Properties

Label 9.11e3_881e3.16t1294.1c1
Dimension 9
Group $S_4\wr C_2$
Conductor $ 11^{3} \cdot 881^{3}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$9$
Group:$S_4\wr C_2$
Conductor:$910134926371= 11^{3} \cdot 881^{3} $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 9 x^{6} - 43 x^{5} + 84 x^{4} - 173 x^{3} + 218 x^{2} - 137 x + 52 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 16T1294
Parity: Odd
Determinant: 1.11_881.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 95.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{2} + 138 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 118 + 130\cdot 139 + 98\cdot 139^{2} + 27\cdot 139^{3} + 55\cdot 139^{4} + 78\cdot 139^{5} + 24\cdot 139^{6} + 5\cdot 139^{7} + 26\cdot 139^{8} + 61\cdot 139^{9} + 13\cdot 139^{10} + 62\cdot 139^{11} + 133\cdot 139^{12} + 99\cdot 139^{13} + 100\cdot 139^{14} + 42\cdot 139^{15} + 20\cdot 139^{16} + 113\cdot 139^{17} + 53\cdot 139^{18} + 59\cdot 139^{19} + 111\cdot 139^{20} + 75\cdot 139^{21} + 14\cdot 139^{22} + 132\cdot 139^{23} + 91\cdot 139^{24} + 103\cdot 139^{25} + 46\cdot 139^{26} + 105\cdot 139^{27} + 98\cdot 139^{28} + 33\cdot 139^{29} + 125\cdot 139^{30} + 77\cdot 139^{31} + 18\cdot 139^{32} + 118\cdot 139^{33} + 3\cdot 139^{34} + 30\cdot 139^{35} + 34\cdot 139^{36} + 54\cdot 139^{37} + 28\cdot 139^{38} + 121\cdot 139^{39} + 112\cdot 139^{40} + 83\cdot 139^{41} + 112\cdot 139^{42} + 60\cdot 139^{43} + 93\cdot 139^{44} + 110\cdot 139^{45} + 19\cdot 139^{46} + 84\cdot 139^{47} + 83\cdot 139^{48} + 89\cdot 139^{49} + 109\cdot 139^{50} + 112\cdot 139^{51} + 55\cdot 139^{52} + 111\cdot 139^{53} + 58\cdot 139^{54} + 108\cdot 139^{55} + 90\cdot 139^{56} + 104\cdot 139^{57} + 94\cdot 139^{58} + 50\cdot 139^{59} + 128\cdot 139^{60} + 64\cdot 139^{61} + 98\cdot 139^{62} + 45\cdot 139^{63} + 36\cdot 139^{64} + 30\cdot 139^{65} + 42\cdot 139^{66} + 25\cdot 139^{67} + 21\cdot 139^{68} + 12\cdot 139^{69} + 36\cdot 139^{70} + 22\cdot 139^{71} + 115\cdot 139^{72} + 24\cdot 139^{73} + 62\cdot 139^{74} + 86\cdot 139^{75} + 116\cdot 139^{76} + 90\cdot 139^{77} + 122\cdot 139^{78} + 42\cdot 139^{79} + 72\cdot 139^{80} + 99\cdot 139^{81} + 90\cdot 139^{82} + 22\cdot 139^{83} + 38\cdot 139^{84} + 108\cdot 139^{85} + 72\cdot 139^{86} + 24\cdot 139^{87} + 99\cdot 139^{88} + 67\cdot 139^{89} + 83\cdot 139^{90} + 86\cdot 139^{91} + 8\cdot 139^{92} + 46\cdot 139^{93} + 78\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 2 }$ $=$ $ 136 a + 71 + \left(67 a + 23\right)\cdot 139 + \left(48 a + 81\right)\cdot 139^{2} + \left(7 a + 78\right)\cdot 139^{3} + \left(11 a + 114\right)\cdot 139^{4} + \left(57 a + 113\right)\cdot 139^{5} + \left(8 a + 94\right)\cdot 139^{6} + \left(106 a + 30\right)\cdot 139^{7} + \left(32 a + 45\right)\cdot 139^{8} + \left(55 a + 107\right)\cdot 139^{9} + \left(53 a + 53\right)\cdot 139^{10} + \left(130 a + 38\right)\cdot 139^{11} + \left(91 a + 126\right)\cdot 139^{12} + \left(31 a + 69\right)\cdot 139^{13} + \left(61 a + 72\right)\cdot 139^{14} + \left(18 a + 44\right)\cdot 139^{15} + \left(83 a + 30\right)\cdot 139^{16} + \left(103 a + 52\right)\cdot 139^{17} + \left(12 a + 58\right)\cdot 139^{18} + \left(41 a + 102\right)\cdot 139^{19} + \left(97 a + 89\right)\cdot 139^{20} + \left(91 a + 86\right)\cdot 139^{21} + \left(78 a + 106\right)\cdot 139^{22} + \left(105 a + 133\right)\cdot 139^{23} + \left(134 a + 6\right)\cdot 139^{24} + \left(94 a + 112\right)\cdot 139^{25} + \left(78 a + 85\right)\cdot 139^{26} + \left(107 a + 14\right)\cdot 139^{27} + \left(65 a + 75\right)\cdot 139^{28} + \left(41 a + 76\right)\cdot 139^{29} + \left(42 a + 40\right)\cdot 139^{30} + \left(100 a + 102\right)\cdot 139^{31} + \left(30 a + 105\right)\cdot 139^{32} + \left(54 a + 51\right)\cdot 139^{33} + \left(2 a + 44\right)\cdot 139^{34} + \left(45 a + 41\right)\cdot 139^{35} + \left(130 a + 33\right)\cdot 139^{36} + \left(120 a + 71\right)\cdot 139^{37} + \left(38 a + 136\right)\cdot 139^{38} + \left(29 a + 50\right)\cdot 139^{39} + \left(133 a + 63\right)\cdot 139^{40} + \left(101 a + 59\right)\cdot 139^{41} + \left(56 a + 138\right)\cdot 139^{42} + \left(121 a + 117\right)\cdot 139^{43} + \left(104 a + 116\right)\cdot 139^{44} + \left(128 a + 58\right)\cdot 139^{45} + \left(107 a + 101\right)\cdot 139^{46} + \left(30 a + 61\right)\cdot 139^{47} + \left(120 a + 96\right)\cdot 139^{48} + \left(92 a + 124\right)\cdot 139^{49} + \left(131 a + 40\right)\cdot 139^{50} + \left(93 a + 19\right)\cdot 139^{51} + \left(50 a + 124\right)\cdot 139^{52} + \left(97 a + 32\right)\cdot 139^{53} + \left(28 a + 85\right)\cdot 139^{54} + \left(33 a + 14\right)\cdot 139^{55} + \left(37 a + 54\right)\cdot 139^{56} + \left(110 a + 46\right)\cdot 139^{57} + \left(114 a + 39\right)\cdot 139^{58} + \left(81 a + 75\right)\cdot 139^{59} + \left(8 a + 37\right)\cdot 139^{60} + \left(53 a + 58\right)\cdot 139^{61} + \left(105 a + 116\right)\cdot 139^{62} + \left(98 a + 127\right)\cdot 139^{63} + \left(86 a + 53\right)\cdot 139^{64} + \left(37 a + 121\right)\cdot 139^{65} + \left(59 a + 16\right)\cdot 139^{66} + \left(32 a + 138\right)\cdot 139^{67} + \left(14 a + 126\right)\cdot 139^{68} + \left(91 a + 9\right)\cdot 139^{69} + \left(78 a + 2\right)\cdot 139^{70} + \left(77 a + 128\right)\cdot 139^{71} + \left(7 a + 43\right)\cdot 139^{72} + \left(13 a + 55\right)\cdot 139^{73} + \left(83 a + 69\right)\cdot 139^{74} + \left(99 a + 85\right)\cdot 139^{75} + \left(108 a + 30\right)\cdot 139^{76} + 19 a\cdot 139^{77} + \left(40 a + 105\right)\cdot 139^{78} + \left(92 a + 19\right)\cdot 139^{79} + \left(138 a + 66\right)\cdot 139^{80} + \left(88 a + 134\right)\cdot 139^{81} + \left(34 a + 64\right)\cdot 139^{82} + \left(16 a + 91\right)\cdot 139^{83} + \left(46 a + 96\right)\cdot 139^{84} + \left(131 a + 8\right)\cdot 139^{85} + \left(13 a + 107\right)\cdot 139^{86} + \left(106 a + 74\right)\cdot 139^{87} + \left(50 a + 107\right)\cdot 139^{88} + \left(88 a + 2\right)\cdot 139^{89} + \left(122 a + 11\right)\cdot 139^{90} + \left(66 a + 99\right)\cdot 139^{91} + \left(107 a + 56\right)\cdot 139^{92} + \left(110 a + 101\right)\cdot 139^{93} + \left(64 a + 72\right)\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 3 }$ $=$ $ 41 a + 119 + \left(38 a + 11\right)\cdot 139 + \left(110 a + 101\right)\cdot 139^{2} + \left(64 a + 103\right)\cdot 139^{3} + \left(73 a + 87\right)\cdot 139^{4} + \left(115 a + 50\right)\cdot 139^{5} + \left(69 a + 91\right)\cdot 139^{6} + \left(59 a + 64\right)\cdot 139^{7} + \left(133 a + 93\right)\cdot 139^{8} + \left(29 a + 2\right)\cdot 139^{9} + \left(130 a + 36\right)\cdot 139^{10} + \left(62 a + 26\right)\cdot 139^{11} + \left(138 a + 133\right)\cdot 139^{12} + \left(87 a + 54\right)\cdot 139^{13} + \left(37 a + 7\right)\cdot 139^{14} + \left(78 a + 26\right)\cdot 139^{15} + \left(111 a + 129\right)\cdot 139^{16} + 123 