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Properties

Label 6.2e12_5e5_43e4.12t160.2c1
Dimension 6
Group $A_4^2:C_4$
Conductor $ 2^{12} \cdot 5^{5} \cdot 43^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$A_4^2:C_4$
Conductor:$43760652800000= 2^{12} \cdot 5^{5} \cdot 43^{4} $
Artin number field: Splitting field of $f= x^{8} - x^{6} - 10 x^{5} + x^{4} + 19 x^{2} - 10 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 12T160
Parity: Even
Determinant: 1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 84.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{3} + x + 35 $
Roots:
$r_{ 1 }$ $=$ $ 40 + 27\cdot 41 + 30\cdot 41^{2} + 15\cdot 41^{3} + 11\cdot 41^{4} + 29\cdot 41^{6} + 28\cdot 41^{7} + 21\cdot 41^{8} + 8\cdot 41^{9} + 24\cdot 41^{10} + 40\cdot 41^{11} + 29\cdot 41^{12} + 7\cdot 41^{13} + 37\cdot 41^{14} + 21\cdot 41^{15} + 30\cdot 41^{17} + 11\cdot 41^{20} + 25\cdot 41^{21} + 8\cdot 41^{22} + 9\cdot 41^{23} + 34\cdot 41^{24} + 4\cdot 41^{25} + 7\cdot 41^{26} + 36\cdot 41^{27} + 19\cdot 41^{28} + 11\cdot 41^{29} + 41^{30} + 12\cdot 41^{31} + 29\cdot 41^{32} + 32\cdot 41^{33} + 41^{34} + 40\cdot 41^{35} + 13\cdot 41^{36} + 10\cdot 41^{37} + 32\cdot 41^{38} + 8\cdot 41^{39} + 38\cdot 41^{40} + 31\cdot 41^{41} + 34\cdot 41^{42} + 7\cdot 41^{43} + 25\cdot 41^{44} + 14\cdot 41^{45} + 38\cdot 41^{46} + 39\cdot 41^{47} + 31\cdot 41^{48} + 16\cdot 41^{49} + 27\cdot 41^{50} + 15\cdot 41^{51} + 33\cdot 41^{52} + 7\cdot 41^{53} + 32\cdot 41^{54} + 34\cdot 41^{55} + 4\cdot 41^{56} + 24\cdot 41^{57} + 15\cdot 41^{58} + 37\cdot 41^{59} + 35\cdot 41^{60} + 10\cdot 41^{61} + 30\cdot 41^{62} + 16\cdot 41^{63} + 19\cdot 41^{64} + 29\cdot 41^{65} + 13\cdot 41^{66} + 26\cdot 41^{67} + 13\cdot 41^{68} + 31\cdot 41^{69} + 13\cdot 41^{70} + 41^{71} + 38\cdot 41^{72} + 32\cdot 41^{73} + 24\cdot 41^{74} + 38\cdot 41^{75} + 3\cdot 41^{76} + 17\cdot 41^{77} + 5\cdot 41^{79} + 33\cdot 41^{80} + 30\cdot 41^{81} + 36\cdot 41^{82} + 32\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 2 }$ $=$ $ 31 a^{2} + 11 a + 13 + \left(32 a^{2} + 29 a + 17\right)\cdot 41 + \left(35 a^{2} + 8 a + 7\right)\cdot 41^{2} + \left(6 a^{2} + 10 a + 19\right)\cdot 41^{3} + \left(4 a^{2} + 31 a + 23\right)\cdot 41^{4} + \left(40 a^{2} + 30 a + 2\right)\cdot 41^{5} + \left(23 a^{2} + 16 a + 2\right)\cdot 41^{6} + \left(26 a^{2} + 40 a + 20\right)\cdot 41^{7} + \left(9 a + 2\right)\cdot 41^{8} + \left(23 a^{2} + 20 a + 38\right)\cdot 41^{9} + \left(22 a^{2} + 17 a + 25\right)\cdot 41^{10} + \left(16 a^{2} + 33 a + 29\right)\cdot 41^{11} + \left(34 a^{2} + 34 a + 13\right)\cdot 41^{12} + \left(34 a^{2} + 2 a + 14\right)\cdot 41^{13} + \left(34 a^{2} + 20 a + 36\right)\cdot 41^{14} + \left(25 a^{2} + 3 a + 29\right)\cdot 41^{15} + \left(32 a^{2} + 11 a + 32\right)\cdot 41^{16} + \left(32 a^{2} + 3 a + 33\right)\cdot 41^{17} + \left(26 a^{2} + 29 a + 22\right)\cdot 41^{18} + \left(14 a^{2} + 29\right)\cdot 41^{19} + \left(19 a^{2} + 36 a + 40\right)\cdot 41^{20} + \left(18 a^{2} + 24 a + 37\right)\cdot 41^{21} + \left(39 a^{2} + 33 a + 38\right)\cdot 41^{22} + \left(5 a^{2} + 22\right)\cdot 41^{23} + \left(32 a^{2} + 38 a + 20\right)\cdot 41^{24} + \left(15 a^{2} + 24 a + 28\right)\cdot 41^{25} + \left(19 a^{2} + 18 a + 17\right)\cdot 41^{26} + \left(31 a + 22\right)\cdot 41^{27} + \left(2 a^{2} + 8 a + 19\right)\cdot 41^{28} + \left(2 a^{2} + 30 a + 29\right)\cdot 41^{29} + \left(6 a^{2} + 25 a + 23\right)\cdot 41^{30} + \left(6 