Properties

 Label 9.746636341248.12t165.a.a Dimension $9$ Group $(A_4\wr C_2):C_2$ Conductor $746636341248$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $9$ Group: $(A_4\wr C_2):C_2$ Conductor: $$746636341248$$$$\medspace = 2^{12} \cdot 3^{12} \cdot 7^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.4.3186376704.2 Galois orbit size: $1$ Smallest permutation container: 12T165 Parity: even Determinant: 1.28.2t1.a.a Projective image: $(A_4\wr C_2):C_2$ Projective field: Galois closure of 8.4.3186376704.2

Defining polynomial

 $f(x)$ $=$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 75.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{3} + 2 x + 27$

Roots:
 $r_{ 1 }$ $=$ $26 a^{2} + 19 a + 22 + \left(5 a^{2} + 7 a + 2\right)\cdot 29 + \left(15 a^{2} + 19 a + 16\right)\cdot 29^{2} + \left(14 a^{2} + 23 a + 1\right)\cdot 29^{3} + \left(9 a + 15\right)\cdot 29^{4} + \left(22 a^{2} + a + 24\right)\cdot 29^{5} + \left(14 a^{2} + 16 a + 7\right)\cdot 29^{6} + \left(24 a^{2} + 13 a + 11\right)\cdot 29^{7} + \left(10 a^{2} + 21 a + 22\right)\cdot 29^{8} + \left(23 a^{2} + 15 a + 17\right)\cdot 29^{9} + \left(4 a^{2} + 27 a + 8\right)\cdot 29^{10} + \left(12 a^{2} + 18 a + 8\right)\cdot 29^{11} + \left(2 a^{2} + 13 a + 13\right)\cdot 29^{12} + \left(5 a^{2} + 19 a + 15\right)\cdot 29^{13} + \left(13 a^{2} + 7 a + 2\right)\cdot 29^{14} + \left(23 a^{2} + 5 a + 2\right)\cdot 29^{15} + \left(8 a^{2} + 16 a + 26\right)\cdot 29^{16} + \left(15 a^{2} + 4 a + 10\right)\cdot 29^{17} + \left(7 a^{2} + 2 a + 24\right)\cdot 29^{18} + \left(21 a^{2} + 2 a + 21\right)\cdot 29^{19} + \left(24 a^{2} + 11 a + 4\right)\cdot 29^{20} + \left(25 a^{2} + 5 a + 22\right)\cdot 29^{21} + \left(19 a^{2} + 5 a + 26\right)\cdot 29^{22} + \left(18 a^{2} + 5 a + 5\right)\cdot 29^{23} + \left(13 a^{2} + 22 a + 1\right)\cdot 29^{24} + \left(2 a^{2} + 19 a + 7\right)\cdot 29^{25} + \left(9 a^{2} + 16 a + 10\right)\cdot 29^{26} + \left(12 a^{2} + 16 a + 10\right)\cdot 29^{27} + \left(26 a^{2} + 7 a + 2\right)\cdot 29^{28} + \left(15 a^{2} + 15 a + 9\right)\cdot 29^{29} + \left(14 a^{2} + 25 a + 25\right)\cdot 29^{30} + \left(28 a^{2} + 25 a + 9\right)\cdot 29^{31} + \left(24 a^{2} + 16 a + 26\right)\cdot 29^{32} + \left(15 a + 8\right)\cdot 29^{33} + \left(17 a^{2} + 12 a + 23\right)\cdot 29^{34} + \left(19 a^{2} + 26 a + 20\right)\cdot 29^{35} + \left(20 a^{2} + 19 a + 17\right)\cdot 29^{36} + \left(24 a^{2} + 3 a + 1\right)\cdot 29^{37} + \left(11 a^{2} + 27 a + 19\right)\cdot 29^{38} + \left(10 a^{2} + 13 a + 12\right)\cdot 29^{39} + \left(16 a^{2} + 26 a + 9\right)\cdot 29^{40} + \left(17 a + 24\right)\cdot 29^{41} + \left(24 a^{2} + 7 a + 1\right)\cdot 29^{42} + \left(18 a^{2} + 8 a + 14\right)\cdot 29^{43} + \left(7 a^{2} + 23\right)\cdot 29^{44} + \left(24 a^{2} + 23 a + 13\right)\cdot 29^{45} + \left(3 a^{2} + 19 a + 24\right)\cdot 29^{46} + \left(4 a^{2} + 5 a + 25\right)\cdot 29^{47} + \left(3 a^{2} + 13 a + 11\right)\cdot 29^{48} + \left(27 a^{2} + 11 a + 24\right)\cdot 29^{49} + \left(15 a^{2} + 6 a + 15\right)\cdot 29^{50} + \left(16 a^{2} + 15 a\right)\cdot 29^{51} + \left(2 a^{2} + 