a\cdot 139^{17} + \left(106 a + 65\right)\cdot 139^{18} + \left(5 a + 3\right)\cdot 139^{19} + \left(24 a + 12\right)\cdot 139^{20} + \left(39 a + 117\right)\cdot 139^{21} + \left(52 a + 101\right)\cdot 139^{22} + \left(128 a + 92\right)\cdot 139^{23} + \left(131 a + 115\right)\cdot 139^{24} + \left(133 a + 45\right)\cdot 139^{25} + \left(96 a + 10\right)\cdot 139^{26} + \left(110 a + 103\right)\cdot 139^{27} + \left(105 a + 17\right)\cdot 139^{28} + \left(40 a + 107\right)\cdot 139^{29} + \left(2 a + 47\right)\cdot 139^{30} + \left(108 a + 24\right)\cdot 139^{31} + \left(98 a + 3\right)\cdot 139^{32} + \left(113 a + 68\right)\cdot 139^{33} + \left(16 a + 99\right)\cdot 139^{34} + \left(27 a + 1\right)\cdot 139^{35} + \left(129 a + 12\right)\cdot 139^{36} + \left(100 a + 17\right)\cdot 139^{37} + \left(108 a + 109\right)\cdot 139^{38} + \left(35 a + 59\right)\cdot 139^{39} + \left(120 a + 120\right)\cdot 139^{40} + \left(115 a + 27\right)\cdot 139^{41} + \left(93 a + 34\right)\cdot 139^{42} + \left(130 a + 109\right)\cdot 139^{43} + \left(122 a + 103\right)\cdot 139^{44} + \left(136 a + 130\right)\cdot 139^{45} + \left(128 a + 51\right)\cdot 139^{46} + \left(24 a + 98\right)\cdot 139^{47} + \left(85 a + 106\right)\cdot 139^{48} + \left(65 a + 37\right)\cdot 139^{49} + \left(65 a + 9\right)\cdot 139^{50} + \left(7 a + 98\right)\cdot 139^{51} + \left(75 a + 2\right)\cdot 139^{52} + \left(102 a + 69\right)\cdot 139^{53} + \left(115 a + 81\right)\cdot 139^{54} + \left(112 a + 123\right)\cdot 139^{55} + \left(134 a + 71\right)\cdot 139^{56} + \left(135 a + 55\right)\cdot 139^{57} + \left(49 a + 1\right)\cdot 139^{58} + \left(101 a + 124\right)\cdot 139^{59} + \left(a + 48\right)\cdot 139^{60} + \left(26 a + 46\right)\cdot 139^{61} + \left(36 a + 130\right)\cdot 139^{62} + \left(57 a + 3\right)\cdot 139^{63} + \left(41 a + 99\right)\cdot 139^{64} + \left(17 a + 123\right)\cdot 139^{65} + \left(4 a + 117\right)\cdot 139^{66} + \left(60 a + 55\right)\cdot 139^{67} + \left(116 a + 62\right)\cdot 139^{68} + \left(32 a + 132\right)\cdot 139^{69} + \left(128 a + 25\right)\cdot 139^{70} + \left(115 a + 87\right)\cdot 139^{71} + \left(110 a + 132\right)\cdot 139^{72} + \left(57 a + 37\right)\cdot 139^{73} + \left(69 a + 98\right)\cdot 139^{74} + \left(30 a + 64\right)\cdot 139^{75} + \left(114 a + 131\right)\cdot 139^{76} + \left(78 a + 61\right)\cdot 139^{77} + \left(117 a + 4\right)\cdot 139^{78} + \left(57 a + 123\right)\cdot 139^{79} + \left(29 a + 63\right)\cdot 139^{80} + \left(27 a + 30\right)\cdot 139^{81} + \left(30 a + 30\right)\cdot 139^{82} + \left(73 a + 35\right)\cdot 139^{83} + \left(111 a + 8\right)\cdot 139^{84} + \left(44 a + 121\right)\cdot 139^{85} + \left(119 a + 122\right)\cdot 139^{86} + \left(118 a + 87\right)\cdot 139^{87} + \left(100 a + 137\right)\cdot 139^{88} + \left(44 a + 75\right)\cdot 139^{89} + \left(104 a + 11\right)\cdot 139^{90} + \left(107 a + 66\right)\cdot 139^{91} + \left(107 a + 131\right)\cdot 139^{92} + \left(100 a + 108\right)\cdot 139^{93} + \left(29 a + 124\right)\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 4 }$ $=$ $ 85 a + 62 + \left(3 a + 52\right)\cdot 139 + \left(72 a + 125\right)\cdot 139^{2} + \left(18 a + 102\right)\cdot 139^{3} + \left(18 a + 117\right)\cdot 139^{4} + \left(a + 55\right)\cdot 139^{5} + \left(98 a + 56\right)\cdot 139^{6} + \left(70 a + 45\right)\cdot 139^{7} + \left(86 a + 18\right)\cdot 139^{8} + \left(29 