a^{2} + 20 a + 7\right)\cdot 41^{31} + \left(16 a^{2} + 15 a + 11\right)\cdot 41^{32} + \left(5 a^{2} + 27 a + 15\right)\cdot 41^{33} + \left(9 a^{2} + 2 a\right)\cdot 41^{34} + \left(9 a^{2} + 15 a + 3\right)\cdot 41^{35} + \left(15 a^{2} + 40 a + 13\right)\cdot 41^{36} + \left(33 a^{2} + 23 a + 13\right)\cdot 41^{37} + \left(35 a^{2} + 7 a + 35\right)\cdot 41^{38} + \left(23 a^{2} + 11 a + 30\right)\cdot 41^{39} + \left(4 a^{2} + 8 a + 36\right)\cdot 41^{40} + \left(12 a^{2} + 16 a + 2\right)\cdot 41^{41} + \left(10 a^{2} + 23 a + 7\right)\cdot 41^{42} + \left(12 a^{2} + 37 a + 15\right)\cdot 41^{43} + \left(23 a^{2} + 5 a + 24\right)\cdot 41^{44} + \left(38 a^{2} + 15 a + 2\right)\cdot 41^{45} + \left(23 a^{2} + 19 a + 16\right)\cdot 41^{46} + \left(33 a^{2} + 7 a + 2\right)\cdot 41^{47} + \left(26 a^{2} + 35 a + 29\right)\cdot 41^{48} + \left(24 a^{2} + 23 a + 11\right)\cdot 41^{49} + \left(24 a^{2} + 9 a + 11\right)\cdot 41^{50} + \left(37 a^{2} + 23 a + 17\right)\cdot 41^{51} + \left(28 a^{2} + 39 a + 9\right)\cdot 41^{52} + \left(27 a^{2} + 9 a + 15\right)\cdot 41^{53} + \left(24 a^{2} + 14 a + 2\right)\cdot 41^{54} + \left(11 a^{2} + 36 a + 15\right)\cdot 41^{55} + \left(36 a^{2} + 25 a + 31\right)\cdot 41^{56} + \left(5 a^{2} + 20 a + 8\right)\cdot 41^{57} + \left(29 a^{2} + 17 a + 38\right)\cdot 41^{58} + \left(15 a^{2} + 28 a + 9\right)\cdot 41^{59} + \left(32 a^{2} + 11 a + 16\right)\cdot 41^{60} + \left(31 a^{2} + a + 3\right)\cdot 41^{61} + \left(22 a^{2} + 10 a + 26\right)\cdot 41^{62} + \left(36 a^{2} + 12 a + 38\right)\cdot 41^{63} + \left(12 a^{2} + 21 a + 23\right)\cdot 41^{64} + \left(34 a^{2} + 22 a + 35\right)\cdot 41^{65} + \left(35 a^{2} + 10 a + 40\right)\cdot 41^{66} + \left(6 a^{2} + 19 a + 18\right)\cdot 41^{67} + \left(32 a^{2} + 19 a + 25\right)\cdot 41^{68} + \left(30 a^{2} + 33 a + 4\right)\cdot 41^{69} + \left(11 a^{2} + 11 a + 26\right)\cdot 41^{70} + \left(7 a^{2} + 33 a + 13\right)\cdot 41^{71} + \left(16 a^{2} + 24 a + 39\right)\cdot 41^{72} + \left(22 a^{2} + 9 a + 19\right)\cdot 41^{73} + \left(12 a^{2} + 31 a + 18\right)\cdot 41^{74} + \left(11 a^{2} + 36 a + 13\right)\cdot 41^{75} + \left(30 a^{2} + 36 a + 31\right)\cdot 41^{76} + \left(5 a^{2} + 24 a + 9\right)\cdot 41^{77} + \left(30 a^{2} + 15 a + 22\right)\cdot 41^{78} + \left(a^{2} + 4 a + 24\right)\cdot 41^{79} + \left(23 a^{2} + 29 a + 33\right)\cdot 41^{80} + \left(28 a^{2} + 25 a + 35\right)\cdot 41^{81} + \left(20 a^{2} + 26 a + 39\right)\cdot 41^{82} + \left(32 a^{2} + 4 a + 3\right)\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 3 }$ $=$ $ 32 a^{2} + 15 a + \left(14 a^{2} + 18 a + 19\right)\cdot 41 + \left(35 a^{2} + 34 a + 34\right)\cdot 41^{2} + \left(16 a^{2} + 27 a + 25\right)\cdot 41^{3} + \left(29 a^{2} + 37 a + 26\right)\cdot 41^{4} + \left(11 a^{2} + 30 a + 24\right)\cdot 41^{5} + \left(5 a^{2} + 18 a + 30\right)\cdot 41^{6} + \left(36 a^{2} + 23 a + 12\right)\cdot 41^{7} + \left(27 a^{2} + 7 a + 34\right)\cdot 41^{8} + \left(12 a^{2} + 6 a + 3\right)\cdot 41^{9} + \left(35 a^{2} + 39 a + 7\right)\cdot 41^{10} + \left(11 a^{2} + 21 a + 40\right)\cdot 41^{11} + \left(a^{2} + a + 18\right)\cdot 41^{12} + \left(15 a^{2} + 17 a + 28\right)\cdot 41^{13} + \left(16 a + 40\right)\cdot 41^{14} + \left(9 a^{2} + 17 a + 4\right)\cdot 41^{15} + \left(25 a^{2} + 39 a + 14\right)\cdot 41^{16} + \left(28 a^{2} + 2 a + 17\right)\cdot 41^{17} + \left(14 a^{2} + 30 a + 28\right)\cdot 41^{18} + \left(36 a^{2} + 