4 a + 8\right)\cdot 29^{52} + \left(2 a^{2} + 26 a + 2\right)\cdot 29^{53} + \left(21 a^{2} + 4 a + 6\right)\cdot 29^{54} + \left(28 a^{2} + 20 a + 12\right)\cdot 29^{55} + \left(25 a^{2} + a + 10\right)\cdot 29^{56} + \left(3 a + 20\right)\cdot 29^{57} + \left(27 a + 17\right)\cdot 29^{58} + \left(13 a^{2} + 6 a + 26\right)\cdot 29^{59} + \left(20 a^{2} + 6 a + 18\right)\cdot 29^{60} + \left(5 a^{2} + 8\right)\cdot 29^{61} + \left(10 a^{2} + 14\right)\cdot 29^{62} + \left(a^{2} + 11 a + 7\right)\cdot 29^{63} + \left(15 a^{2} + 15 a + 8\right)\cdot 29^{64} + \left(12 a^{2} + 11 a + 14\right)\cdot 29^{65} + \left(27 a^{2} + a + 5\right)\cdot 29^{66} + \left(9 a^{2} + 21 a + 23\right)\cdot 29^{67} + \left(13 a^{2} + 25 a + 15\right)\cdot 29^{68} + \left(4 a^{2} + 14 a + 17\right)\cdot 29^{69} + \left(15 a^{2} + 24 a + 18\right)\cdot 29^{70} + \left(7 a^{2} + 10 a + 6\right)\cdot 29^{71} + \left(25 a^{2} + 21 a + 9\right)\cdot 29^{72} + \left(25 a^{2} + 2 a + 10\right)\cdot 29^{73} + \left(a^{2} + 28 a + 14\right)\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 2 }$ $=$ $16 + 27\cdot 29 + 2\cdot 29^{2} + 27\cdot 29^{3} + 22\cdot 29^{4} + 24\cdot 29^{5} + 22\cdot 29^{6} + 10\cdot 29^{7} + 27\cdot 29^{8} + 5\cdot 29^{9} + 13\cdot 29^{10} + 13\cdot 29^{11} + 29^{12} + 21\cdot 29^{13} + 17\cdot 29^{14} + 20\cdot 29^{15} + 3\cdot 29^{16} + 3\cdot 29^{17} + 6\cdot 29^{18} + 27\cdot 29^{19} + 12\cdot 29^{20} + 27\cdot 29^{21} + 4\cdot 29^{22} + 11\cdot 29^{23} + 29^{24} + 24\cdot 29^{25} + 26\cdot 29^{26} + 22\cdot 29^{27} + 16\cdot 29^{28} + 13\cdot 29^{29} + 2\cdot 29^{30} + 28\cdot 29^{31} + 6\cdot 29^{32} + 17\cdot 29^{33} + 27\cdot 29^{34} + 26\cdot 29^{35} + 20\cdot 29^{36} + 14\cdot 29^{37} + 3\cdot 29^{38} + 6\cdot 29^{39} + 23\cdot 29^{40} + 27\cdot 29^{41} + 3\cdot 29^{42} + 3\cdot 29^{43} + 15\cdot 29^{44} + 29^{45} + 18\cdot 29^{46} + 16\cdot 29^{47} + 10\cdot 29^{48} + 17\cdot 29^{49} + 11\cdot 29^{50} + 26\cdot 29^{51} + 8\cdot 29^{52} + 14\cdot 29^{53} + 21\cdot 29^{54} + 14\cdot 29^{55} + 11\cdot 29^{56} + 10\cdot 29^{58} + 27\cdot 29^{59} + 8\cdot 29^{60} + 10\cdot 29^{61} + 27\cdot 29^{62} + 21\cdot 29^{63} + 15\cdot 29^{64} + 18\cdot 29^{65} + 14\cdot 29^{66} + 8\cdot 29^{67} + 27\cdot 29^{68} + 17\cdot 29^{69} + 21\cdot 29^{70} + 22\cdot 29^{71} + 12\cdot 29^{72} + 4\cdot 29^{73} + 19\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 3 }$ $=$ $25 a^{2} + 5 + \left(28 a^{2} + 17 a + 16\right)\cdot 29 + \left(2 a^{2} + 28 a + 23\right)\cdot 29^{2} + \left(8 a^{2} + 28 a + 11\right)\cdot 29^{3} + \left(15 a^{2} + 24 a + 8\right)\cdot 29^{4} + \left(13 a^{2} + 17 a + 11\right)\cdot 29^{5} + \left(6 a + 2\right)\cdot 29^{6} + \left(26 a^{2} + 12 a + 10\right)\cdot 29^{7} + \left(27 a^{2} + 10 a + 22\right)\cdot 29^{8} + \left(9 a^{2} + 5 a + 20\right)\cdot 29^{9} + \left(23 a^{2} + 22 a + 3\right)\cdot 29^{10} + \left(8 a^{2} + 14 a\right)\cdot 29^{11} + \left(5 a^{2} + 10 a + 15\right)\cdot 29^{12} + \left(13 a^{2} + 18 a + 13\right)\cdot 29^{13} + \left(6 a^{2} + 14 a + 10\right)\cdot 29^{14} + \left(3 a^{2} + 2 a + 22\right)\cdot 29^{15} + \left(17 a^{2} + 