a + 39\right)\cdot 139^{9} + \left(135 a + 137\right)\cdot 139^{10} + \left(2 a + 58\right)\cdot 139^{11} + \left(57 a + 88\right)\cdot 139^{12} + \left(71 a + 134\right)\cdot 139^{13} + \left(27 a + 4\right)\cdot 139^{14} + \left(30 a + 28\right)\cdot 139^{15} + \left(126 a + 39\right)\cdot 139^{16} + \left(120 a + 102\right)\cdot 139^{17} + \left(52 a + 84\right)\cdot 139^{18} + \left(89 a + 51\right)\cdot 139^{19} + \left(118 a + 34\right)\cdot 139^{20} + \left(66 a + 109\right)\cdot 139^{21} + \left(123 a + 20\right)\cdot 139^{22} + \left(33 a + 89\right)\cdot 139^{23} + \left(26 a + 22\right)\cdot 139^{24} + \left(72 a + 90\right)\cdot 139^{25} + \left(56 a + 108\right)\cdot 139^{26} + \left(112 a + 60\right)\cdot 139^{27} + \left(72 a + 106\right)\cdot 139^{28} + \left(133 a + 21\right)\cdot 139^{29} + \left(94 a + 39\right)\cdot 139^{30} + \left(9 a + 134\right)\cdot 139^{31} + \left(76 a + 20\right)\cdot 139^{32} + \left(134 a + 128\right)\cdot 139^{33} + \left(72 a + 71\right)\cdot 139^{34} + \left(105 a + 84\right)\cdot 139^{35} + \left(101 a + 64\right)\cdot 139^{36} + \left(124 a + 108\right)\cdot 139^{37} + \left(118 a + 111\right)\cdot 139^{38} + \left(83 a + 20\right)\cdot 139^{39} + \left(111 a + 33\right)\cdot 139^{40} + \left(130 a + 89\right)\cdot 139^{41} + \left(36 a + 48\right)\cdot 139^{42} + \left(35 a + 114\right)\cdot 139^{43} + \left(30 a + 113\right)\cdot 139^{44} + \left(106 a + 21\right)\cdot 139^{45} + \left(29 a + 116\right)\cdot 139^{46} + \left(43 a + 68\right)\cdot 139^{47} + \left(40 a + 35\right)\cdot 139^{48} + \left(85 a + 137\right)\cdot 139^{49} + \left(117 a + 41\right)\cdot 139^{50} + \left(29 a + 37\right)\cdot 139^{51} + \left(28 a + 120\right)\cdot 139^{52} + \left(6 a + 133\right)\cdot 139^{53} + \left(116 a + 101\right)\cdot 139^{54} + \left(85 a + 102\right)\cdot 139^{55} + \left(80 a + 70\right)\cdot 139^{56} + \left(95 a + 21\right)\cdot 139^{57} + \left(89 a + 15\right)\cdot 139^{58} + \left(106 a + 96\right)\cdot 139^{59} + \left(15 a + 95\right)\cdot 139^{60} + \left(27 a + 135\right)\cdot 139^{61} + \left(92 a + 37\right)\cdot 139^{62} + \left(121 a + 46\right)\cdot 139^{63} + \left(85 a + 3\right)\cdot 139^{64} + \left(113 a + 36\right)\cdot 139^{65} + \left(56 a + 45\right)\cdot 139^{66} + \left(69 a + 108\right)\cdot 139^{67} + \left(28 a + 53\right)\cdot 139^{68} + \left(83 a + 129\right)\cdot 139^{69} + \left(73 a + 45\right)\cdot 139^{70} + \left(74 a + 87\right)\cdot 139^{71} + \left(22 a + 28\right)\cdot 139^{72} + \left(20 a + 7\right)\cdot 139^{73} + \left(119 a + 40\right)\cdot 139^{74} + \left(105 a + 6\right)\cdot 139^{75} + \left(33 a + 54\right)\cdot 139^{76} + \left(101 a + 62\right)\cdot 139^{77} + \left(40 a + 129\right)\cdot 139^{78} + \left(92 a + 52\right)\cdot 139^{79} + \left(46 a + 84\right)\cdot 139^{80} + \left(11 a + 124\right)\cdot 139^{81} + \left(75 a + 23\right)\cdot 139^{82} + \left(102 a + 46\right)\cdot 139^{83} + \left(135 a + 54\right)\cdot 139^{84} + \left(55 a + 93\right)\cdot 139^{85} + \left(42 a + 4\right)\cdot 139^{86} + \left(106 a + 23\right)\cdot 139^{87} + \left(133 a + 128\right)\cdot 139^{88} + \left(71 a + 27\right)\cdot 139^{89} + \left(15 a + 133\right)\cdot 139^{90} + \left(78 a + 101\right)\cdot 139^{91} + \left(69 a + 117\right)\cdot 139^{92} + \left(37 a + 127\right)\cdot 139^{93} + \left(120 a + 100\right)\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 