(225,-122,308,-167) -> Matrix(1,4,-2,-7) Matrix(195,-107,308,-169) -> Matrix(3,5,-2,-3) Matrix(113,-64,196,-111) -> Matrix(1,2,-2,-3) Matrix(673,-390,868,-503) -> Matrix(3,2,-2,-1) Matrix(1819,-1065,2632,-1541) -> Matrix(1,1,0,1) Matrix(2241,-1313,3248,-1903) -> Matrix(1,1,-2,-1) Matrix(113,-67,140,-83) -> Matrix(1,1,-2,-1) Matrix(477,-289,784,-475) -> Matrix(1,-5,0,1) Matrix(253,-155,364,-223) -> Matrix(1,1,0,1) Matrix(281,-175,448,-279) -> Matrix(3,5,-2,-3) Matrix(1317,-830,1820,-1147) -> Matrix(3,4,-4,-5) Matrix(141,-100,196,-139) -> Matrix(3,4,-4,-5) Matrix(9073,-6561,12544,-9071) -> Matrix(7,5,-10,-7) Matrix(2297,-1681,3136,-2295) -> Matrix(21,11,-44,-23) Matrix(85,-63,112,-83) -> Matrix(1,1,-2,-1) Matrix(307,-237,364,-281) -> Matrix(1,1,0,1) Matrix(5461,-4225,7056,-5459) -> Matrix(3,5,-2,-3) Matrix(645,-529,784,-643) -> Matrix(1,1,-2,-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): 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 0/1 (-1/1,0/1) 0 14 1/7 0/1 1 2 1/6 (-1/2,1/0) 0 7 5/28 (-1/1,0/1).(-1/2,1/0) 0 1 2/11 (-1/1,0/1) 0 14 1/5 (-1/2,0/1) 0 14 3/14 0/1 3 1 2/9 (0/1,1/1) 0 14 5/22 (1/2,1/0) 0 7 3/13 (2/1,1/0) 0 14 1/4 (-1/1,0/1).(-1/2,1/0) 0 7 4/15 (-3/5,-1/2) 0 14 3/11 (-1/3,0/1) 0 14 2/7 0/1 2 2 5/17 (0/1,1/1) 0 14 8/27 (0/1,1/1) 0 14 3/10 (1/2,1/0) 0 7 4/13 (-1/1,1/0) 0 14 5/16 (-1/1,0/1).(-1/2,1/0) 0 7 1/3 (-1/1,0/1) 0 14 5/14 0/1 4 1 4/11 (0/1,1/3) 0 14 7/19 (0/1,1/1) 0 14 31/84 (0/1,1/1).(1/2,1/0) 0 1 17/46 (1/2,1/0) 0 7 10/27 (0/1,1/1) 0 14 3/8 (0/1,1/1).(1/2,1/0) 0 7 11/29 (0/1,1/1) 0 14 8/21 1/1 1 2 13/34 (3/2,1/0) 0 7 5/13 (0/1,1/0) 0 14 7/18 (3/2,1/0) 0 7 11/28 1/0 5 1 2/5 (-1/1,1/0) 0 14 7/17 (-1/1,0/1) 0 14 12/29 (-1/1,0/1) 0 14 29/70 0/1 1 1 17/41 (0/1,1/0) 0 14 5/12 (-1/1,0/1).(-1/2,1/0) 0 7 3/7 0/1 1 2 7/16 (0/1,1/1).(1/2,1/0) 0 7 11/25 (0/1,1/1) 0 14 37/84 (0/1,1/1).(1/2,1/0) 0 1 15/34 (1/2,1/0) 0 7 4/9 (0/1,1/1) 0 14 5/11 (1/1,2/1) 0 14 6/13 (5/1,1/0) 0 14 13/28 1/0 11 1 1/2 (-1/2,1/0) 0 7 1/0 (-1/1,0/1).(-1/2,1/0) 0 1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(85,-11,224,-29) (0/1,1/7) -> (11/29,8/21) Glide Reflection Matrix(139,-21,364,-55) (1/7,1/6) -> (8/21,13/34) Glide Reflection Matrix(29,-5,168,-29) (1/6,5/28) -> (1/6,5/28) Reflection Matrix(111,-20,616,-111) (5/28,2/11) -> (5/28,2/11) Reflection Matrix(57,-11,140,-27) (2/11,1/5) -> (2/5,7/17) Hyperbolic Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(55,-12,252,-55) (3/14,2/9) -> (3/14,2/9) Reflection Matrix(223,-50,504,-113) (2/9,5/22) -> (15/34,4/9) Hyperbolic Matrix(279,-64,728,-167) (5/22,3/13) -> (13/34,5/13) Glide Reflection Matrix(29,-7,112,-27) (3/13,1/4) -> (1/4,4/15) Parabolic Matrix(141,-38,308,-83) (4/15,3/11) -> (5/11,6/13) Hyperbolic Matrix(57,-16,196,-55) (3/11,2/7) -> (2/7,5/17) Parabolic Matrix(393,-116,952,-281) (5/17,8/27) -> (7/17,12/29) Glide Reflection Matrix(559,-166,1512,-449) (8/27,3/10) -> (17/46,10/27) Hyperbolic Matrix(141,-43,364,-111) (3/10,4/13) -> (5/13,7/18) Hyperbolic Matrix(337,-105,812,-253) (4/13,5/16) -> (17/41,5/12) Glide Reflection Matrix(197,-62,448,-141) (5/16,1/3) -> (7/16,11/25) Glide