16 a + 21\right)\cdot 29^{16} + \left(18 a^{2} + 4 a + 19\right)\cdot 29^{17} + \left(7 a^{2} + 21 a + 27\right)\cdot 29^{18} + \left(13 a^{2} + 3 a + 26\right)\cdot 29^{19} + \left(11 a^{2} + 4 a + 6\right)\cdot 29^{20} + \left(6 a^{2} + 17 a + 10\right)\cdot 29^{21} + \left(a^{2} + 13 a + 19\right)\cdot 29^{22} + \left(19 a^{2} + 16 a + 10\right)\cdot 29^{23} + \left(15 a^{2} + 5 a + 14\right)\cdot 29^{24} + \left(17 a^{2} + 17 a + 27\right)\cdot 29^{25} + \left(5 a^{2} + 21 a + 28\right)\cdot 29^{26} + \left(14 a^{2} + 28 a + 22\right)\cdot 29^{27} + \left(11 a^{2} + 24 a + 13\right)\cdot 29^{28} + \left(5 a^{2} + 20 a + 4\right)\cdot 29^{29} + \left(9 a^{2} + 5 a + 25\right)\cdot 29^{30} + \left(15 a^{2} + 26 a + 14\right)\cdot 29^{31} + \left(10 a^{2} + 4 a + 18\right)\cdot 29^{32} + \left(25 a^{2} + 11 a + 13\right)\cdot 29^{33} + \left(2 a^{2} + 2 a + 24\right)\cdot 29^{34} + \left(6 a^{2} + 17 a + 7\right)\cdot 29^{35} + \left(28 a^{2} + 8 a + 24\right)\cdot 29^{36} + \left(15 a^{2} + 25 a + 8\right)\cdot 29^{37} + \left(5 a^{2} + 21 a + 27\right)\cdot 29^{38} + \left(23 a^{2} + 21 a + 11\right)\cdot 29^{39} + \left(5 a^{2} + 9 a + 3\right)\cdot 29^{40} + \left(27 a^{2} + 18 a + 27\right)\cdot 29^{41} + \left(18 a^{2} + a + 6\right)\cdot 29^{42} + \left(5 a^{2} + 20 a + 16\right)\cdot 29^{43} + \left(6 a^{2} + 25 a + 15\right)\cdot 29^{44} + \left(4 a^{2} + 23 a + 27\right)\cdot 29^{45} + \left(13 a^{2} + 26 a + 21\right)\cdot 29^{46} + \left(10 a^{2} + 12 a + 10\right)\cdot 29^{47} + \left(13 a^{2} + 2 a + 25\right)\cdot 29^{48} + \left(26 a^{2} + 3 a + 6\right)\cdot 29^{49} + \left(28 a^{2} + 19\right)\cdot 29^{50} + \left(18 a^{2} + 24 a + 15\right)\cdot 29^{51} + \left(18 a^{2} + 5 a + 17\right)\cdot 29^{52} + \left(22 a^{2} + 26 a + 7\right)\cdot 29^{53} + \left(6 a^{2} + 10 a + 5\right)\cdot 29^{54} + \left(10 a^{2} + 19 a + 5\right)\cdot 29^{55} + \left(27 a + 1\right)\cdot 29^{56} + \left(9 a^{2} + 9 a + 21\right)\cdot 29^{57} + \left(17 a^{2} + 11 a + 12\right)\cdot 29^{58} + \left(5 a^{2} + 5 a + 4\right)\cdot 29^{59} + \left(21 a^{2} + 17 a + 26\right)\cdot 29^{60} + \left(8 a^{2} + 9 a + 11\right)\cdot 29^{61} + \left(10 a^{2} + 2 a + 23\right)\cdot 29^{62} + \left(15 a^{2} + 25 a + 24\right)\cdot 29^{63} + \left(25 a^{2} + 6 a + 15\right)\cdot 29^{64} + \left(24 a^{2} + 20 a + 9\right)\cdot 29^{65} + \left(21 a^{2} + 17 a + 27\right)\cdot 29^{66} + \left(24 a^{2} + 26 a + 10\right)\cdot 29^{67} + \left(6 a^{2} + 25 a + 10\right)\cdot 29^{68} + \left(3 a^{2} + 13 a + 7\right)\cdot 29^{69} + \left(14 a^{2} + 15 a + 11\right)\cdot 29^{70} + \left(8 a^{2} + 21 a + 4\right)\cdot 29^{71} + \left(25 a^{2} + 22 a + 14\right)\cdot 29^{72} + \left(26 a^{2} + 28 a + 27\right)\cdot 29^{73} + \left(6 a^{2} + 7\right)\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 4 }$ $=$ $19 a^{2} + 9 a + 3 + \left(a^{2} + 16 a + 26\right)\cdot 29 + \left(15 a + 24\right)\cdot 29^{2} + \left(10 a^{2} + 22 a + 14\right)\cdot 29^{3} + \left(20 a^{2} + 10 a + 12\right)\cdot 29^{4} + \left(17 a^{2} + 15 a + 28\right)\cdot 29^{5} + \left(8 a^{2} + 2 a + 18\right)\cdot 29^{6} + \left(6 