68 + \left(71 a + 94\right)\cdot 139 + \left(90 a + 61\right)\cdot 139^{2} + \left(131 a + 37\right)\cdot 139^{3} + \left(127 a + 118\right)\cdot 139^{4} + \left(81 a + 20\right)\cdot 139^{5} + \left(130 a + 46\right)\cdot 139^{6} + \left(32 a + 128\right)\cdot 139^{7} + \left(106 a + 110\right)\cdot 139^{8} + \left(83 a + 129\right)\cdot 139^{9} + \left(85 a + 51\right)\cdot 139^{10} + \left(8 a + 115\right)\cdot 139^{11} + \left(47 a + 87\right)\cdot 139^{12} + \left(107 a + 9\right)\cdot 139^{13} + \left(77 a + 102\right)\cdot 139^{14} + \left(120 a + 1\right)\cdot 139^{15} + \left(55 a + 95\right)\cdot 139^{16} + \left(35 a + 72\right)\cdot 139^{17} + \left(126 a + 106\right)\cdot 139^{18} + \left(97 a + 130\right)\cdot 139^{19} + \left(41 a + 6\right)\cdot 139^{20} + \left(47 a + 81\right)\cdot 139^{21} + \left(60 a + 93\right)\cdot 139^{22} + \left(33 a + 21\right)\cdot 139^{23} + \left(4 a + 36\right)\cdot 139^{24} + \left(44 a + 72\right)\cdot 139^{25} + \left(60 a + 69\right)\cdot 139^{26} + \left(31 a + 43\right)\cdot 139^{27} + \left(73 a + 33\right)\cdot 139^{28} + \left(97 a + 52\right)\cdot 139^{29} + \left(96 a + 41\right)\cdot 139^{30} + \left(38 a + 21\right)\cdot 139^{31} + \left(108 a + 36\right)\cdot 139^{32} + \left(84 a + 75\right)\cdot 139^{33} + \left(136 a + 131\right)\cdot 139^{34} + \left(93 a + 83\right)\cdot 139^{35} + \left(8 a + 118\right)\cdot 139^{36} + \left(18 a + 61\right)\cdot 139^{37} + \left(100 a + 54\right)\cdot 139^{38} + \left(109 a + 41\right)\cdot 139^{39} + \left(5 a + 28\right)\cdot 139^{40} + \left(37 a + 28\right)\cdot 139^{41} + \left(82 a + 93\right)\cdot 139^{42} + \left(17 a + 43\right)\cdot 139^{43} + \left(34 a + 100\right)\cdot 139^{44} + \left(10 a + 82\right)\cdot 139^{45} + \left(31 a + 80\right)\cdot 139^{46} + \left(108 a + 123\right)\cdot 139^{47} + \left(18 a + 46\right)\cdot 139^{48} + \left(46 a + 97\right)\cdot 139^{49} + \left(7 a + 79\right)\cdot 139^{50} + \left(45 a + 120\right)\cdot 139^{51} + \left(88 a + 80\right)\cdot 139^{52} + \left(41 a + 79\right)\cdot 139^{53} + \left(110 a + 16\right)\cdot 139^{54} + \left(105 a + 19\right)\cdot 139^{55} + \left(101 a + 58\right)\cdot 139^{56} + \left(28 a + 119\right)\cdot 139^{57} + \left(24 a + 43\right)\cdot 139^{58} + \left(57 a + 42\right)\cdot 139^{59} + \left(130 a + 103\right)\cdot 139^{60} + \left(85 a + 102\right)\cdot 139^{61} + \left(33 a + 29\right)\cdot 139^{62} + \left(40 a + 121\right)\cdot 139^{63} + \left(52 a + 41\right)\cdot 139^{64} + \left(101 a + 72\right)\cdot 139^{65} + \left(79 a + 38\right)\cdot 139^{66} + \left(106 a + 111\right)\cdot 139^{67} + \left(124 a + 108\right)\cdot 139^{68} + \left(47 a + 86\right)\cdot 139^{69} + \left(60 a + 128\right)\cdot 139^{70} + \left(61 a + 126\right)\cdot 139^{71} + \left(131 a + 112\right)\cdot 139^{72} + \left(125 a + 60\right)\cdot 139^{73} + 55 a\cdot 139^{74} + \left(39 a + 102\right)\cdot 139^{75} + \left(30 a + 39\right)\cdot 139^{76} + \left(119 a + 50\right)\cdot 139^{77} + \left(98 a + 125\right)\cdot 139^{78} + \left(46 a + 71\right)\cdot 139^{79} + 112\cdot 139^{80} + \left(50 a + 84\right)\cdot 139^{81} + \left(104 a + 10\right)\cdot 139^{82} + \left(122 a + 73\right)\cdot 139^{83} + \left(92 a + 126\right)\cdot 139^{84} + \left(7 a + 93\right)\cdot 139^{85} + \left(125 a + 128\right)\cdot 139^{86} + \left(32 a + 27\right)\cdot 139^{87} + \left(88 a + 52\right)\cdot 