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 0/1 1/1 -6/7 0/1 -5/6 1/1 -9/11 0/1 1/5 -4/5 1/2 1/1 -11/14 1/1 -7/9 1/1 2/1 -17/22 1/1 -10/13 1/1 1/0 -3/4 -1/1 1/1 -11/15 -1/1 -1/2 -8/11 -1/3 0/1 -5/7 0/1 -12/17 0/1 1/5 -19/27 0/1 1/5 -7/10 1/5 -9/13 3/10 1/3 -11/16 1/3 5/13 -2/3 1/2 1/1 -9/14 1/1 -7/11 1/1 4/3 -12/19 5/3 2/1 -29/46 1/1 -17/27 1/1 2/1 -5/8 1/1 3/1 -13/21 1/0 -8/13 1/1 1/0 -11/18 1/1 -3/5 -1/1 1/0 -10/17 0/1 1/1 -17/29 0/1 1/1 -7/12 -1/1 1/1 -4/7 0/1 -9/16 1/5 1/3 -14/25 3/7 1/2 -19/34 1/1 -5/9 0/1 1/1 -6/11 0/1 1/1 -7/13 1/3 1/2 -1/2 1/1 -5/11 0/1 1/1 -4/9 2/1 3/1 -3/7 1/0 -8/19 -2/1 -1/1 -13/31 -3/1 1/0 -5/12 -3/1 -1/1 -17/41 -7/6 -1/1 -29/70 -1/1 -12/29 -1/1 -4/5 -7/17 -1/3 0/1 -2/5 1/1 1/0 -11/28 1/0 -9/23 -5/1 1/0 -7/18 -3/1 -5/13 -1/1 -1/2 -13/34 -1/3 -8/21 0/1 -11/29 0/1 1/1 -3/8 -1/1 1/1 -10/27 -2/1 -1/1 -7/19 0/1 1/1 -4/11 0/1 1/1 -5/14 1/1 -1/3 1/1 1/0 -5/16 -1/1 1/1 -9/29 0/1 1/1 -13/42 1/1 -4/13 1/1 1/0 -3/10 1/1 -2/7 1/0 -5/18 -7/1 -18/65 -19/3 -6/1 -31/112 -6/1 -13/47 -6/1 -17/3 -8/29 -16/3 -5/1 -3/11 -4/1 -3/1 -7/26 -3/1 -11/41 -1/1 1/0 -15/56 1/0 -4/15 -3/1 1/0 -1/4 -3/1 -1/1 -3/13 -3/2 -1/1 -5/22 -5/3 -12/53 -17/11 -3/2 -19/84 -3/2 -7/31 -3/2 -7/5 -2/9 -4/3 -1/1 -3/14 -1/1 -1/5 -1/1 -1/2 -2/11 -1/3 0/1 -5/28 0/1 -3/17 0/1 1/5 -1/6 1/1 -2/13 1/1 1/0 -1/7 1/0 -1/8 -3/1 -1/1 0/1 -1/1 0/1 1/7 -1/2 1/6 -1/3 2/11 0/1 1/1 1/5 -1/1 1/0 3/14 -1/1 2/9 -1/1 -4/5 5/22 -5/7 3/13 -1/1 -3/4 1/4 -1/1 -3/5 4/15 -3/5 -1/2 3/11 -3/5 -4/7 2/7 -1/2 5/17 -5/11 -4/9 8/27 -3/7 -8/19 3/10 -1/3 4/13 -1/2 -1/3 5/16 -1/1 -1/3 1/3 -1/2 -1/3 5/14 -1/3 4/11 -1/3 0/1 7/19 -1/3 0/1 17/46 1/1 10/27 -1/1 -2/3 3/8 -1/1 -1/3 8/21 0/1 5/13 -1/1 1/0 7/18 -3/5 2/5 -1/2 -1/3 7/17 0/1 1/1 12/29 -4/3 -1/1 5/12 -1/1 -3/5 3/7 -1/2 7/16 -3/7 -1/3 11/25 -1/2 -1/3 15/34 -5/11 4/9 -3/7 -2/5 5/11 -1/3 0/1 6/13 -1/2 -1/3 1/2 -1/3 6/11 -1/3 0/1 5/9 -1/3 0/1 4/7 0/1 11/19 0/1 1/1 18/31 -3/1 1/0 7/12 -1/1 -1/3 24/41 -1/2 -1/3 41/70 -1/3 17/29 -1/3 0/1 10/17 -1/3 0/1 3/5 -1/1 -1/2 17/28 -1/2 14/23 -1/2 -5/11 11/18 -1/3 8/13 -1/2 -1/3 21/34 -1/3 13/21 -1/2 18/29 -4/9 -3/7 5/8 -3/7 -1/3 17/27 -2/5 -1/3 12/19 -2/5 -5/13 7/11 -4/11 -1/3 9/14 -1/3 2/3 -1/3 -1/4 11/16 -5/23 -1/5 20/29 -8/39 -1/5 29/42 -1/5 9/13 -1/5 -3/16 7/10 -1/7 5/7 0/1 13/18 1/1 47/65 -1/3 0/1 81/112 0/1 34/47 0/1 1/3 21/29 0/1 1/1 8/11 0/1 1/1 19/26 1/1 30/41 1/1 1/0 41/56 1/0 11/15 -1/1 1/0 3/4 -1/1 -1/3 10/13 -1/2 -1/3 17/22 -1/3 41/53 -3/5 -1/2 65/84 -1/2 24/31 -1/2 -5/11 7/9 -2/5 -1/3 11/14 -1/3 4/5 -1/3 -1/4 9/11 -1/7 0/1 23/28 0/1 14/17 0/1 1/1 5/6 -1/3 11/13 -1/4 -1/5 6/7 0/1 7/8 -1/1 -1/3 1/1 -1/3 0/1 1/0 0/1 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,-4,1) Matrix(85,74,-224,-195) -> Matrix(1,0,0,1) Matrix(139,118,-364,-309) -> Matrix(1,0,-4,1) Matrix(29,24,-168,-139) -> Matrix(1,0,0,1) Matrix(57,46,140,113) -> Matrix(1,0,-4,1) Matrix(111,88,140,111) -> Matrix(3,-2,-10,7) Matrix(197,154,252,197) -> Matrix(3,-4,-8,11) Matrix(281,218,-504,-391) -> Matrix(1,-2,2,-3) Matrix(449,346,728,561) -> Matrix(1,-2,-2,5) Matrix(29,22,112,85) -> Matrix(1,2,-2,-3) Matrix(27,20,112,83) -> Matrix(1,2,-2,-3) Matrix(167,122,-308,-225) -> Matrix(1,0,4,1) Matrix(139,100,-196,-141) -> Matrix(1,0,8,1) Matrix(559,394,952,671) -> Matrix(1,0,-8,1) Matrix(953,670,-1512,-1063) -> Matrix(11,-2,6,-1) Matrix(141,98,364,253) -> Matrix(7,-2,-10,3) Matrix(337,232,-812,-559) -> Matrix(11,-4,-8,3) Matrix(309,212,532,365) -> Matrix(5,-2,-2,1) Matrix(55,36,84,55) -> Matrix(3,-2,-10,7) Matrix(197,126,308,197) -> Matrix(7,-8,-20,23) Matrix(139,88,308,195) -> Matrix(3,-4,-8,11) Matrix(923,582,-3332,-2101) -> Matrix(1,-8,0,1) Matrix(197,124,224,141) -> Matrix(1,-2,-2,5) Matrix(29,18,-224,-139) -> Matrix(1,-4,0,1) Matrix(55,34,-364,-225) -> Matrix(1,0,0,1) Matrix(111,68,364,223) -> Matrix(1,-2,-2,5) Matrix(197,120,-504,-307) -> Matrix(1,-4,0,1) Matrix(27,16,140,83) -> Matrix(1,0,0,1) Matrix(589,346,812,477) -> Matrix(1,0,0,1) Matrix(253,148,-812,-475) -> Matrix(1,0,0,1) Matrix(111,64,-196,-113) -> Matrix(1,0,4,1) Matrix(307,172,448,251) -> Matrix(5,-2,-22,9) Matrix(533,298,-2352,-1315) -> Matrix(13,-8,-8,5) Matrix(113,62,308,169) -> Matrix(1,0,-4,1) Matrix(113,60,-420,-223) -> Matrix(5,-2,-2,1) Matrix(83,38,-308,-141) -> Matrix(1,-4,0,1) Matrix(195,88,308,139) -> Matrix(3,-4,-8,11) Matrix(83,36,-196,-85) -> Matrix(1,-4,0,1) Matrix(195,82,-868,-365) -> Matrix(3,2,-2,-1) Matrix(167,70,532,223) -> Matrix(1,2,-2,-3) Matrix(1819,754,2632,1091) -> Matrix(9,10,-46,-51) Matrix(2241,928,3248,