a^{2} + 26 a + 25\right)\cdot 29^{7} + \left(9 a^{2} + 2 a\right)\cdot 29^{8} + \left(14 a^{2} + 27 a + 25\right)\cdot 29^{9} + \left(23 a^{2} + 7 a + 23\right)\cdot 29^{10} + \left(7 a^{2} + 25 a + 21\right)\cdot 29^{11} + \left(17 a^{2} + 2 a + 13\right)\cdot 29^{12} + \left(6 a^{2} + 7 a + 17\right)\cdot 29^{13} + \left(26 a^{2} + 16 a\right)\cdot 29^{14} + \left(3 a^{2} + 27 a + 5\right)\cdot 29^{15} + \left(2 a^{2} + 5 a + 17\right)\cdot 29^{16} + \left(12 a^{2} + 8 a + 6\right)\cdot 29^{17} + \left(24 a^{2} + 12 a + 8\right)\cdot 29^{18} + \left(2 a^{2} + 14 a + 26\right)\cdot 29^{19} + \left(8 a^{2} + 11 a + 1\right)\cdot 29^{20} + \left(21 a^{2} + 17 a + 16\right)\cdot 29^{21} + \left(27 a^{2} + 19 a + 27\right)\cdot 29^{22} + \left(23 a^{2} + 4 a + 12\right)\cdot 29^{23} + \left(18 a^{2} + 21 a + 27\right)\cdot 29^{24} + \left(6 a^{2} + 16 a + 2\right)\cdot 29^{25} + \left(25 a^{2} + 6 a + 22\right)\cdot 29^{26} + \left(18 a^{2} + 25 a + 28\right)\cdot 29^{27} + \left(27 a^{2} + 21 a + 3\right)\cdot 29^{28} + \left(2 a^{2} + 21 a + 11\right)\cdot 29^{29} + \left(13 a^{2} + 3 a + 23\right)\cdot 29^{30} + \left(2 a^{2} + 7 a + 13\right)\cdot 29^{31} + \left(22 a^{2} + 13 a + 22\right)\cdot 29^{32} + \left(17 a^{2} + 26 a + 21\right)\cdot 29^{33} + \left(24 a^{2} + 16 a + 23\right)\cdot 29^{34} + \left(19 a^{2} + 5 a + 1\right)\cdot 29^{35} + \left(12 a^{2} + 28 a + 7\right)\cdot 29^{36} + \left(25 a^{2} + 22 a + 12\right)\cdot 29^{37} + \left(12 a^{2} + 20\right)\cdot 29^{38} + \left(25 a^{2} + 16 a + 3\right)\cdot 29^{39} + \left(22 a + 8\right)\cdot 29^{40} + \left(20 a^{2} + 5 a + 21\right)\cdot 29^{41} + \left(21 a^{2} + 2 a + 27\right)\cdot 29^{42} + \left(17 a^{2} + 4 a + 2\right)\cdot 29^{43} + \left(13 a^{2} + 28 a + 12\right)\cdot 29^{44} + \left(11 a^{2} + 3 a + 6\right)\cdot 29^{45} + \left(9 a^{2} + 25 a + 22\right)\cdot 29^{46} + \left(25 a^{2} + 9 a + 5\right)\cdot 29^{47} + \left(10 a^{2} + 10 a + 22\right)\cdot 29^{48} + \left(20 a^{2} + 27 a + 5\right)\cdot 29^{49} + \left(3 a^{2} + 24 a + 9\right)\cdot 29^{50} + \left(12 a^{2} + 26 a + 4\right)\cdot 29^{51} + \left(a^{2} + 15 a + 16\right)\cdot 29^{52} + \left(22 a^{2} + 9\right)\cdot 29^{53} + \left(5 a^{2} + 27 a + 24\right)\cdot 29^{54} + \left(14 a^{2} + 9 a + 21\right)\cdot 29^{55} + \left(9 a^{2} + a + 7\right)\cdot 29^{56} + \left(25 a^{2} + 2 a + 14\right)\cdot 29^{57} + \left(7 a^{2} + 22 a + 18\right)\cdot 29^{58} + \left(12 a^{2} + 4 a + 25\right)\cdot 29^{59} + \left(16 a^{2} + 6 a + 3\right)\cdot 29^{60} + \left(3 a^{2} + 2 a + 25\right)\cdot 29^{61} + \left(26 a^{2} + 17 a + 25\right)\cdot 29^{62} + \left(23 a^{2} + 10 a + 27\right)\cdot 29^{63} + \left(a^{2} + 16 a + 9\right)\cdot 29^{64} + \left(18 a^{2} + 13 a + 2\right)\cdot 29^{65} + \left(8 a^{2} + 19\right)\cdot 29^{66} + \left(16 a^{2} + a + 2\right)\cdot 29^{67} + \left(16 a^{2} + 9 a + 20\right)\cdot 29^{68} + \left(5 a^{2} + 16 a + 28\right)\cdot 29^{69} + \left(4 a^{2} + 23 a + 3\right)\cdot 29^{70} + \left(a^{2} + 2 a + 27\right)\cdot 29^{71} + \left(12 a^{2} + 16 a + 10\right)\cdot 29^{72} + \left(6 a^{2} + 20 a + 