# Properties

 Label 9.363994344000.12t165.a.a Dimension $9$ Group $(A_4\wr C_2):C_2$ Conductor $363994344000$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $9$ Group: $(A_4\wr C_2):C_2$ Conductor: $$363994344000$$$$\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 17^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.4.40608119002500.1 Galois orbit size: $1$ Smallest permutation container: 12T165 Parity: even Determinant: 1.1785.2t1.a.a Projective image: $(A_4\wr C_2):C_2$ Projective field: Galois closure of 8.4.40608119002500.1

## Defining polynomial

 $f(x)$ $=$ $x^{8} - 2 x^{7} - 11 x^{6} + 36 x^{5} - x^{4} - 131 x^{3} + 222 x^{2} - 156 x - 404$.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 89.

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

Roots:
 $r_{ 1 }$ $=$ $25 a^{2} + 9 a + 11 + \left(3 a^{2} + 2 a + 13\right)\cdot 29 + \left(15 a + 1\right)\cdot 29^{2} + \left(6 a^{2} + 24 a + 16\right)\cdot 29^{3} + \left(6 a^{2} + 17 a + 2\right)\cdot 29^{4} + \left(21 a^{2} + 8 a + 5\right)\cdot 29^{5} + \left(13 a^{2} + 8 a + 18\right)\cdot 29^{6} + \left(26 a^{2} + a + 17\right)\cdot 29^{7} + \left(a^{2} + 4 a + 5\right)\cdot 29^{8} + \left(17 a^{2} + 23\right)\cdot 29^{9} + \left(26 a^{2} + 5 a + 23\right)\cdot 29^{10} + \left(6 a^{2} + 15 a + 21\right)\cdot 29^{11} + \left(16 a^{2} + a + 28\right)\cdot 29^{12} + \left(28 a^{2} + 27 a + 16\right)\cdot 29^{13} + \left(5 a^{2} + 4 a + 27\right)\cdot 29^{14} + \left(21 a^{2} + 19 a + 6\right)\cdot 29^{15} + \left(18 a^{2} + 9 a + 7\right)\cdot 29^{16} + \left(a^{2} + 26 a + 14\right)\cdot 29^{17} + \left(7 a^{2} + 27 a + 3\right)\cdot 29^{18} + \left(22 a^{2} + 3 a + 13\right)\cdot 29^{19} + \left(7 a^{2} + 5 a + 3\right)\cdot 29^{20} + \left(4 a^{2} + 23 a + 16\right)\cdot 29^{21} + \left(7 a^{2} + 12 a + 7\right)\cdot 29^{22} + \left(5 a^{2} + 17 a + 15\right)\cdot 29^{23} + \left(a^{2} + 3 a + 22\right)\cdot 29^{24} + \left(10 a^{2} + 9 a + 14\right)\cdot 29^{25} + \left(26 a^{2} + 27 a + 16\right)\cdot 29^{26} + \left(a^{2} + a + 6\right)\cdot 29^{27} + \left(9 a^{2} + 18 a + 22\right)\cdot 29^{28} + \left(11 a^{2} + 26 a + 23\right)\cdot 29^{29} + \left(26 a^{2} + 18\right)\cdot 29^{30} + \left(11 a^{2} + 14 a + 20\right)\cdot 29^{31} + \left(12 a^{2} + 10 a + 12\right)\cdot 29^{32} + \left(4 a^{2} + 9 a + 13\right)\cdot 29^{33} + \left(17 a^{2} + 3 a + 17\right)\cdot 29^{34} + \left(4 a^{2} + 3 a + 1\right)\cdot 29^{35} + \left(20 a^{2} + 2 a\right)\cdot 29^{36} + \left(25 