21 a + 16\right)\cdot 41^{19} + \left(3 a^{2} + 17 a + 30\right)\cdot 41^{20} + \left(33 a^{2} + a + 6\right)\cdot 41^{21} + \left(13 a^{2} + 10 a + 8\right)\cdot 41^{22} + \left(34 a^{2} + 22 a + 28\right)\cdot 41^{23} + \left(18 a^{2} + 19 a + 11\right)\cdot 41^{24} + \left(25 a^{2} + 36 a + 21\right)\cdot 41^{25} + \left(21 a^{2} + 22 a + 5\right)\cdot 41^{26} + \left(23 a^{2} + 35 a + 24\right)\cdot 41^{27} + \left(33 a^{2} + 27 a + 40\right)\cdot 41^{28} + \left(18 a^{2} + 7 a + 26\right)\cdot 41^{29} + \left(13 a^{2} + 17 a + 28\right)\cdot 41^{30} + \left(7 a^{2} + 16 a + 35\right)\cdot 41^{31} + \left(23 a^{2} + 22 a + 15\right)\cdot 41^{32} + \left(19 a^{2} + 35 a + 38\right)\cdot 41^{33} + \left(11 a^{2} + 34 a + 1\right)\cdot 41^{34} + \left(10 a^{2} + 38 a + 31\right)\cdot 41^{35} + \left(22 a^{2} + 11 a + 17\right)\cdot 41^{36} + \left(13 a^{2} + 32 a + 27\right)\cdot 41^{37} + \left(a^{2} + 38 a + 39\right)\cdot 41^{38} + \left(28 a^{2} + 35 a + 19\right)\cdot 41^{39} + \left(26 a + 20\right)\cdot 41^{40} + \left(26 a^{2} + 10 a + 39\right)\cdot 41^{41} + \left(23 a + 27\right)\cdot 41^{42} + \left(17 a^{2} + 31 a + 4\right)\cdot 41^{43} + \left(22 a^{2} + 16 a + 10\right)\cdot 41^{44} + \left(19 a^{2} + 38 a + 17\right)\cdot 41^{45} + \left(12 a^{2} + 17 a + 8\right)\cdot 41^{46} + \left(38 a^{2} + 12 a + 19\right)\cdot 41^{47} + \left(17 a^{2} + 36 a + 9\right)\cdot 41^{48} + \left(21 a^{2} + 24 a + 23\right)\cdot 41^{49} + \left(38 a^{2} + 17 a + 20\right)\cdot 41^{50} + \left(3 a^{2} + 33 a + 8\right)\cdot 41^{51} + \left(7 a^{2} + 27 a + 22\right)\cdot 41^{52} + \left(30 a^{2} + a + 30\right)\cdot 41^{53} + \left(6 a^{2} + 3 a + 17\right)\cdot 41^{54} + \left(18 a^{2} + a + 19\right)\cdot 41^{55} + \left(35 a^{2} + 40 a + 3\right)\cdot 41^{56} + \left(19 a^{2} + 25 a + 18\right)\cdot 41^{57} + \left(13 a^{2} + 21 a\right)\cdot 41^{58} + 27\cdot 41^{59} + \left(16 a^{2} + 3 a + 32\right)\cdot 41^{60} + \left(24 a^{2} + 24 a + 25\right)\cdot 41^{61} + \left(8 a^{2} + 8 a + 16\right)\cdot 41^{62} + \left(21 a^{2} + 26 a + 28\right)\cdot 41^{63} + \left(35 a^{2} + 39 a + 11\right)\cdot 41^{64} + \left(16 a^{2} + 25 a + 10\right)\cdot 41^{65} + \left(18 a^{2} + 13 a + 29\right)\cdot 41^{66} + \left(27 a^{2} + 28 a + 32\right)\cdot 41^{67} + \left(40 a^{2} + 36 a + 3\right)\cdot 41^{68} + \left(22 a^{2} + 9 a + 13\right)\cdot 41^{69} + \left(35 a^{2} + 40 a + 28\right)\cdot 41^{70} + \left(12 a^{2} + 12 a + 3\right)\cdot 41^{71} + \left(36 a^{2} + a + 39\right)\cdot 41^{72} + \left(37 a^{2} + 5 a + 2\right)\cdot 41^{73} + \left(22 a^{2} + 39 a + 39\right)\cdot 41^{74} + \left(18 a^{2} + 28 a + 31\right)\cdot 41^{75} + \left(27 a^{2} + 8 a + 15\right)\cdot 41^{76} + \left(22 a + 6\right)\cdot 41^{77} + \left(6 a^{2} + 39 a + 6\right)\cdot 41^{78} + \left(29 a^{2} + 29 a + 29\right)\cdot 41^{79} + \left(34 a^{2} + 4 a + 27\right)\cdot 41^{80} + \left(20 a^{2} + 20 a + 30\right)\cdot 41^{81} + \left(7 a^{2} + 30 a + 3\right)\cdot 41^{82} + \left(a^{2} + 19 a + 24\right)\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 4 }$ $=$ $ 37 a^{2} + 28 a + 25 + \left(40 a^{2} + 15 a + 31\right)\cdot 41 + \left(3 a^{2} + 36 a + 19\right)\cdot 41^{2} + \left(9 a^{2} + 15 a + 14\right)\cdot 41^{3} + \left(26 a^{2} + 33 a + 27\right)\cdot 41^{4} + \left(a^{2} + a + 14\right)\cdot 41^{5} + \left(30 a^{2} + 40 a + 10\right)\cdot 41^{6} + \left(a^{2} + 39 a + 5\right)\cdot 41^{7} + \left(25 a^{2} + 15 a + 