13\right)\cdot 29^{73} + \left(17 a^{2} + 6 a + 15\right)\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 5 }$ $=$ $17 a + 20 + \left(6 a^{2} + 11 a + 14\right)\cdot 29 + \left(26 a^{2} + 25\right)\cdot 29^{2} + \left(4 a^{2} + 28 a + 26\right)\cdot 29^{3} + \left(28 a^{2} + 19 a + 15\right)\cdot 29^{4} + \left(15 a^{2} + 13 a + 14\right)\cdot 29^{5} + \left(8 a^{2} + 2 a + 3\right)\cdot 29^{6} + \left(3 a^{2} + 27 a + 28\right)\cdot 29^{7} + \left(5 a^{2} + 15 a + 20\right)\cdot 29^{8} + \left(19 a^{2} + 5 a + 13\right)\cdot 29^{9} + \left(14 a^{2} + 16 a + 11\right)\cdot 29^{10} + \left(16 a^{2} + 19 a + 10\right)\cdot 29^{11} + \left(15 a^{2} + 12 a + 9\right)\cdot 29^{12} + \left(26 a^{2} + 10 a + 2\right)\cdot 29^{13} + \left(16 a^{2} + 9 a + 5\right)\cdot 29^{14} + \left(5 a^{2} + 21 a + 6\right)\cdot 29^{15} + \left(16 a^{2} + 13 a + 1\right)\cdot 29^{16} + \left(a^{2} + 12 a + 26\right)\cdot 29^{17} + \left(15 a^{2} + 14 a + 27\right)\cdot 29^{18} + \left(13 a + 9\right)\cdot 29^{19} + \left(7 a^{2} + 20\right)\cdot 29^{20} + \left(13 a^{2} + 22 a + 9\right)\cdot 29^{21} + \left(4 a^{2} + 23 a + 4\right)\cdot 29^{22} + \left(12 a^{2} + 17 a + 11\right)\cdot 29^{23} + \left(11 a^{2} + 9 a + 18\right)\cdot 29^{24} + \left(a^{2} + 8 a + 15\right)\cdot 29^{25} + \left(2 a^{2} + a + 14\right)\cdot 29^{26} + \left(8 a^{2} + 24\right)\cdot 29^{27} + \left(13 a^{2} + 26 a + 25\right)\cdot 29^{28} + \left(19 a^{2} + 4 a + 3\right)\cdot 29^{29} + \left(26 a^{2} + 2 a\right)\cdot 29^{30} + \left(20 a^{2} + 23 a + 3\right)\cdot 29^{31} + \left(20 a^{2} + 3\right)\cdot 29^{32} + \left(9 a^{2} + 7 a + 12\right)\cdot 29^{33} + \left(7 a^{2} + 10 a + 1\right)\cdot 29^{34} + \left(4 a^{2} + 9 a + 15\right)\cdot 29^{35} + \left(3 a^{2} + 25 a\right)\cdot 29^{36} + \left(4 a^{2} + 25 a + 22\right)\cdot 29^{37} + \left(3 a^{2} + 16 a + 4\right)\cdot 29^{38} + \left(24 a^{2} + 23 a + 13\right)\cdot 29^{39} + \left(2 a^{2} + 27 a + 28\right)\cdot 29^{40} + \left(13 a^{2} + 2 a + 17\right)\cdot 29^{41} + \left(20 a^{2} + 26 a + 18\right)\cdot 29^{42} + \left(18 a^{2} + a + 4\right)\cdot 29^{43} + \left(25 a^{2} + 20 a + 22\right)\cdot 29^{44} + \left(10 a^{2} + 10 a + 26\right)\cdot 29^{45} + \left(5 a^{2} + 14 a + 1\right)\cdot 29^{46} + \left(17 a^{2} + 12 a + 10\right)\cdot 29^{47} + \left(8 a^{2} + 12 a + 9\right)\cdot 29^{48} + \left(23 a^{2} + 16 a + 12\right)\cdot 29^{49} + \left(9 a^{2} + 17 a + 3\right)\cdot 29^{50} + \left(17 a^{2} + 19 a + 23\right)\cdot 29^{51} + \left(20 a^{2} + 11 a\right)\cdot 29^{52} + \left(24 a^{2} + 14 a + 20\right)\cdot 29^{53} + \left(28 a^{2} + 18 a + 5\right)\cdot 29^{54} + \left(6 a^{2} + 11 a + 20\right)\cdot 29^{55} + \left(8 a^{2} + 23 a + 11\right)\cdot 29^{56} + \left(7 a^{2} + 13 a + 28\right)\cdot 29^{57} + \left(4 a^{2} + 15 a + 4\right)\cdot 29^{58} + \left(16 a + 26\right)\cdot 29^{59} + \left(a^{2} + 28 a + 8\right)\cdot 29^{60} + \left(3 a^{2} + 13 a + 4\right)\cdot 29^{61} + \left(13 a^{2} + 20 a + 27\right)\cdot 29^{62} + \left(11 a^{2} + 12 a + 9\right)\cdot 29^{63} + \left(27 a^{2} + 6 a + 18\right)\cdot 29^{64} + \left(22 a^{2} + 21 a + 16\right)\cdot 29^{65} + \left(18 a^{2} + 13\right)\cdot 29^{66} + \left(11 a^{2} + 26 a + 22\right)\cdot 29^{67} + \left(16 a^{2} + 5 a + 3\right)\cdot 29^{68} + \left(a^{2} + 2 a + 5\right)\cdot 29^{69} + \left(19 a^{2} + 23 a + 8\right)\cdot 29^{70} + \left(8 a^{2} + 17 a + 14\right)\cdot 29^{71} + \left(5 a^{2} + 17 a + 16\right)\cdot 29^{72} + \left(a^{2} + 3 a + 12\right)\cdot 29^{73} + \left(18 a^{2} + 16 a + 3\right)\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 6 }$ $=$ $4 a^{2} + 12 a + 6 + \left(23 a^{2} + 18\right)\cdot 29 + \left(28 a^{2} + 9\right)\cdot 29^{2} + \left(15 a^{2} + a + 22\right)\cdot 29^{3} + \left(14 a^{2} + 13 a + 26\right)\cdot 29^{4} + \left(28 a^{2} + 26 a + 11\right)\cdot 29^{5} + \left(19 a^{2} + 19 a + 28\right)\cdot 29^{6} + \left(28 a^{2} + 18 a + 3\right)\cdot 29^{7} + \left(24 a^{2} + 2 a + 28\right)\cdot 29^{8} + \left(28 a^{2} + 18 a + 16\right)\cdot 29^{9} + \left(19 a^{2} + 19 a + 18\right)\cdot 29^{10} + \left(3 a^{2} + 23 a + 12\right)\cdot 29^{11} + \left(8 a^{2} + 5 a + 28\right)\cdot 29^{12} + 18 a^{2}29^{13} + \left(5 a^{2} + 5 a + 19\right)\cdot 29^{14} + \left(20 a^{2} + 5 a + 25\right)\cdot 29^{15} + \left(24 a^{2} + 28 a + 2\right)\cdot 29^{16} + \left(8 a^{2} + 11 a + 26\right)\cdot 29^{17} + \left(6 a^{2} + 22 a + 25\right)\cdot 29^{18} + \left(15 a^{2} + 11 a + 19\right)\cdot 29^{19} + \left(10 a^{2} + 24 a + 5\right)\cdot 29^{20} + \left(9 a^{2} + 18 a + 14\right)\cdot 29^{21} + \left(23 a^{2} + 20 a\right)\cdot 29^{22} + \left(26 a^{2} + 23 a + 21\right)\cdot 29^{23} + \left(a^{2} + 13 a + 5\right)\cdot 29^{24} + \left(10 a^{2} + 3 a + 27\right)\cdot 29^{25} + \left(21 a^{2} + 6 a + 20\right)\cdot 29^{26} + \left(6 a^{2} + 22\right)\cdot 29^{27} + \left(4 a^{2} + 7 a + 13\right)\cdot 29^{28} + \left(4 a^{2} + 3 a + 12\right)\cdot 29^{29} + \left(22 a^{2} + 21 a + 13\right)\cdot 29^{30} + \left(21 a^{2} + 8 a + 23\right)\cdot 29^{31} + \left(26 a^{2} + 23 a + 20\right)\cdot 29^{32} + \left(22 a^{2} + 10 a\right)\cdot 29^{33} + \left(18 a^{2} + 16 a + 7\right)\cdot 29^{34} + \left(18 a^{2} + 2 a + 5\right)\cdot 29^{35} + \left(26 a^{2} + 24 a + 22\right)\cdot 29^{36} + \left(8 a^{2} + 6 a + 18\right)\cdot 29^{37} + \left(20 a^{2} + 19 a + 27\right)\cdot 29^{38} + \left(10 a^{2} + 12 a + 4\right)\cdot 29^{39} + \left(20 a^{2} + 20 a + 13\right)\cdot 29^{40} + \left(17 a^{2} + 7 a + 14\right)\cdot 29^{41} + \left(18 a^{2} + a + 6\right)\cdot 29^{42} + \left(4 a^{2} + 7 a + 5\right)\cdot 29^{43} + \left(26 a^{2} + 12 a + 13\right)\cdot 29^{44} + \left(13 a^{2} + 23 a + 11\right)\cdot 29^{45} + \left(10 a^{2} + 16 a + 18\right)\cdot 29^{46} + \left(a^{2} + 3 a + 27\right)\cdot 29^{47} + \left(7 a^{2} + 14 a + 16\right)\cdot 29^{48} + \left(8 a^{2} + 9 a + 11\right)\cdot 29^{49} + \left(19 a^{2} + 11 a + 6\right)\cdot 29^{50} + \left(21 a^{2} + 14 a + 19\right)\cdot 29^{51} + \left(18 a^{2} + 11 a + 17\right)\cdot 29^{52} + \left(10 a^{2} + 17 a + 20\right)\cdot 29^{53} + \left(22 a^{2} + 28 a + 6\right)\cdot 29^{54} + \left(11 a^{2} + 26 a + 7\right)\cdot 29^{55} + \left(20 a^{2} + 6 a + 18\right)\cdot 29^{56} + \left(12 