Reflection Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(111,-40,308,-111) (5/14,4/11) -> (5/14,4/11) Reflection Matrix(139,-51,308,-113) (4/11,7/19) -> (4/9,5/11) Hyperbolic Matrix(1177,-434,3192,-1177) (7/19,31/84) -> (7/19,31/84) Reflection Matrix(1427,-527,3864,-1427) (31/84,17/46) -> (31/84,17/46) Reflection Matrix(169,-63,448,-167) (10/27,3/8) -> (3/8,11/29) Parabolic Matrix(197,-77,504,-197) (7/18,11/28) -> (7/18,11/28) Reflection Matrix(111,-44,280,-111) (11/28,2/5) -> (11/28,2/5) Reflection Matrix(1681,-696,4060,-1681) (12/29,29/70) -> (12/29,29/70) Reflection Matrix(2379,-986,5740,-2379) (29/70,17/41) -> (29/70,17/41) Reflection Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(1849,-814,4200,-1849) (11/25,37/84) -> (11/25,37/84) Reflection Matrix(1259,-555,2856,-1259) (37/84,15/34) -> (37/84,15/34) Reflection Matrix(337,-156,728,-337) (6/13,13/28) -> (6/13,13/28) Reflection Matrix(27,-13,56,-27) (13/28,1/2) -> (13/28,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(85,-11,224,-29) -> Matrix(1,1,2,1) Matrix(139,-21,364,-55) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(29,-5,168,-29) -> Matrix(1,1,0,-1) (1/6,5/28) -> (-1/2,1/0) Matrix(111,-20,616,-111) -> Matrix(-1,0,2,1) (5/28,2/11) -> (-1/1,0/1) Matrix(57,-11,140,-27) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(29,-6,140,-29) -> Matrix(-1,0,4,1) (1/5,3/14) -> (-1/2,0/1) Matrix(55,-12,252,-55) -> Matrix(1,0,2,-1) (3/14,2/9) -> (0/1,1/1) Matrix(223,-50,504,-113) -> Matrix(1,0,0,1) Matrix(279,-64,728,-167) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(29,-7,112,-27) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(141,-38,308,-83) -> Matrix(5,2,2,1) Matrix(57,-16,196,-55) -> Matrix(1,0,4,1) 0/1 Matrix(393,-116,952,-281) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(559,-166,1512,-449) -> Matrix(1,0,0,1) Matrix(141,-43,364,-111) -> Matrix(1,1,0,1) 1/0 Matrix(337,-105,812,-253) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(197,-62,448,-141) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(29,-10,84,-29) -> Matrix(-1,0,2,1) (1/3,5/14) -> (-1/1,0/1) Matrix(111,-40,308,-111) -> Matrix(1,0,6,-1) (5/14,4/11) -> (0/1,1/3) Matrix(139,-51,308,-113) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(1177,-434,3192,-1177) -> Matrix(1,0,2,-1) (7/19,31/84) -> (0/1,1/1) Matrix(1427,-527,3864,-1427) -> Matrix(-1,1,0,1) (31/84,17/46) -> (1/2,1/0) Matrix(169,-63,448,-167) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(197,-77,504,-197) -> Matrix(-1,3,0,1) (7/18,11/28) -> (3/2,1/0) Matrix(111,-44,280,-111) -> Matrix(1,2,0,-1) (11/28,2/5) -> (-1/1,1/0) Matrix(1681,-696,4060,-1681) -> Matrix(-1,0,2,1) (12/29,29/70) -> (-1/1,0/1) Matrix(2379,-986,5740,-2379) -> Matrix(1,0,0,-1) (29/70,17/41) -> (0/1,1/0) Matrix(85,-36,196,-83) -> Matrix(1,0,2,1) 0/1 Matrix(1849,-814,4200,-1849) -> Matrix(1,0,2,-1) (11/25,37/84) -> (0/1,1/1) Matrix(1259,-555,2856,-1259) -> Matrix(-1,1,0,1) (37/84,15/34) -> (1/2,1/0) Matrix(337,-156,728,-337) -> Matrix(-1,10,0,1) (6/13,13/28) -> (5/1,1/0) Matrix(27,-13,56,-27) -> Matrix(1,1,0,-1) (13/28,1/2) -> (-1/2,1/0) Matrix(-1,1,0,1) -> Matrix(1,1,0,-1) (1/2,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.