a^{2} + 24 a + 26\right)\cdot 29^{37} + \left(11 a^{2} + 15 a + 7\right)\cdot 29^{38} + \left(9 a^{2} + 22 a + 11\right)\cdot 29^{39} + \left(13 a^{2} + 13 a + 21\right)\cdot 29^{40} + \left(12 a^{2} + 14 a + 5\right)\cdot 29^{41} + \left(14 a^{2} + 7 a + 19\right)\cdot 29^{42} + \left(17 a^{2} + 5 a + 12\right)\cdot 29^{43} + \left(a^{2} + 12 a + 17\right)\cdot 29^{44} + \left(5 a^{2} + 28 a + 16\right)\cdot 29^{45} + \left(16 a^{2} + 19 a + 17\right)\cdot 29^{46} + \left(22 a^{2} + 26 a + 10\right)\cdot 29^{47} + \left(4 a^{2} + a + 22\right)\cdot 29^{48} + \left(6 a^{2} + 6 a + 10\right)\cdot 29^{49} + \left(20 a^{2} + 19 a + 3\right)\cdot 29^{50} + \left(10 a^{2} + 16 a + 4\right)\cdot 29^{51} + \left(24 a^{2} + 8 a + 6\right)\cdot 29^{52} + \left(28 a^{2} + 10 a + 14\right)\cdot 29^{53} + \left(10 a^{2} + 5\right)\cdot 29^{54} + \left(26 a^{2} + 5 a + 28\right)\cdot 29^{55} + \left(21 a^{2} + 8 a + 5\right)\cdot 29^{56} + \left(a^{2} + a + 13\right)\cdot 29^{57} + \left(6 a^{2} + 16 a + 20\right)\cdot 29^{58} + \left(3 a^{2} + a + 16\right)\cdot 29^{59} + \left(13 a^{2} + 21 a + 1\right)\cdot 29^{60} + \left(6 a^{2} + 12 a + 7\right)\cdot 29^{61} + \left(9 a^{2} + 28 a + 14\right)\cdot 29^{62} + \left(10 a^{2} + 15 a + 11\right)\cdot 29^{63} + \left(19 a^{2} + 23 a + 4\right)\cdot 29^{64} + \left(16 a^{2} + 2 a + 8\right)\cdot 29^{65} + \left(21 a^{2} + 6 a + 23\right)\cdot 29^{66} + \left(15 a^{2} + 21 a + 24\right)\cdot 29^{67} + \left(17 a^{2} + 25 a + 12\right)\cdot 29^{68} + \left(13 a^{2} + 19 a + 21\right)\cdot 29^{69} + \left(13 a^{2} + 21 a + 17\right)\cdot 29^{70} + \left(7 a^{2} + 15 a + 28\right)\cdot 29^{71} + \left(25 a^{2} + 18 a + 19\right)\cdot 29^{72} + \left(11 a^{2} + 19 a + 17\right)\cdot 29^{73} + \left(13 a + 28\right)\cdot 29^{74} + \left(a^{2} + 6 a + 24\right)\cdot 29^{75} + \left(25 a + 16\right)\cdot 29^{76} + \left(13 a^{2} + 19 a + 1\right)\cdot 29^{77} + \left(4 a^{2} + 26 a + 27\right)\cdot 29^{78} + \left(13 a^{2} + 14 a + 15\right)\cdot 29^{79} + \left(8 a^{2} + 20 a\right)\cdot 29^{80} + \left(23 a^{2} + 3 a + 13\right)\cdot 29^{81} + \left(21 a^{2} + 7 a + 20\right)\cdot 29^{82} + \left(27 a^{2} + 20 a + 26\right)\cdot 29^{83} + \left(27 a^{2} + 22 a + 27\right)\cdot 29^{84} + \left(7 a^{2} + 5 a + 18\right)\cdot 29^{85} + \left(25 a^{2} + 26 a + 1\right)\cdot 29^{86} + \left(4 a^{2} + 11 a + 9\right)\cdot 29^{87} + \left(20 a + 25\right)\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 2 }$ $=$ $23 a^{2} + 19 a + 9 + \left(7 a^{2} + 7 a + 18\right)\cdot 29 + \left(25 a^{2} + 20 a + 4\right)\cdot 29^{2} + \left(9 a^{2} + 27 a + 22\right)\cdot 29^{3} + \left(10 a^{2} + 19 a + 16\right)\cdot 29^{4} + \left(13 a^{2} + 18 a + 24\right)\cdot 29^{5} + \left(24 a^{2} + 17 a + 25\right)\cdot 29^{6} + \left(18 a^{2} + 5 a + 19\right)\cdot 29^{7} + \left(10 a^{2} + 26 a + 7\right)\cdot 29^{8} + \left(20 a^{2} + 10 a + 28\right)\cdot 29^{9} + \left(4 a^{2} + 12 a + 1\right)\cdot 29^{10} + \left(8 a^{2} + 15 a + 17\right)\cdot 29^{11} + \left(5 a^{2} + 18 a + 24\right)\cdot 29^{12} + \left(28 a^{2} + 5 a + 4\right)\cdot 29^{13} + \left(9 a^{2} + 10 a + 16\right)\cdot 29^{14} + \left(14 a^{2} + 19 a + 24\right)\cdot 29^{15} + \left(21 a^{2} + 19 a + 21\right)\cdot 29^{16} + \left(15 a^{2} + 22 a + 8\right)\cdot 29^{17} + \left(15 a + 8\right)\cdot 29^{18} + \left(28 a^{2} + 12 a + 8\right)\cdot 29^{19} + \left(8 a^{2} + 10 a + 6\right)\cdot 29^{20} + \left(9 a^{2} + 23 a + 16\right)\cdot 29^{21} + 12 a\cdot 29^{22} + \left(10 a^{2} + 16 a + 4\right)\cdot 29^{23} + \left(21 a^{2} + 14 a + 10\right)\cdot 29^{24} + \left(2 a^{2} + 22 a + 3\right)\cdot 29^{25} + \left(18 a^{2} + 25 a + 7\right)\cdot 29^{26} + \left(7 a^{2} + 23 a + 5\right)\cdot 29^{27} + \left(4 a^{2} + 12 a + 19\right)\cdot 29^{28} + \left(26 a^{2} + 23 a + 2\right)\cdot 29^{29} + \left(3 a^{2} + 10 a + 3\right)\cdot 29^{30} + \left(15 a^{2} + 8 a + 20\right)\cdot 29^{31} + \left(10 a^{2} + 11 a + 22\right)\cdot 29^{32} + \left(4 a^{2} + 18 a + 21\right)\cdot 29^{33} + \left(25 a^{2} + 14 a + 15\right)\cdot 29^{34} + \left(3 a^{2} + 23 a + 21\right)\cdot 29^{35} + \left(10 a^{2} + 26 a + 5\right)\cdot 29^{36} + \left(27 a^{2} + 26 a + 23\right)\cdot 29^{37} + \left(13 a^{2} + 6 a + 9\right)\cdot 29^{38} + 15\cdot 29^{39} + \left(a + 15\right)\cdot 29^{40} + \left(2 a^{2} + 12 a + 6\right)\cdot 29^{41} + \left(14 a^{2} + 5 a + 11\right)\cdot 29^{42} + \left(20 a^{2} + 14 a + 16\right)\cdot 29^{43} + \left(24 a^{2} + 28 a + 16\right)\cdot 29^{44} + \left(12 a^{2} + 2 a + 26\right)\cdot 29^{45} + \left(23 a^{2} + 9 a + 26\right)\cdot 29^{46} + \left(23 a^{2} + 27 a + 10\right)\cdot 29^{47} + \left(20 a^{2} + 14 a + 19\right)\cdot 29^{48} + \left(10 a^{2} + 8 a + 8\right)\cdot 29^{49} + \left(8 a^{2} + 2 a + 7\right)\cdot 29^{50} + \left(5 a^{2} + 11 a\right)\cdot 29^{51} + \left(3 a^{2} + 2 a + 28\right)\cdot 29^{52} + \left(28 a^{2} + 4 a + 23\right)\cdot 29^{53} + \left(14 a^{2} + a + 