139^{88} + \left(50 a + 40\right)\cdot 139^{89} + \left(16 a + 45\right)\cdot 139^{90} + \left(72 a + 43\right)\cdot 139^{91} + \left(31 a + 97\right)\cdot 139^{92} + \left(28 a + 104\right)\cdot 139^{93} + \left(74 a + 26\right)\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 6 }$ $=$ $ 54 a + 8 + \left(135 a + 110\right)\cdot 139 + \left(66 a + 54\right)\cdot 139^{2} + \left(120 a + 49\right)\cdot 139^{3} + \left(120 a + 117\right)\cdot 139^{4} + \left(137 a + 38\right)\cdot 139^{5} + \left(40 a + 14\right)\cdot 139^{6} + \left(68 a + 18\right)\cdot 139^{7} + \left(52 a + 34\right)\cdot 139^{8} + \left(109 a + 121\right)\cdot 139^{9} + \left(3 a + 103\right)\cdot 139^{10} + \left(136 a + 65\right)\cdot 139^{11} + \left(81 a + 3\right)\cdot 139^{12} + \left(67 a + 10\right)\cdot 139^{13} + \left(111 a + 100\right)\cdot 139^{14} + \left(108 a + 30\right)\cdot 139^{15} + \left(12 a + 135\right)\cdot 139^{16} + \left(18 a + 96\right)\cdot 139^{17} + \left(86 a + 16\right)\cdot 139^{18} + \left(49 a + 88\right)\cdot 139^{19} + \left(20 a + 63\right)\cdot 139^{20} + \left(72 a + 57\right)\cdot 139^{21} + \left(15 a + 77\right)\cdot 139^{22} + \left(105 a + 138\right)\cdot 139^{23} + \left(112 a + 14\right)\cdot 139^{24} + \left(66 a + 136\right)\cdot 139^{25} + \left(82 a + 92\right)\cdot 139^{26} + \left(26 a + 116\right)\cdot 139^{27} + \left(66 a + 66\right)\cdot 139^{28} + \left(5 a + 82\right)\cdot 139^{29} + 44 a\cdot 139^{30} + \left(129 a + 49\right)\cdot 139^{31} + \left(62 a + 87\right)\cdot 139^{32} + \left(4 a + 47\right)\cdot 139^{33} + \left(66 a + 10\right)\cdot 139^{34} + \left(33 a + 117\right)\cdot 139^{35} + \left(37 a + 60\right)\cdot 139^{36} + \left(14 a + 131\right)\cdot 139^{37} + \left(20 a + 105\right)\cdot 139^{38} + \left(55 a + 124\right)\cdot 139^{39} + \left(27 a + 60\right)\cdot 139^{40} + \left(8 a + 108\right)\cdot 139^{41} + \left(102 a + 93\right)\cdot 139^{42} + \left(103 a + 112\right)\cdot 139^{43} + \left(108 a + 108\right)\cdot 139^{44} + \left(32 a + 97\right)\cdot 139^{45} + \left(109 a + 39\right)\cdot 139^{46} + \left(95 a + 82\right)\cdot 139^{47} + \left(98 a + 32\right)\cdot 139^{48} + \left(53 a + 43\right)\cdot 139^{49} + \left(21 a + 74\right)\cdot 139^{50} + \left(109 a + 88\right)\cdot 139^{51} + \left(110 a + 118\right)\cdot 139^{52} + \left(132 a + 111\right)\cdot 139^{53} + \left(22 a + 72\right)\cdot 139^{54} + \left(53 a + 72\right)\cdot 139^{55} + \left(58 a + 65\right)\cdot 139^{56} + \left(43 a + 36\right)\cdot 139^{57} + \left(49 a + 9\right)\cdot 139^{58} + \left(32 a + 113\right)\cdot 139^{59} + \left(123 a + 4\right)\cdot 139^{60} + \left(111 a + 8\right)\cdot 139^{61} + \left(46 a + 103\right)\cdot 139^{62} + \left(17 a + 75\right)\cdot 139^{63} + \left(53 a + 106\right)\cdot 139^{64} + \left(25 a + 63\right)\cdot 139^{65} + \left(82 a + 127\right)\cdot 139^{66} + \left(69 a + 120\right)\cdot 139^{67} + \left(110 a + 12\right)\cdot 139^{68} + \left(55 a + 45\right)\cdot 139^{69} + \left(65 a + 36\right)\cdot 139^{70} + \left(64 a + 88\right)\cdot 139^{71} + \left(116 a + 115\right)\cdot 139^{72} + \left(118 a + 4\right)\cdot 139^{73} + 19 a\cdot 139^{74} + \left(33 a + 132\right)\cdot 139^{75} + \left(105 a + 120\right)\cdot 139^{76} + \left(37 a + 129\right)\cdot 139^{77} + \left(98 a + 68\right)\cdot 139^{78} + \left(46 a + 104\right)\cdot 139^{79} + \left(92 a + 38\right)\cdot 