23\right)\cdot 41^{8} + \left(21 a^{2} + 14 a + 11\right)\cdot 41^{9} + \left(39 a^{2} + 22 a + 18\right)\cdot 41^{10} + \left(37 a^{2} + 19 a + 25\right)\cdot 41^{11} + \left(37 a^{2} + 40 a + 1\right)\cdot 41^{12} + \left(25 a^{2} + 40 a + 1\right)\cdot 41^{13} + \left(32 a^{2} + a + 23\right)\cdot 41^{14} + \left(20 a^{2} + 38 a + 6\right)\cdot 41^{15} + \left(34 a^{2} + 19 a + 9\right)\cdot 41^{16} + \left(18 a^{2} + 31 a + 16\right)\cdot 41^{17} + \left(9 a^{2} + 23 a + 33\right)\cdot 41^{18} + \left(32 a^{2} + 23 a + 7\right)\cdot 41^{19} + \left(16 a^{2} + 17 a + 21\right)\cdot 41^{20} + \left(2 a^{2} + 38 a + 20\right)\cdot 41^{21} + \left(9 a^{2} + 30\right)\cdot 41^{22} + \left(35 a^{2} + 21 a + 6\right)\cdot 41^{23} + \left(28 a^{2} + 15 a + 35\right)\cdot 41^{24} + \left(35 a^{2} + 17 a + 35\right)\cdot 41^{25} + \left(10 a^{2} + 26 a + 4\right)\cdot 41^{26} + \left(18 a + 29\right)\cdot 41^{27} + \left(30 a^{2} + 27 a + 40\right)\cdot 41^{28} + \left(8 a^{2} + 36 a + 1\right)\cdot 41^{29} + \left(35 a^{2} + 19 a + 23\right)\cdot 41^{30} + \left(11 a^{2} + 15 a + 17\right)\cdot 41^{31} + \left(19 a^{2} + 32 a + 30\right)\cdot 41^{32} + \left(4 a^{2} + 11 a + 5\right)\cdot 41^{33} + \left(6 a^{2} + 15 a + 17\right)\cdot 41^{34} + 18 a\cdot 41^{35} + \left(18 a^{2} + 5 a + 21\right)\cdot 41^{36} + \left(14 a^{2} + 11 a + 33\right)\cdot 41^{37} + \left(4 a^{2} + 2 a + 5\right)\cdot 41^{38} + \left(36 a^{2} + 34 a + 21\right)\cdot 41^{39} + \left(28 a^{2} + 5 a + 6\right)\cdot 41^{40} + \left(19 a^{2} + 32 a + 16\right)\cdot 41^{41} + \left(24 a^{2} + 3 a + 18\right)\cdot 41^{42} + \left(5 a^{2} + 38 a + 28\right)\cdot 41^{43} + \left(24 a^{2} + 25 a + 7\right)\cdot 41^{44} + \left(38 a^{2} + 23 a + 7\right)\cdot 41^{45} + \left(16 a^{2} + 39 a + 12\right)\cdot 41^{46} + \left(2 a^{2} + 22 a + 29\right)\cdot 41^{47} + \left(2 a^{2} + 31 a + 31\right)\cdot 41^{48} + \left(4 a^{2} + 34 a + 10\right)\cdot 41^{49} + \left(8 a^{2} + 21 a + 37\right)\cdot 41^{50} + \left(27 a^{2} + 4 a + 12\right)\cdot 41^{51} + \left(35 a^{2} + 31 a + 26\right)\cdot 41^{52} + \left(38 a^{2} + 27 a + 9\right)\cdot 41^{53} + \left(38 a^{2} + 17 a + 15\right)\cdot 41^{54} + \left(14 a^{2} + 26 a + 39\right)\cdot 41^{55} + \left(10 a^{2} + 4 a + 18\right)\cdot 41^{56} + \left(38 a^{2} + 37 a + 17\right)\cdot 41^{57} + \left(12 a^{2} + 12 a + 3\right)\cdot 41^{58} + \left(29 a^{2} + 14 a + 7\right)\cdot 41^{59} + \left(10 a^{2} + 10 a + 36\right)\cdot 41^{60} + \left(33 a^{2} + 28 a + 4\right)\cdot 41^{61} + \left(15 a^{2} + 34 a + 14\right)\cdot 41^{62} + \left(22 a^{2} + 3 a + 9\right)\cdot 41^{63} + \left(6 a^{2} + 35 a + 25\right)\cdot 41^{64} + \left(25 a^{2} + 19 a + 20\right)\cdot 41^{65} + \left(24 a^{2} + 8 a + 25\right)\cdot 41^{66} + \left(37 a^{2} + 29 a + 2\right)\cdot 41^{67} + \left(30 a^{2} + 11 a + 16\right)\cdot 41^{68} + \left(39 a^{2} + 25 a + 2\right)\cdot 41^{69} + \left(5 a^{2} + 23 a + 13\right)\cdot 41^{70} + \left(3 a^{2} + 34 a + 15\right)\cdot 41^{71} + \left(12 a^{2} + 19 a + 36\right)\cdot 41^{72} + \left(30 a^{2} + 28 a + 22\right)\cdot 41^{73} + \left(34 a^{2} + 17 a + 28\right)\cdot 41^{74} + \left(10 a^{2} + 37 a + 21\right)\cdot 41^{75} + \left(7 a^{2} + 16 a + 3\right)\cdot 41^{76} + \left(19 a^{2} + 14 a + 7\right)\cdot 41^{77} + \left(18 a^{2} + 36 a + 12\right)\cdot 41^{78} + \left(2 a^{2} + 25 a + 27\right)\cdot 41^{79} + \left(19 a^{2} + 38 a + 1\right)\cdot 41^{80} + \left(35 a^{2} + 