a^{2} + 5 a + 6\right)\cdot 29^{57} + \left(7 a^{2} + 2 a + 9\right)\cdot 29^{58} + \left(23 a^{2} + 7 a + 18\right)\cdot 29^{59} + \left(6 a^{2} + 12 a + 16\right)\cdot 29^{60} + \left(17 a^{2} + 5 a + 13\right)\cdot 29^{61} + \left(5 a^{2} + 6 a + 7\right)\cdot 29^{62} + \left(2 a^{2} + 20 a + 7\right)\cdot 29^{63} + \left(5 a^{2} + 15 a + 27\right)\cdot 29^{64} + \left(10 a^{2} + 16 a + 18\right)\cdot 29^{65} + \left(17 a^{2} + 10 a + 11\right)\cdot 29^{66} + \left(21 a^{2} + 5 a + 16\right)\cdot 29^{67} + \left(5 a^{2} + 26 a + 18\right)\cdot 29^{68} + \left(24 a^{2} + 12 a + 25\right)\cdot 29^{69} + \left(24 a^{2} + 19 a + 15\right)\cdot 29^{70} + \left(11 a^{2} + 18 a + 18\right)\cdot 29^{71} + \left(27 a^{2} + 17 a + 26\right)\cdot 29^{72} + \left(25 a + 21\right)\cdot 29^{73} + \left(4 a^{2} + 11 a + 13\right)\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 7 }$ $=$ $13 a^{2} + a + 24 + \left(21 a^{2} + 5 a + 13\right)\cdot 29 + \left(13 a^{2} + 23 a + 4\right)\cdot 29^{2} + \left(4 a^{2} + 11 a + 17\right)\cdot 29^{3} + \left(8 a^{2} + 8 a + 15\right)\cdot 29^{4} + \left(18 a^{2} + 12 a + 19\right)\cdot 29^{5} + \left(5 a^{2} + 10 a + 24\right)\cdot 29^{6} + \left(27 a^{2} + 18 a + 14\right)\cdot 29^{7} + \left(8 a^{2} + 4 a\right)\cdot 29^{8} + \left(20 a^{2} + 15 a + 4\right)\cdot 29^{9} + \left(22 a + 3\right)\cdot 29^{10} + \left(9 a^{2} + 13 a + 4\right)\cdot 29^{11} + \left(9 a^{2} + 12 a + 3\right)\cdot 29^{12} + \left(17 a^{2} + 2 a + 22\right)\cdot 29^{13} + \left(18 a^{2} + 5 a + 9\right)\cdot 29^{14} + \left(a^{2} + 25 a + 21\right)\cdot 29^{15} + \left(18 a^{2} + 6 a + 28\right)\cdot 29^{16} + \left(a^{2} + 16 a + 11\right)\cdot 29^{17} + \left(26 a^{2} + 14 a + 10\right)\cdot 29^{18} + \left(4 a^{2} + 12 a + 19\right)\cdot 29^{19} + \left(25 a^{2} + 6 a + 24\right)\cdot 29^{20} + \left(10 a^{2} + 6 a + 11\right)\cdot 29^{21} + \left(10 a^{2} + 4 a + 4\right)\cdot 29^{22} + \left(15 a^{2} + 19 a + 11\right)\cdot 29^{23} + \left(25 a^{2} + 14 a + 7\right)\cdot 29^{24} + \left(19 a^{2} + 21 a + 1\right)\cdot 29^{25} + \left(23 a^{2} + 5 a + 20\right)\cdot 29^{26} + \left(26 a^{2} + 16 a\right)\cdot 29^{27} + \left(3 a^{2} + 28 a + 11\right)\cdot 29^{28} + \left(10 a^{2} + 20 a + 1\right)\cdot 29^{29} + \left(a^{2} + 28 a + 27\right)\cdot 29^{30} + \left(27 a^{2} + 24 a + 7\right)\cdot 29^{31} + \left(10 a^{2} + 27 a + 17\right)\cdot 29^{32} + \left(10 a^{2} + 15 a + 21\right)\cdot 29^{33} + \left(16 a^{2} + 28 a + 12\right)\cdot 29^{34} + \left(18 a^{2} + 25 a + 19\right)\cdot 29^{35} + \left(24 a^{2} + 9 a + 3\right)\cdot 29^{36} + \left(7 a^{2} + 2 a + 8\right)\cdot 29^{37} + \left(4 a^{2} + a + 28\right)\cdot 29^{38} + \left(22 a^{2} + 28 a + 8\right)\cdot 29^{39} + \left(11 a^{2} + 8 a + 3\right)\cdot 29^{40} + \left(8 a^{2} + 5 a + 25\right)\cdot 29^{41} + \left(12 a^{2} + 19 a + 24\right)\cdot 29^{42} + \left(21 a^{2} + 16 a + 7\right)\cdot 29^{43} + \left(7 a^{2} + 4\right)\cdot 29^{44} + \left(22 a^{2} + 2 a + 11\right)\cdot 29^{45} + \left(15 a^{2} + 13 a + 11\right)\cdot 29^{46} + \left(28 a^{2} + 13 a\right)\cdot 29^{47} + \left(14 