139^{80} + \left(127 a + 89\right)\cdot 139^{81} + \left(63 a + 87\right)\cdot 139^{82} + \left(36 a + 73\right)\cdot 139^{83} + \left(3 a + 87\right)\cdot 139^{84} + \left(83 a + 13\right)\cdot 139^{85} + \left(96 a + 130\right)\cdot 139^{86} + \left(32 a + 86\right)\cdot 139^{87} + \left(5 a + 16\right)\cdot 139^{88} + \left(67 a + 105\right)\cdot 139^{89} + \left(123 a + 76\right)\cdot 139^{90} + \left(60 a + 25\right)\cdot 139^{91} + \left(69 a + 109\right)\cdot 139^{92} + \left(101 a + 95\right)\cdot 139^{93} + \left(18 a + 44\right)\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 7 }$ $=$ $ 98 a + 21 + \left(100 a + 9\right)\cdot 139 + \left(28 a + 34\right)\cdot 139^{2} + \left(74 a + 58\right)\cdot 139^{3} + \left(65 a + 96\right)\cdot 139^{4} + \left(23 a + 92\right)\cdot 139^{5} + \left(69 a + 45\right)\cdot 139^{6} + \left(79 a + 54\right)\cdot 139^{7} + \left(5 a + 28\right)\cdot 139^{8} + \left(109 a + 38\right)\cdot 139^{9} + \left(8 a + 136\right)\cdot 139^{10} + \left(76 a + 97\right)\cdot 139^{11} + 69\cdot 139^{12} + \left(51 a + 4\right)\cdot 139^{13} + \left(101 a + 96\right)\cdot 139^{14} + \left(60 a + 66\right)\cdot 139^{15} + \left(27 a + 23\right)\cdot 139^{16} + \left(15 a + 13\right)\cdot 139^{17} + \left(32 a + 48\right)\cdot 139^{18} + \left(133 a + 41\right)\cdot 139^{19} + \left(114 a + 30\right)\cdot 139^{20} + \left(99 a + 132\right)\cdot 139^{21} + \left(86 a + 114\right)\cdot 139^{22} + \left(10 a + 29\right)\cdot 139^{23} + \left(7 a + 119\right)\cdot 139^{24} + \left(5 a + 47\right)\cdot 139^{25} + \left(42 a + 112\right)\cdot 139^{26} + \left(28 a + 116\right)\cdot 139^{27} + \left(33 a + 12\right)\cdot 139^{28} + \left(98 a + 42\right)\cdot 139^{29} + \left(136 a + 9\right)\cdot 139^{30} + \left(30 a + 130\right)\cdot 139^{31} + \left(40 a + 132\right)\cdot 139^{32} + \left(25 a + 82\right)\cdot 139^{33} + \left(122 a + 2\right)\cdot 139^{34} + \left(111 a + 12\right)\cdot 139^{35} + \left(9 a + 114\right)\cdot 139^{36} + \left(38 a + 127\right)\cdot 139^{37} + \left(30 a + 116\right)\cdot 139^{38} + \left(103 a + 125\right)\cdot 139^{39} + \left(18 a + 65\right)\cdot 139^{40} + \left(23 a + 23\right)\cdot 139^{41} + \left(45 a + 12\right)\cdot 139^{42} + \left(8 a + 7\right)\cdot 139^{43} + \left(16 a + 96\right)\cdot 139^{44} + \left(2 a + 5\right)\cdot 139^{45} + \left(10 a + 44\right)\cdot 139^{46} + \left(114 a + 133\right)\cdot 139^{47} + \left(53 a + 27\right)\cdot 139^{48} + \left(73 a + 18\right)\cdot 139^{49} + \left(73 a + 9\right)\cdot 139^{50} + \left(131 a + 40\right)\cdot 139^{51} + \left(63 a + 70\right)\cdot 139^{52} + \left(36 a + 96\right)\cdot 139^{53} + \left(23 a + 94\right)\cdot 139^{54} + \left(26 a + 120\right)\cdot 139^{55} + \left(4 a + 93\right)\cdot 139^{56} + \left(3 a + 56\right)\cdot 139^{57} + \left(89 a + 54\right)\cdot 139^{58} + \left(37 a + 36\right)\cdot 139^{59} + \left(137 a + 88\right)\cdot 139^{60} + \left(112 a + 70\right)\cdot 139^{61} + \left(102 a + 1\right)\cdot 139^{62} + \left(81 a + 25\right)\cdot 139^{63} + \left(97 a + 83\right)\cdot 139^{64} + \left(121 a + 99\right)\cdot 139^{65} + \left(134 a + 104\right)\cdot 139^{66} + \left(78 a + 111\right)\cdot 139^{67} + \left(22 a + 118\right)\cdot 139^{68} + \left(106 a + 48\right)\cdot 139^{69} + \left(10 a + 121\right)\cdot 139^{70} + \left(23 a + 74\right)\cdot 139^{71} + \left(28 a + 127\right)\cdot 139^{72} + \left(81 a + 