31 a + 27\right)\cdot 41^{81} + \left(38 a^{2} + 40 a + 13\right)\cdot 41^{82} + \left(12 a^{2} + 19 a + 11\right)\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 5 }$ $=$ $ 19 a^{2} + 15 a + 5 + \left(34 a^{2} + 34 a + 32\right)\cdot 41 + \left(10 a^{2} + 38 a + 31\right)\cdot 41^{2} + \left(17 a^{2} + 2 a + 39\right)\cdot 41^{3} + \left(7 a^{2} + 13 a + 11\right)\cdot 41^{4} + \left(30 a^{2} + 20 a + 23\right)\cdot 41^{5} + \left(11 a^{2} + 5 a + 7\right)\cdot 41^{6} + \left(19 a^{2} + 18 a + 15\right)\cdot 41^{7} + \left(12 a^{2} + 23 a + 10\right)\cdot 41^{8} + \left(5 a^{2} + 14 a + 26\right)\cdot 41^{9} + \left(24 a^{2} + 25 a + 40\right)\cdot 41^{10} + \left(12 a^{2} + 26 a + 26\right)\cdot 41^{11} + \left(5 a^{2} + 4 a + 21\right)\cdot 41^{12} + \left(32 a^{2} + 21 a + 12\right)\cdot 41^{13} + \left(5 a^{2} + 4 a + 3\right)\cdot 41^{14} + \left(6 a^{2} + 20 a + 3\right)\cdot 41^{15} + \left(24 a^{2} + 31 a + 27\right)\cdot 41^{16} + \left(20 a^{2} + 34 a + 25\right)\cdot 41^{17} + \left(40 a^{2} + 22 a + 4\right)\cdot 41^{18} + \left(30 a^{2} + 18 a + 13\right)\cdot 41^{19} + \left(17 a^{2} + 28 a + 12\right)\cdot 41^{20} + \left(30 a^{2} + 14 a + 32\right)\cdot 41^{21} + \left(28 a^{2} + 38 a + 31\right)\cdot 41^{22} + \left(17 a + 5\right)\cdot 41^{23} + \left(31 a^{2} + 24 a + 6\right)\cdot 41^{24} + \left(40 a^{2} + 20 a + 4\right)\cdot 41^{25} + \left(40 a^{2} + 40 a + 32\right)\cdot 41^{26} + \left(16 a^{2} + 14 a + 19\right)\cdot 41^{27} + \left(5 a^{2} + 4 a + 35\right)\cdot 41^{28} + \left(20 a^{2} + 3 a + 27\right)\cdot 41^{29} + \left(21 a^{2} + 39 a + 6\right)\cdot 41^{30} + \left(27 a^{2} + 3 a + 8\right)\cdot 41^{31} + \left(a^{2} + 3 a + 15\right)\cdot 41^{32} + \left(16 a^{2} + 19 a + 22\right)\cdot 41^{33} + \left(20 a^{2} + 3 a + 21\right)\cdot 41^{34} + \left(21 a^{2} + 28 a + 38\right)\cdot 41^{35} + \left(3 a^{2} + 29 a + 18\right)\cdot 41^{36} + \left(35 a^{2} + 25 a + 14\right)\cdot 41^{37} + \left(3 a^{2} + 35 a\right)\cdot 41^{38} + \left(30 a^{2} + 34 a + 35\right)\cdot 41^{39} + \left(35 a^{2} + 5 a + 2\right)\cdot 41^{40} + \left(2 a^{2} + 14 a + 24\right)\cdot 41^{41} + \left(30 a^{2} + 35 a + 6\right)\cdot 41^{42} + \left(11 a^{2} + 12 a + 1\right)\cdot 41^{43} + \left(36 a^{2} + 18 a + 33\right)\cdot 41^{44} + \left(23 a^{2} + 28 a + 33\right)\cdot 41^{45} + \left(4 a^{2} + 3 a + 16\right)\cdot 41^{46} + \left(10 a^{2} + 21 a\right)\cdot 41^{47} + \left(37 a^{2} + 10 a + 36\right)\cdot 41^{48} + \left(35 a^{2} + 33 a + 32\right)\cdot 41^{49} + \left(18 a^{2} + 13 a + 34\right)\cdot 41^{50} + \left(40 a^{2} + 25 a + 32\right)\cdot 41^{51} + \left(4 a^{2} + 14 a + 20\right)\cdot 41^{52} + \left(24 a^{2} + 29 a + 26\right)\cdot 41^{53} + \left(9 a^{2} + 23 a + 19\right)\cdot 41^{54} + \left(11 a^{2} + 3 a + 28\right)\cdot 41^{55} + \left(10 a^{2} + 16 a + 27\right)\cdot 41^{56} + \left(15 a^{2} + 35 a + 28\right)\cdot 41^{57} + \left(39 a^{2} + a + 17\right)\cdot 41^{58} + \left(24 a^{2} + 12 a + 2\right)\cdot 41^{59} + \left(33 a^{2} + 26 a + 17\right)\cdot 41^{60} + \left(25 a^{2} + 15 a + 40\right)\cdot 41^{61} + \left(9 a^{2} + 22 a + 30\right)\cdot 41^{62} + \left(24 a^{2} + 2 a + 16\right)\cdot 41^{63} + \left(33 a^{2} + 21 a + 10\right)\cdot 41^{64} + \left(30 a^{2} + 33 a + 33\right)\cdot 41^{65} + \left(27 a^{2} + 16 a + 21\right)\cdot 41^{66} + \left(6 a^{2} + 34 a + 32\right)\cdot 41^{67} + \left(9 a^{2} + 25 a + 23\right)\cdot 41^{68} + \left(28 a^{2} + 