1345) -> Matrix(13,12,-64,-59) Matrix(281,116,952,393) -> Matrix(7,4,-16,-9) Matrix(113,46,140,57) -> Matrix(1,0,-4,1) Matrix(477,188,784,309) -> Matrix(1,-6,-2,13) Matrix(475,186,784,307) -> Matrix(1,6,-2,-11) Matrix(253,98,364,141) -> Matrix(1,2,-6,-11) Matrix(167,64,728,279) -> Matrix(1,2,-2,-3) Matrix(281,106,448,169) -> Matrix(1,-2,-2,5) Matrix(279,104,448,167) -> Matrix(1,-2,-2,5) Matrix(503,186,-1820,-673) -> Matrix(11,6,-2,-1) Matrix(169,62,308,113) -> Matrix(1,0,-4,1) Matrix(111,40,308,111) -> Matrix(1,0,-4,1) Matrix(29,10,84,29) -> Matrix(1,-2,-2,5) Matrix(197,62,448,141) -> Matrix(1,-2,-2,5) Matrix(1903,590,3248,1007) -> Matrix(1,0,-4,1) Matrix(1541,476,2632,813) -> Matrix(1,-2,-2,5) Matrix(223,68,364,111) -> Matrix(1,-2,-2,5) Matrix(55,16,-196,-57) -> Matrix(1,-8,0,1) Matrix(9073,2512,12544,3473) -> Matrix(1,6,6,37) Matrix(9071,2510,12544,3471) -> Matrix(1,6,-6,-35) Matrix(335,92,812,223) -> Matrix(1,4,0,1) Matrix(2297,616,3136,841) -> Matrix(1,0,0,1) Matrix(2295,614,3136,839) -> Matrix(1,4,0,1) Matrix(85,22,112,29) -> Matrix(1,2,-2,-3) Matrix(83,20,112,27) -> Matrix(1,2,-2,-3) Matrix(307,70,364,83) -> Matrix(1,2,-6,-11) Matrix(5461,1236,7056,1597) -> Matrix(21,32,-44,-67) Matrix(5459,1234,7056,1595) -> Matrix(11,16,-20,-29) Matrix(55,12,252,55) -> Matrix(7,8,-8,-9) Matrix(29,6,140,29) -> Matrix(3,2,-2,-1) Matrix(83,16,140,27) -> Matrix(1,0,0,1) Matrix(645,116,784,141) -> Matrix(1,0,4,1) Matrix(643,114,784,139) -> Matrix(1,0,-12,1) Matrix(281,44,364,57) -> Matrix(1,-2,-2,5) Matrix(83,10,224,27) -> Matrix(1,2,-2,-3) Matrix(139,-18,224,-29) -> Matrix(7,4,-16,-9) Matrix(225,-34,364,-55) -> Matrix(1,0,0,1) Matrix(139,-24,168,-29) -> Matrix(1,0,0,1) Matrix(223,-50,504,-113) -> Matrix(13,10,-30,-23) Matrix(141,-38,308,-83) -> Matrix(7,4,-16,-9) Matrix(57,-16,196,-55) -> Matrix(15,8,-32,-17) Matrix(559,-166,1512,-449) -> Matrix(5,2,2,1) Matrix(475,-148,812,-253) -> Matrix(1,0,0,1) Matrix(2409,-890,3332,-1231) -> Matrix(1,0,0,1) Matrix(195,-74,224,-85) -> Matrix(1,0,0,1) Matrix(309,-118,364,-139) -> Matrix(1,0,-4,1) Matrix(307,-120,504,-197) -> Matrix(7,4,-16,-9) Matrix(559,-232,812,-337) -> Matrix(5,4,-24,-19) Matrix(85,-36,196,-83) -> Matrix(7,4,-16,-9) Matrix(1819,-802,2352,-1037) -> Matrix(9,4,-16,-7) Matrix(307,-144,420,-197) -> Matrix(5,2,2,1) Matrix(225,-122,308,-167) -> Matrix(1,0,4,1) Matrix(113,-64,196,-111) -> Matrix(1,0,4,1) Matrix(673,-390,868,-503) -> Matrix(1,-2,-2,5) Matrix(1317,-830,1820,-1147) -> Matrix(5,2,2,1) Matrix(141,-100,196,-139) -> Matrix(1,0,8,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/2 -3/7 -11/28 -3/10 -2/7 -5/28 -1/6 -1/7 0/1 1/7 1/6 1/5 3/14 1/4 2/7 3/10 1/3 5/14 3/8 2/5 5/12 3/7 1/2 41/70 9/14 2/3 29/42 11/14 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -6/7 0/1 -5/6 1/1 -9/11 0/1 1/5 -4/5 1/2 1/1 -3/4 -1/1 1/1 -5/7 0/1 -7/10 1/5 -9/13 3/10 1/3 -2/3 1/2 1/1 -5/8 1/1 3/1 -13/21 1/0 -8/13 1/1 1/0 -11/18 1/1 -3/5 -1/1 1/0 -10/17 0/1 1/1 -7/12 -1/1 1/1 -4/7 0/1 -1/2 1/1 -3/7 1/0 -5/12 -3/1 -1/1 -7/17 -1/3 0/1 -2/5 1/1 1/0 -11/28 1/0 -9/23 -5/1 1/0 -7/18 -3/1 -5/13 -1/1 -1/2 -8/21 0/1 -3/8 -1/1 1/1 -1/3 1/1 1/0 -4/13 1/1 1/0 -3/10 1/1 -2/7 1/0 -1/4 -3/1 -1/1 -1/5 -1/1 -1/2 -2/11 -1/3 0/1 -5/28 0/1 -3/17 0/1 1/5 -1/6 1/1 -1/7 1/0 0/1 -1/1 0/1 1/7 -1/2 1/6 -1/3 2/11 0/1 1/1 1/5 -1/1 1/0 3/14 -1/1 2/9 -1/1 -4/5 1/4 -1/1 -3/5 2/7 -1/2 3/10 -1/3 4/13 -1/2 -1/3 5/16 -1/1 -1/3 1/3 -1/2 -1/3 5/14 -1/3 4/11 -1/3 0/1 3/8 -1/1 -1/3 8/21 0/1 5/13 -1/1 1/0 7/18 -3/5 2/5 -1/2 -1/3 7/17 0/1 1/1 12/29 -4/3 -1/1 5/12 -1/1 -3/5 3/7 -1/2 1/2 -1/3 4/7 0/1 7/12 -1/1 -1/3 24/41 -1/2 -1/3 41/70 -1/3 17/29 -1/3 0/1 10/17 -1/3 0/1 3/5 -1/1 -1/2 17/28 -1/2 14/23 -1/2 -5/11 11/18 -1/3 8/13 -1/2 -1/3 13/21 -1/2 5/8 -3/7 -1/3 7/11 -4/11 -1/3 9/14 -1/3 2/3 -1/3 -1/4 11/16 -5/23 -1/5 20/29 -8/39 -1/5 29/42 -1/5 9/13 -1/5 -3/16 7/10 -1/7 5/7 0/1 3/4 -1/1 -1/3 7/9 -2/5 -1/3 11/14 -1/3 4/5 -1/3 -1/4 9/11 -1/7 0/1 23/28 0/1 14/17 0/1 1/1 5/6 -1/3 6/7 0/1 1/1 -1/3 0/1 1/0 0/1 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(69,61,-112,-99) (-1/1,-6/7) -> (-13/21,-8/13) Hyperbolic Matrix(13,11,-84,-71) (-6/7,-5/6) -> (-1/6,-1/7) 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(43,33,56,43) (-4/5,-3/4) -> (3/4,7/9) Hyperbolic Matrix(15,11,-56,-41) (-3/4,-5/7) -> (-2/7,-1/4) Hyperbolic Matrix(41,29,-140,-99) (-5/7,-7/10) -> (-3/10,-2/7) Hyperbolic Matrix(141,98,364,253) (-7/10,-9/13) -> (5/13,7/18) Hyperbolic Matrix(127,87,308,211) (-9/13,-2/3) -> (7/17,12/29) Hyperbolic