123\right)\cdot 139^{73} + \left(69 a + 109\right)\cdot 139^{74} + \left(108 a + 25\right)\cdot 139^{75} + \left(24 a + 76\right)\cdot 139^{76} + \left(60 a + 26\right)\cdot 139^{77} + \left(21 a + 43\right)\cdot 139^{78} + \left(81 a + 63\right)\cdot 139^{79} + \left(109 a + 35\right)\cdot 139^{80} + \left(111 a + 28\right)\cdot 139^{81} + \left(108 a + 33\right)\cdot 139^{82} + \left(65 a + 78\right)\cdot 139^{83} + \left(27 a + 46\right)\cdot 139^{84} + \left(94 a + 54\right)\cdot 139^{85} + \left(19 a + 58\right)\cdot 139^{86} + \left(20 a + 87\right)\cdot 139^{87} + \left(38 a + 119\right)\cdot 139^{88} + \left(94 a + 19\right)\cdot 139^{89} + \left(34 a + 71\right)\cdot 139^{90} + \left(31 a + 69\right)\cdot 139^{91} + \left(31 a + 131\right)\cdot 139^{92} + \left(38 a + 101\right)\cdot 139^{93} + \left(109 a + 53\right)\cdot 139^{94} +O\left(139^{ 95 }\right)$
$r_{ 8 }$ $=$ $ 91 + 123\cdot 139 + 137\cdot 139^{2} + 97\cdot 139^{3} + 126\cdot 139^{4} + 104\cdot 139^{5} + 43\cdot 139^{6} + 70\cdot 139^{7} + 60\cdot 139^{8} + 56\cdot 139^{9} + 23\cdot 139^{10} + 91\cdot 139^{11} + 52\cdot 139^{12} + 33\cdot 139^{13} + 72\cdot 139^{14} + 37\cdot 139^{15} + 83\cdot 139^{16} + 104\cdot 139^{17} + 122\cdot 139^{18} + 78\cdot 139^{19} + 68\cdot 139^{20} + 35\cdot 139^{21} + 26\cdot 139^{22} + 57\cdot 139^{23} + 9\cdot 139^{24} + 87\cdot 139^{25} + 29\cdot 139^{26} + 134\cdot 139^{27} + 5\cdot 139^{28} + 139^{29} + 113\cdot 139^{30} + 16\cdot 139^{31} + 12\cdot 139^{32} + 123\cdot 139^{33} + 52\cdot 139^{34} + 46\cdot 139^{35} + 118\cdot 139^{36} + 122\cdot 139^{37} + 31\cdot 139^{38} + 11\cdot 139^{39} + 71\cdot 139^{40} + 135\cdot 139^{41} + 22\cdot 139^{42} + 129\cdot 139^{43} + 100\cdot 139^{44} + 47\cdot 139^{45} + 102\cdot 139^{46} + 42\cdot 139^{47} + 126\cdot 139^{48} + 7\cdot 139^{49} + 52\cdot 139^{50} + 39\cdot 139^{51} + 122\cdot 139^{52} + 59\cdot 139^{53} + 44\cdot 139^{54} + 133\cdot 139^{55} + 50\cdot 139^{56} + 115\cdot 139^{57} + 19\cdot 139^{58} + 18\cdot 139^{59} + 49\cdot 139^{60} + 69\cdot 139^{61} + 38\cdot 139^{62} + 110\cdot 139^{63} + 131\cdot 139^{64} + 8\cdot 139^{65} + 63\cdot 139^{66} + 23\cdot 139^{67} + 51\cdot 139^{68} + 91\cdot 139^{69} + 20\cdot 139^{70} + 80\cdot 139^{71} + 18\cdot 139^{72} + 102\cdot 139^{73} + 36\cdot 139^{74} + 53\cdot 139^{75} + 125\cdot 139^{76} + 133\cdot 139^{77} + 95\cdot 139^{78} + 77\cdot 139^{79} + 82\cdot 139^{80} + 103\cdot 139^{81} + 75\cdot 139^{82} + 135\cdot 139^{83} + 97\cdot 139^{84} + 62\cdot 139^{85} + 70\cdot 139^{86} + 4\cdot 139^{87} + 34\cdot 139^{88} + 77\cdot 139^{89} + 123\cdot 139^{90} + 63\cdot 139^{91} + 42\cdot 139^{92} + 8\cdot 139^{93} + 54\cdot 139^{94} +O\left(139^{ 95 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(2,3,5,7)$
$(1,2)(3,4)(5,6)(7,8)$
$(2,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$9$
$6$$2$$(1,6)(4,8)$$-3$
$9$$2$$(1,6)(2,5)(3,7)(4,8)$$1$
$12$$2$$(2,3)$$3$
$24$$2$$(1,2)(3,4)(5,6)(7,8)$$3$
$36$$2$$(1,4)(2,3)$$1$
$36$$2$$(1,6)(2,3)(4,8)$$-1$
$16$$3$$(2,5,7)$$0$
$64$$3$$(2,5,7)(4,6,8)$$0$
$12$$4$$(1,4,6,8)$$-3$
$36$$4$$(1,4,6,8)(2,3,5,7)$$1$
$36$$4$$(1,6)(2,3,5,7)(4,8)$$1$
$72$$4$$(1,5,6,2)(3,4,7,8)$$-1$
$72$$4$$(1,4,6,8)(2,3)$$-1$
$144$$4$$(1,2,4,3)(5,6)(7,8)$$1$
$48$$6$$(1,6)(2,7,5)(4,8)$$0$
$96$$6$$(2,3)(4,8,6)$$0$
$192$$6$$(1,3)(2,4,5,6,7,8)$$0$
$144$$8$$(1,3,4,5,6,7,8,2)$$-1$
$96$$12$$(1,4,6,8)(2,5,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.