38 a + 16\right)\cdot 41^{69} + \left(34 a^{2} + 29 a\right)\cdot 41^{70} + \left(20 a^{2} + 35 a + 9\right)\cdot 41^{71} + \left(29 a^{2} + 14 a + 7\right)\cdot 41^{72} + \left(21 a^{2} + 26 a + 33\right)\cdot 41^{73} + \left(5 a^{2} + 11 a + 13\right)\cdot 41^{74} + \left(11 a^{2} + 16 a + 13\right)\cdot 41^{75} + \left(24 a^{2} + 36 a + 27\right)\cdot 41^{76} + \left(34 a^{2} + 34 a + 1\right)\cdot 41^{77} + \left(4 a^{2} + 26 a + 19\right)\cdot 41^{78} + \left(10 a^{2} + 6 a + 16\right)\cdot 41^{79} + \left(24 a^{2} + 7 a + 34\right)\cdot 41^{80} + \left(32 a^{2} + 36 a + 24\right)\cdot 41^{81} + \left(12 a^{2} + 24 a + 34\right)\cdot 41^{82} + \left(7 a^{2} + 16 a\right)\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 6 }$ $=$ $ 6 a^{2} + 29 a + 18 + \left(19 a^{2} + 9 a + 3\right)\cdot 41 + \left(3 a^{2} + 30 a + 33\right)\cdot 41^{2} + \left(12 a^{2} + 37 a + 2\right)\cdot 41^{3} + \left(35 a^{2} + 29 a + 6\right)\cdot 41^{4} + \left(39 a^{2} + 17 a + 40\right)\cdot 41^{5} + \left(17 a^{2} + 18 a + 15\right)\cdot 41^{6} + \left(11 a^{2} + 10 a + 25\right)\cdot 41^{7} + \left(30 a^{2} + 12 a + 26\right)\cdot 41^{8} + \left(6 a^{2} + a + 1\right)\cdot 41^{9} + \left(19 a^{2} + 25 a + 32\right)\cdot 41^{10} + \left(30 a^{2} + 19 a + 6\right)\cdot 41^{11} + \left(6 a^{2} + 8\right)\cdot 41^{12} + \left(36 a^{2} + 11 a + 35\right)\cdot 41^{13} + \left(40 a^{2} + 8 a + 14\right)\cdot 41^{14} + \left(23 a^{2} + 16 a + 22\right)\cdot 41^{15} + \left(30 a^{2} + a + 6\right)\cdot 41^{16} + \left(37 a^{2} + 15\right)\cdot 41^{17} + \left(8 a^{2} + 8 a + 19\right)\cdot 41^{18} + \left(25 a^{2} + 14 a + 30\right)\cdot 41^{19} + \left(39 a^{2} + 23 a + 22\right)\cdot 41^{20} + \left(29 a^{2} + 25 a + 11\right)\cdot 41^{21} + \left(9 a^{2} + 20 a + 17\right)\cdot 41^{22} + \left(16 a^{2} + 27 a + 21\right)\cdot 41^{23} + \left(7 a^{2} + 26 a + 34\right)\cdot 41^{24} + \left(6 a^{2} + 18 a + 29\right)\cdot 41^{25} + \left(5 a^{2} + 10 a + 14\right)\cdot 41^{26} + \left(14 a^{2} + 20 a + 38\right)\cdot 41^{27} + \left(21 a + 20\right)\cdot 41^{28} + \left(36 a^{2} + 35 a + 6\right)\cdot 41^{29} + \left(27 a^{2} + 36 a + 18\right)\cdot 41^{30} + \left(6 a^{2} + 38 a\right)\cdot 41^{31} + \left(32 a^{2} + 29 a + 39\right)\cdot 41^{32} + \left(10 a^{2} + 22 a + 9\right)\cdot 41^{33} + \left(6 a^{2} + 28 a + 17\right)\cdot 41^{34} + \left(16 a^{2} + 40 a + 38\right)\cdot 41^{35} + \left(35 a^{2} + 38 a + 18\right)\cdot 41^{36} + \left(5 a^{2} + 19 a\right)\cdot 41^{37} + \left(31 a^{2} + 30 a + 10\right)\cdot 41^{38} + \left(39 a^{2} + 25 a + 37\right)\cdot 41^{39} + \left(8 a^{2} + 27 a + 6\right)\cdot 41^{40} + \left(4 a^{2} + 15 a + 33\right)\cdot 41^{41} + \left(25 a^{2} + 6 a + 18\right)\cdot 41^{42} + \left(36 a^{2} + 35\right)\cdot 41^{43} + \left(24 a^{2} + 6 a + 21\right)\cdot 41^{44} + \left(8 a^{2} + 7 a + 14\right)\cdot 41^{45} + \left(12 a^{2} + 8 a + 36\right)\cdot 41^{46} + \left(31 a^{2} + 17 a + 34\right)\cdot 41^{47} + \left(3 a^{2} + 4 a + 32\right)\cdot 41^{48} + \left(35 a^{2} + 24 a + 17\right)\cdot 41^{49} + \left(24 a^{2} + 35 a + 7\right)\cdot 41^{50} + \left(16 a^{2} + 16 a + 33\right)\cdot 41^{51} + \left(38 a^{2} + 24 a\right)\cdot 41^{52} + \left(15 a^{2} + 33 a + 8\right)\cdot 41^{53} + \left(34 a^{2} + 34 a + 12\right)\cdot 41^{54} + \left(14 a^{2} + 25 a + 39\right)\cdot 41^{55} + \left(7 a^{2} + 32 a + 16\right)\cdot 41^{56} + \left(4 a^{2} + 12 a + 8\right)\cdot 41^{57} + \left(15 