Matrix(71,45,112,71) (-2/3,-5/8) -> (5/8,7/11) Hyperbolic Matrix(127,79,-336,-209) (-5/8,-13/21) -> (-8/21,-3/8) 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(97,57,308,181) (-10/17,-7/12) -> (5/16,1/3) Hyperbolic Matrix(71,41,-168,-97) (-7/12,-4/7) -> (-3/7,-5/12) Hyperbolic Matrix(13,7,-28,-15) (-4/7,-1/2) -> (-1/2,-3/7) Parabolic Matrix(211,87,308,127) (-5/12,-7/17) -> (2/3,11/16) 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(13,5,-112,-43) (-5/13,-8/21) -> (-1/7,0/1) Hyperbolic Matrix(41,15,112,41) (-3/8,-1/3) -> (4/11,3/8) Hyperbolic Matrix(181,57,308,97) (-1/3,-4/13) -> (17/29,10/17) Hyperbolic Matrix(223,68,364,111) (-4/13,-3/10) -> (11/18,8/13) Hyperbolic Matrix(13,3,56,13) (-1/4,-1/5) -> (2/9,1/4) 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(43,-5,112,-13) (0/1,1/7) -> (8/21,5/13) Hyperbolic Matrix(71,-11,84,-13) (1/7,1/6) -> (5/6,6/7) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(43,-9,196,-41) (1/5,3/14) -> (3/14,2/9) Parabolic Matrix(41,-11,56,-15) (1/4,2/7) -> (5/7,3/4) Hyperbolic Matrix(99,-29,140,-41) (2/7,3/10) -> (7/10,5/7) Hyperbolic Matrix(475,-148,812,-253) (4/13,5/16) -> (7/12,24/41) Hyperbolic Matrix(71,-25,196,-69) (1/3,5/14) -> (5/14,4/11) Parabolic Matrix(209,-79,336,-127) (3/8,8/21) -> (13/21,5/8) 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(97,-41,168,-71) (5/12,3/7) -> (4/7,7/12) Hyperbolic Matrix(15,-7,28,-13) (3/7,1/2) -> (1/2,4/7) Parabolic Matrix(2871,-1681,4900,-2869) (24/41,41/70) -> (41/70,17/29) Parabolic Matrix(99,-61,112,-69) (8/13,13/21) -> (6/7,1/1) Hyperbolic Matrix(127,-81,196,-125) (7/11,9/14) -> (9/14,2/3) Parabolic Matrix(1219,-841,1764,-1217) (20/29,29/42) -> (29/42,9/13) Parabolic Matrix(155,-121,196,-153) (7/9,11/14) -> (11/14,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(69,61,-112,-99) -> Matrix(2,-1,1,0) Matrix(13,11,-84,-71) -> Matrix(2,-1,1,0) Matrix(29,24,-168,-139) -> Matrix(1,0,0,1) Matrix(57,46,140,113) -> Matrix(1,0,-4,1) Matrix(43,33,56,43) -> Matrix(0,-1,1,2) Matrix(15,11,-56,-41) -> Matrix(2,1,-1,0) Matrix(41,29,-140,-99) -> Matrix(6,-1,1,0) Matrix(141,98,364,253) -> Matrix(7,-2,-10,3) Matrix(127,87,308,211) -> Matrix(2,-1,1,0) Matrix(71,45,112,71) -> Matrix(2,-3,-5,8) Matrix(127,79,-336,-209) -> Matrix(0,1,-1,2) Matrix(111,68,364,223) -> Matrix(1,-2,-2,5) Matrix(197,120,-504,-307) -> Matrix(1,-4,0,1) Matrix(27,16,140,83) -> Matrix(1,0,0,1) Matrix(97,57,308,181) -> Matrix(0,-1,1,2) Matrix(71,41,-168,-97) -> Matrix(2,1,-1,0) Matrix(13,7,-28,-15) -> Matrix(2,-1,1,0) Matrix(211,87,308,127) -> Matrix(2,1,-9,-4) Matrix(113,46,140,57) -> Matrix(1,0,-4,1) Matrix(477,188,784,309) -> Matrix(1,-6,-2,13) Matrix(475,186,784,307) -> Matrix(1,6,-2,-11) Matrix(253,98,364,141) -> Matrix(1,2,-6,-11) Matrix(13,5,-112,-43) -> Matrix(2,1,-1,0) Matrix(41,15,112,41) -> Matrix(0,-1,1,2) Matrix(181,57,308,97) -> Matrix(0,-1,1,2) Matrix(223,68,364,111) -> Matrix(1,-2,-2,5) Matrix(13,3,56,13) -> Matrix(2,3,-3,-4) Matrix(83,16,140,27) -> Matrix(1,0,0,1) Matrix(645,116,784,141) -> Matrix(1,0,4,1) Matrix(643,114,784,139) -> Matrix(1,0,-12,1) Matrix(43,-5,112,-13) -> Matrix(2,1,-1,0) Matrix(71,-11,84,-13) -> Matrix(2,1,-9,-4) Matrix(139,-24,168,-29) -> Matrix(1,0,0,1) Matrix(43,-9,196,-41) -> Matrix(4,5,-5,-6) Matrix(41,-11,56,-15) -> Matrix(2,1,-1,0) Matrix(99,-29,140,-41) -> Matrix(2,1,-17,-8) Matrix(475,-148,812,-253) -> Matrix(1,0,0,1) Matrix(71,-25,196,-69) -> Matrix(2,1,-9,-4) Matrix(209,-79,336,-127) -> Matrix(4,1,-9,-2) Matrix(307,-120,504,-197) -> Matrix(7,4,-16,-9) Matrix(559,-232,812,-337) -> Matrix(5,4,-24,-19) Matrix(97,-41,168,-71) -> Matrix(2,1,-1,0) Matrix(15,-7,28,-13) -> Matrix(2,1,-9,-4) Matrix(2871,-1681,4900,-2869) -> Matrix(2,1,-9,-4) Matrix(99,-61,112,-69) -> Matrix(2,1,-9,-4) Matrix(127,-81,196,-125) -> Matrix(14,5,-45,-16) Matrix(1219,-841,1764,-1217) -> Matrix(54,11,-275,-56) Matrix(155,-121,196,-153) -> Matrix(8,3,-27,-10) 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 0/1 (-1/1,0/1) 0 14 1/7 -1/2 2 2 1/6 -1/3 1 7 2/11 (0/1,1/1) 0 14 1/5 (-1/1,1/0) 0 14 3/14 -1/1 5 1 1/4 (-1/1,-3/5) 0 7 2/7 -1/2 4 2 3/10 -1/3 1 7 4/13 (-1/2,-1/3) 0 14 1/3 (-1/2,-1/3) 0 14 5/14 -1/3 1 1 3/8 (-1/1,-1/3) 0 7 8/21 0/1 2 2 5/13 (-1/1,1/0) 0 14 7/18 -3/5 1 7 2/5 (-1/2,-1/3) 0 14 7/17 (0/1,1/1) 0 14 5/12 (-1/1,-3/5) 0 7 3/7 -1/2 2 2 1/2 -1/3 1 7 4/7 0/1 2 2 7/12 (-1/1,-1/3) 0 7 41/70 -1/3 1 1 17/29 (-1/3,0/1) 0 14 10/17 (-1/3,0/1) 0 14 3/5 (-1/1,-1/2) 0 14 17/28 -1/2 6 1 14/23 (-1/2,-5/11) 0 14 11/18 -1/3 1 7 8/13 (-1/2,-1/3) 0 14 13/21 -1/2 2 2 5/8 (-3/7,-1/3) 0 7 9/14 -1/3 5 1 2/3 (-1/3,-1/4) 0 14 11/16 (-5/23,-1/5) 0 7 29/42 -1/5 11 1 9/13 (-1/5,-3/16) 0 14 7/10 -1/7 1 7 5/7 0/1 4 2 3/4 (-1/1,-1/3) 0 7 11/14 -1/3 3 1 4/5 (-1/3,-1/4) 0 14 9/11 (-1/7,0/1) 0 14 23/28 0/1 8 1 14/17 (0/1,1/1) 0 14 5/6 -1/3 1 7 6/7 0/1 2 2 1/1 (-1/3,0/1) 0 14 1/0 0/1 2 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(43,-5,112,-13) (0/1,1/7) -> (8/21,5/13) Hyperbolic Matrix(71,-11,84,-13) (1/7,1/6) -> (5/6,6/7) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(83,-16,140,-27) (2/11,1/5) -> (10/17,3/5) Glide Reflection Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(13,-3,56,-13) (3/14,1/4) -> (3/14,1/4) Reflection Matrix(41,-11,56,-15) (1/4,2/7) -> (5/7,3/4) Hyperbolic Matrix(99,-29,140,-41) (2/7,3/10) -> (7/10,5/7) Hyperbolic Matrix(223,-68,364,-111) (3/10,4/13) -> (11/18,8/13) Glide Reflection Matrix(181,-57,308,-97) (4/13,1/3) -> (17/29,10/17) Glide Reflection Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(41,-15,112,-41) (5/14,3/8) -> (5/14,3/8) Reflection Matrix(209,-79,336,-127) (3/8,8/21) -> (13/21,5/8) Hyperbolic Matrix(253,-98,364,-141) (5/13,7/18) -> (9/13,7/10) Glide Reflection Matrix(307,-120,504,-197) (7/18,2/5) -> (14/23,11/18) Hyperbolic Matrix(113,-46,140,-57) (2/5,7/17) -> (4/5,9/11) Glide Reflection Matrix(211,-87,308,-127) (7/17,5/12) -> (2/3,11/16) Glide Reflection Matrix(97,-41,168,-71) (5/12,3/7) -> (4/7,7/12) Hyperbolic Matrix(15,-7,28,-13) (3/7,1/2) -> (1/2,4/7) Parabolic Matrix(491,-287,840,-491) (7/12,41/70) -> (7/12,41/70) Reflection Matrix(2379,-1394,4060,-2379) (41/70,17/29) -> (41/70,17/29) Reflection Matrix(169,-102,280,-169) (3/5,17/28) -> (3/5,17/28) Reflection Matrix(783,-476,1288,-783) (17/28,14/23) -> (17/28,14/23) Reflection Matrix(99,-61,112,-69) (8/13,13/21) -> (6/7,1/1) Hyperbolic Matrix(71,-45,112,-71) (5/8,9/14) -> (5/8,9/14) Reflection Matrix(55,-36,84,-55) (9/14,2/3) -> (9/14,2/3) Reflection Matrix(463,-319,672,-463) (11/16,29/42) -> (11/16,29/42) Reflection Matrix(755,-522,1092,-755) (29/42,9/13) -> (29/42,9/13) Reflection Matrix(43,-33,56,-43) (3/4,11/14) -> (3/4,11/14) Reflection Matrix(111,-88,140,-111) (11/14,4/5) -> (11/14,4/5) Reflection Matrix(505,-414,616,-505) (9/11,23/28) -> (9/11,23/28) Reflection