a^{2} + 5 a + 18\right)\cdot 29^{48} + \left(10 a^{2} + 19 a + 21\right)\cdot 29^{49} + \left(9 a^{2} + 26 a + 16\right)\cdot 29^{50} + \left(15 a + 17\right)\cdot 29^{51} + \left(25 a^{2} + 8 a + 18\right)\cdot 29^{52} + \left(4 a^{2} + 2 a + 15\right)\cdot 29^{53} + \left(2 a^{2} + 26 a + 19\right)\cdot 29^{54} + \left(15 a^{2} + 27 a + 3\right)\cdot 29^{55} + \left(22 a^{2} + 25 a + 25\right)\cdot 29^{56} + \left(2 a^{2} + 23 a + 22\right)\cdot 29^{57} + \left(21 a^{2} + 8 a + 16\right)\cdot 29^{58} + \left(3 a^{2} + 17 a + 4\right)\cdot 29^{59} + \left(21 a^{2} + 16 a + 10\right)\cdot 29^{60} + \left(19 a^{2} + 26 a + 27\right)\cdot 29^{61} + \left(21 a^{2} + 11 a + 19\right)\cdot 29^{62} + \left(3 a^{2} + 7 a + 10\right)\cdot 29^{63} + \left(12 a^{2} + 26 a + 4\right)\cdot 29^{64} + \left(27 a^{2} + 3 a + 5\right)\cdot 29^{65} + \left(21 a^{2} + 27 a + 27\right)\cdot 29^{66} + \left(2 a^{2} + 6 a + 3\right)\cdot 29^{67} + \left(28 a^{2} + 23 a + 16\right)\cdot 29^{68} + \left(18 a^{2} + 26 a + 17\right)\cdot 29^{69} + \left(9 a^{2} + 9 a + 1\right)\cdot 29^{70} + \left(20 a^{2} + 15 a + 14\right)\cdot 29^{71} + \left(20 a^{2} + 20 a + 22\right)\cdot 29^{72} + \left(25 a^{2} + 5 a + 19\right)\cdot 29^{73} + \left(9 a^{2} + 23 a + 5\right)\cdot 29^{74} +O\left(29^{ 75 }\right)$ $r_{ 8 }$ $=$ $22 + 25\cdot 29 + 8\cdot 29^{2} + 23\cdot 29^{3} + 27\cdot 29^{4} + 9\cdot 29^{5} + 7\cdot 29^{6} + 11\cdot 29^{7} + 22\cdot 29^{8} + 11\cdot 29^{9} + 4\cdot 29^{10} + 16\cdot 29^{11} + 2\cdot 29^{12} + 23\cdot 29^{13} + 21\cdot 29^{14} + 12\cdot 29^{15} + 14\cdot 29^{16} + 11\cdot 29^{17} + 14\cdot 29^{18} + 22\cdot 29^{19} + 9\cdot 29^{20} + 4\cdot 29^{21} + 28\cdot 29^{22} + 2\cdot 29^{23} + 11\cdot 29^{24} + 10\cdot 29^{25} + 29^{26} + 12\cdot 29^{27} + 28\cdot 29^{28} + 29^{29} + 28\cdot 29^{30} + 14\cdot 29^{31} + 20\cdot 29^{33} + 24\cdot 29^{34} + 18\cdot 29^{35} + 19\cdot 29^{36} + 14\cdot 29^{38} + 25\cdot 29^{39} + 26\cdot 29^{40} + 15\cdot 29^{41} + 25\cdot 29^{42} + 3\cdot 29^{43} + 10\cdot 29^{44} + 17\cdot 29^{45} + 26\cdot 29^{46} + 18\cdot 29^{47} + 29^{48} + 16\cdot 29^{49} + 4\cdot 29^{50} + 9\cdot 29^{51} + 28\cdot 29^{52} + 25\cdot 29^{53} + 26\cdot 29^{54} + 29^{55} + 29^{56} + 2\cdot 29^{57} + 26\cdot 29^{58} + 11\cdot 29^{59} + 22\cdot 29^{60} + 14\cdot 29^{61} + 28\cdot 29^{62} + 5\cdot 29^{63} + 16\cdot 29^{64} + 29^{65} + 26\cdot 29^{66} + 27\cdot 29^{67} + 3\cdot 29^{68} + 25\cdot 29^{69} + 5\cdot 29^{70} + 8\cdot 29^{71} + 3\cdot 29^{72} + 6\cdot 29^{73} + 7\cdot 29^{74} +O\left(29^{ 75 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $9$ $6$ $2$ $(3,6)(5,8)$ $-3$ $9$ $2$ $(1,4)(2,7)(3,6)(5,8)$ $1$ $12$ $2$ $(1,3)(2,5)(4,6)(7,8)$ $3$ $12$ $2$ $(1,5)(2,6)(3,7)(4,8)$ $3$ $36$ $2$ $(1,7)(5,8)$ $1$ $16$ $3$ $(3,8,6)$ $0$ $32$ $3$ $(1,2,4)(3,5,6)$ $0$ $32$ $3$ $(1,2,4)(5,6,8)$ $0$ $36$ $4$ $(1,3,4,6)(2,5,7,8)$ $-1$ $36$ $4$ $(1,5,4,8)(2,6,7,3)$ $-1$ $36$ $4$ $(1,7,4,2)(3,5,6,8)$ $1$ $72$ $4$ $(3,5,6,8)(4,7)$ $-1$ $48$ $6$ $(1,4)(2,7)(3,8,6)$ $0$ $96$ $6$ $(1,5,2,6,4,3)(7,8)$ $0$ $96$ $6$ $(1,6,2,8,4,5)(3,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.