a^{2} + 10 a + 32\right)\cdot 41^{58} + \left(a^{2} + 6 a + 15\right)\cdot 41^{59} + \left(25 a^{2} + 19 a + 18\right)\cdot 41^{60} + \left(26 a^{2} + 9 a\right)\cdot 41^{61} + \left(31 a^{2} + 12 a + 11\right)\cdot 41^{62} + \left(29 a^{2} + 23 a + 14\right)\cdot 41^{63} + \left(15 a^{2} + 7 a + 31\right)\cdot 41^{64} + \left(29 a^{2} + 40 a + 9\right)\cdot 41^{65} + \left(37 a^{2} + 18 a + 34\right)\cdot 41^{66} + \left(20 a^{2} + 5 a + 18\right)\cdot 41^{67} + \left(35 a^{2} + 32 a + 5\right)\cdot 41^{68} + \left(2 a^{2} + 7 a + 5\right)\cdot 41^{69} + \left(a^{2} + 11 a + 37\right)\cdot 41^{70} + \left(31 a^{2} + 38 a + 33\right)\cdot 41^{71} + \left(a^{2} + 8 a + 15\right)\cdot 41^{72} + \left(4 a^{2} + 18 a + 5\right)\cdot 41^{73} + \left(27 a^{2} + 36 a + 37\right)\cdot 41^{74} + \left(30 a^{2} + 26 a + 34\right)\cdot 41^{75} + \left(18 a^{2} + a + 24\right)\cdot 41^{76} + \left(17 a^{2} + 31 a + 19\right)\cdot 41^{77} + \left(38 a^{2} + 3 a + 25\right)\cdot 41^{78} + \left(33 a^{2} + 7 a + 34\right)\cdot 41^{79} + \left(23 a^{2} + 36 a + 4\right)\cdot 41^{80} + \left(39 a^{2} + 26 a + 16\right)\cdot 41^{81} + \left(13 a^{2} + 3 a + 24\right)\cdot 41^{82} + \left(25 a^{2} + 2 a + 19\right)\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 7 }$ $=$ $ 23 + 13\cdot 41 + 8\cdot 41^{2} + 38\cdot 41^{3} + 19\cdot 41^{4} + 31\cdot 41^{5} + 34\cdot 41^{7} + 34\cdot 41^{8} + 13\cdot 41^{9} + 8\cdot 41^{10} + 26\cdot 41^{11} + 27\cdot 41^{12} + 26\cdot 41^{13} + 41^{14} + 3\cdot 41^{15} + 8\cdot 41^{16} + 5\cdot 41^{17} + 26\cdot 41^{18} + 22\cdot 41^{19} + 39\cdot 41^{20} + 4\cdot 41^{21} + 3\cdot 41^{22} + 25\cdot 41^{23} + 2\cdot 41^{24} + 28\cdot 41^{25} + 26\cdot 41^{26} + 15\cdot 41^{27} + 27\cdot 41^{28} + 38\cdot 41^{29} + 22\cdot 41^{30} + 30\cdot 41^{31} + 39\cdot 41^{32} + 5\cdot 41^{33} + 17\cdot 41^{34} + 9\cdot 41^{35} + 32\cdot 41^{36} + 26\cdot 41^{37} + 6\cdot 41^{38} + 37\cdot 41^{39} + 21\cdot 41^{40} + 15\cdot 41^{41} + 40\cdot 41^{42} + 19\cdot 41^{43} + 14\cdot 41^{44} + 28\cdot 41^{45} + 40\cdot 41^{46} + 18\cdot 41^{47} + 7\cdot 41^{48} + 14\cdot 41^{49} + 15\cdot 41^{50} + 23\cdot 41^{51} + 29\cdot 41^{52} + 9\cdot 41^{53} + 41^{54} + 19\cdot 41^{55} + 19\cdot 41^{56} + 26\cdot 41^{57} + 25\cdot 41^{58} + 41^{59} + 16\cdot 41^{60} + 12\cdot 41^{61} + 8\cdot 41^{62} + 39\cdot 41^{63} + 35\cdot 41^{64} + 2\cdot 41^{65} + 31\cdot 41^{66} + 38\cdot 41^{67} + 28\cdot 41^{68} + 6\cdot 41^{69} + 27\cdot 41^{70} + 14\cdot 41^{71} + 37\cdot 41^{72} + 25\cdot 41^{73} + 10\cdot 41^{74} + 23\cdot 41^{75} + 7\cdot 41^{76} + 23\cdot 41^{77} + 34\cdot 41^{78} + 11\cdot 41^{79} + 27\cdot 41^{80} + 31\cdot 41^{81} + 3\cdot 41^{82} + 12\cdot 41^{83} +O\left(41^{ 84 }\right)$
$r_{ 8 }$ $=$ $ 39 a^{2} + 25 a + 40 + \left(21 a^{2} + 15 a + 18\right)\cdot 41 + \left(33 a^{2} + 15 a + 39\right)\cdot 41^{2} + \left(19 a^{2} + 28 a + 7\right)\cdot 41^{3} + \left(20 a^{2} + 18 a + 37\right)\cdot 41^{4} + \left(40 a^{2} + 21 a + 26\right)\cdot 41^{5} + \left(33 a^{2} + 23 a + 26\right)\cdot 41^{6} + \left(27 a^{2} + 31 a + 22\right)\cdot 41^{7} + \left(26 a^{2} + 12 a + 10\right)\cdot 41^{8} + \left(12 a^{2} + 25 a + 19\right)\cdot 41^{9} + \left(23 a^{2} + 34 a + 7\right)\cdot 41^{10} + \left(13 a^{2} + a + 9\right)\cdot 41^{11} + \left(37 a^{2} + 1\right)\cdot 41^{12} + \left(19 a^{2} + 30 a + 38\right)\cdot 41^{13} + \left(8 a^{2} + 30 a + 6\right)\cdot 41^{14} + \left(37 a^{2} + 27 a + 31\right)\cdot 