Matrix(783,-644,952,-783) (23/28,14/17) -> (23/28,14/17) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(43,-5,112,-13) -> Matrix(2,1,-1,0) -1/1 Matrix(71,-11,84,-13) -> Matrix(2,1,-9,-4) -1/3 Matrix(139,-24,168,-29) -> Matrix(1,0,0,1) Matrix(83,-16,140,-27) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(29,-6,140,-29) -> Matrix(1,2,0,-1) (1/5,3/14) -> (-1/1,1/0) Matrix(13,-3,56,-13) -> Matrix(4,3,-5,-4) (3/14,1/4) -> (-1/1,-3/5) Matrix(41,-11,56,-15) -> Matrix(2,1,-1,0) -1/1 Matrix(99,-29,140,-41) -> Matrix(2,1,-17,-8) Matrix(223,-68,364,-111) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(181,-57,308,-97) -> Matrix(2,1,-3,-2) *** -> (-1/1,-1/3) Matrix(29,-10,84,-29) -> Matrix(5,2,-12,-5) (1/3,5/14) -> (-1/2,-1/3) Matrix(41,-15,112,-41) -> Matrix(2,1,-3,-2) (5/14,3/8) -> (-1/1,-1/3) Matrix(209,-79,336,-127) -> Matrix(4,1,-9,-2) -1/3 Matrix(253,-98,364,-141) -> Matrix(3,2,-16,-11) Matrix(307,-120,504,-197) -> Matrix(7,4,-16,-9) -1/2 Matrix(113,-46,140,-57) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(211,-87,308,-127) -> Matrix(0,1,1,-4) Matrix(97,-41,168,-71) -> Matrix(2,1,-1,0) -1/1 Matrix(15,-7,28,-13) -> Matrix(2,1,-9,-4) -1/3 Matrix(491,-287,840,-491) -> Matrix(2,1,-3,-2) (7/12,41/70) -> (-1/1,-1/3) Matrix(2379,-1394,4060,-2379) -> Matrix(-1,0,6,1) (41/70,17/29) -> (-1/3,0/1) Matrix(169,-102,280,-169) -> Matrix(3,2,-4,-3) (3/5,17/28) -> (-1/1,-1/2) Matrix(783,-476,1288,-783) -> Matrix(21,10,-44,-21) (17/28,14/23) -> (-1/2,-5/11) Matrix(99,-61,112,-69) -> Matrix(2,1,-9,-4) -1/3 Matrix(71,-45,112,-71) -> Matrix(8,3,-21,-8) (5/8,9/14) -> (-3/7,-1/3) Matrix(55,-36,84,-55) -> Matrix(7,2,-24,-7) (9/14,2/3) -> (-1/3,-1/4) Matrix(463,-319,672,-463) -> Matrix(24,5,-115,-24) (11/16,29/42) -> (-5/23,-1/5) Matrix(755,-522,1092,-755) -> Matrix(31,6,-160,-31) (29/42,9/13) -> (-1/5,-3/16) Matrix(43,-33,56,-43) -> Matrix(2,1,-3,-2) (3/4,11/14) -> (-1/1,-1/3) Matrix(111,-88,140,-111) -> Matrix(7,2,-24,-7) (11/14,4/5) -> (-1/3,-1/4) Matrix(505,-414,616,-505) -> Matrix(-1,0,14,1) (9/11,23/28) -> (-1/7,0/1) Matrix(783,-644,952,-783) -> Matrix(1,0,2,-1) (23/28,14/17) -> (0/1,1/1) Matrix(-1,2,0,1) -> Matrix(-1,0,6,1) (1/1,1/0) -> (-1/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.