41^{15} + \left(16 a^{2} + 19 a + 24\right)\cdot 41^{16} + \left(25 a^{2} + 9 a + 20\right)\cdot 41^{17} + \left(22 a^{2} + 9 a + 28\right)\cdot 41^{18} + \left(24 a^{2} + 3 a + 2\right)\cdot 41^{19} + \left(25 a^{2} + 27\right)\cdot 41^{20} + \left(8 a^{2} + 18 a + 24\right)\cdot 41^{21} + \left(22 a^{2} + 19 a + 25\right)\cdot 41^{22} + \left(30 a^{2} + 33 a + 3\right)\cdot 41^{23} + \left(4 a^{2} + 39 a + 19\right)\cdot 41^{24} + \left(40 a^{2} + 4 a + 11\right)\cdot 41^{25} + \left(24 a^{2} + 4 a + 14\right)\cdot 41^{26} + \left(26 a^{2} + 2 a + 19\right)\cdot 41^{27} + \left(10 a^{2} + 33 a\right)\cdot 41^{28} + \left(37 a^{2} + 9 a + 21\right)\cdot 41^{29} + \left(18 a^{2} + 25 a + 39\right)\cdot 41^{30} + \left(22 a^{2} + 27 a + 10\right)\cdot 41^{31} + \left(30 a^{2} + 19 a + 24\right)\cdot 41^{32} + \left(25 a^{2} + 6 a + 33\right)\cdot 41^{33} + \left(28 a^{2} + 38 a + 4\right)\cdot 41^{34} + \left(24 a^{2} + 22 a + 3\right)\cdot 41^{35} + \left(28 a^{2} + 37 a + 28\right)\cdot 41^{36} + \left(20 a^{2} + 9 a + 37\right)\cdot 41^{37} + \left(5 a^{2} + 8 a + 33\right)\cdot 41^{38} + \left(6 a^{2} + 22 a + 14\right)\cdot 41^{39} + \left(3 a^{2} + 7 a + 30\right)\cdot 41^{40} + \left(17 a^{2} + 34 a\right)\cdot 41^{41} + \left(32 a^{2} + 30 a + 10\right)\cdot 41^{42} + \left(39 a^{2} + 2 a + 10\right)\cdot 41^{43} + \left(32 a^{2} + 9 a + 27\right)\cdot 41^{44} + \left(34 a^{2} + 10 a + 4\right)\cdot 41^{45} + \left(11 a^{2} + 34 a + 36\right)\cdot 41^{46} + \left(7 a^{2} + 18\right)\cdot 41^{47} + \left(35 a^{2} + 5 a + 26\right)\cdot 41^{48} + \left(a^{2} + 23 a + 36\right)\cdot 41^{49} + \left(8 a^{2} + 24 a + 9\right)\cdot 41^{50} + \left(38 a^{2} + 19 a + 20\right)\cdot 41^{51} + \left(7 a^{2} + 26 a + 21\right)\cdot 41^{52} + \left(27 a^{2} + 20 a + 15\right)\cdot 41^{53} + \left(8 a^{2} + 29 a + 22\right)\cdot 41^{54} + \left(11 a^{2} + 29 a + 9\right)\cdot 41^{55} + \left(23 a^{2} + 3 a\right)\cdot 41^{56} + \left(39 a^{2} + 32 a + 32\right)\cdot 41^{57} + \left(12 a^{2} + 17 a + 30\right)\cdot 41^{58} + \left(10 a^{2} + 20 a + 21\right)\cdot 41^{59} + \left(5 a^{2} + 11 a + 32\right)\cdot 41^{60} + \left(22 a^{2} + 3 a + 24\right)\cdot 41^{61} + \left(34 a^{2} + 35 a + 26\right)\cdot 41^{62} + \left(29 a^{2} + 13 a\right)\cdot 41^{63} + \left(18 a^{2} + 39 a + 6\right)\cdot 41^{64} + \left(27 a^{2} + 21 a + 22\right)\cdot 41^{65} + \left(19 a^{2} + 13 a + 8\right)\cdot 41^{66} + \left(23 a^{2} + 6 a + 34\right)\cdot 41^{67} + \left(15 a^{2} + 38 a + 5\right)\cdot 41^{68} + \left(39 a^{2} + 7 a + 2\right)\cdot 41^{69} + \left(33 a^{2} + 6 a + 18\right)\cdot 41^{70} + \left(6 a^{2} + 9 a + 31\right)\cdot 41^{71} + \left(27 a^{2} + 12 a + 32\right)\cdot 41^{72} + \left(6 a^{2} + 35 a + 20\right)\cdot 41^{73} + \left(20 a^{2} + 27 a + 32\right)\cdot 41^{74} + \left(40 a^{2} + 17 a + 27\right)\cdot 41^{75} + \left(14 a^{2} + 22 a + 8\right)\cdot 41^{76} + \left(4 a^{2} + 36 a + 38\right)\cdot 41^{77} + \left(25 a^{2} + 2\right)\cdot 41^{78} + \left(4 a^{2} + 8 a + 15\right)\cdot 41^{79} + \left(39 a^{2} + 7 a + 1\right)\cdot 41^{80} + \left(6 a^{2} + 23 a + 8\right)\cdot 41^{81} + \left(29 a^{2} + 37 a + 7\right)\cdot 41^{82} + \left(2 a^{2} + 18 a + 18\right)\cdot 41^{83} +O\left(41^{ 84 }\right)$

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

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

Character values on conjugacy classes

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