5\right)\cdot 29^{54} + \left(7 a^{2} + 25 a + 14\right)\cdot 29^{55} + \left(26 a^{2} + 5 a + 20\right)\cdot 29^{56} + \left(17 a^{2} + 21 a + 20\right)\cdot 29^{57} + \left(21 a^{2} + 20 a + 28\right)\cdot 29^{58} + \left(25 a^{2} + 19 a\right)\cdot 29^{59} + \left(15 a^{2} + 12 a + 28\right)\cdot 29^{60} + \left(20 a^{2} + 10 a + 6\right)\cdot 29^{61} + \left(5 a^{2} + 23 a + 28\right)\cdot 29^{62} + \left(19 a^{2} + 21 a + 11\right)\cdot 29^{63} + \left(12 a^{2} + 24 a + 7\right)\cdot 29^{64} + \left(3 a^{2} + 7 a + 1\right)\cdot 29^{65} + \left(26 a^{2} + 9 a + 20\right)\cdot 29^{66} + \left(12 a^{2} + 5 a + 22\right)\cdot 29^{67} + \left(24 a^{2} + 22 a + 19\right)\cdot 29^{68} + \left(18 a^{2} + 23 a + 17\right)\cdot 29^{69} + \left(5 a^{2} + 26 a + 7\right)\cdot 29^{70} + \left(20 a^{2} + 4 a + 18\right)\cdot 29^{71} + \left(15 a^{2} + 8 a + 14\right)\cdot 29^{72} + \left(14 a^{2} + 11 a + 27\right)\cdot 29^{73} + \left(3 a^{2} + 3 a + 18\right)\cdot 29^{74} + \left(10 a^{2} + 26 a + 28\right)\cdot 29^{75} + \left(a^{2} + 21\right)\cdot 29^{76} + \left(27 a^{2} + 21 a + 10\right)\cdot 29^{77} + \left(22 a^{2} + 21 a + 4\right)\cdot 29^{78} + \left(14 a^{2} + 22 a + 19\right)\cdot 29^{79} + \left(27 a^{2} + 25 a + 20\right)\cdot 29^{80} + \left(2 a^{2} + 13 a + 3\right)\cdot 29^{81} + \left(23 a^{2} + 28 a + 25\right)\cdot 29^{82} + \left(2 a^{2} + 19\right)\cdot 29^{83} + \left(20 a + 11\right)\cdot 29^{84} + \left(23 a^{2} + 10 a + 25\right)\cdot 29^{85} + \left(24 a^{2} + 22 a + 5\right)\cdot 29^{86} + \left(25 a^{2} + 23 a + 1\right)\cdot 29^{87} + \left(15 a^{2} + 3 a + 7\right)\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 3 }$ $=$ $8 + 5\cdot 29 + 2\cdot 29^{3} + 20\cdot 29^{4} + 8\cdot 29^{5} + 20\cdot 29^{6} + 15\cdot 29^{7} + 19\cdot 29^{8} + 25\cdot 29^{9} + 12\cdot 29^{10} + 10\cdot 29^{11} + 5\cdot 29^{12} + 11\cdot 29^{13} + 20\cdot 29^{14} + 12\cdot 29^{15} + 20\cdot 29^{16} + 7\cdot 29^{17} + 6\cdot 29^{18} + 17\cdot 29^{20} + 17\cdot 29^{21} + 28\cdot 29^{22} + 27\cdot 29^{23} + 25\cdot 29^{24} + 22\cdot 29^{26} + 14\cdot 29^{27} + 17\cdot 29^{28} + 9\cdot 29^{29} + 6\cdot 29^{30} + 3\cdot 29^{32} + 10\cdot 29^{33} + 24\cdot 29^{34} + 8\cdot 29^{35} + 23\cdot 29^{36} + 10\cdot 29^{37} + 26\cdot 29^{38} + 13\cdot 29^{39} + 11\cdot 29^{40} + 17\cdot 29^{41} + 22\cdot 29^{42} + 3\cdot 29^{43} + 20\cdot 29^{44} + 13\cdot 29^{46} + 4\cdot 29^{47} + 25\cdot 29^{48} + 16\cdot 29^{49} + 11\cdot 29^{50} + 20\cdot 29^{51} + 15\cdot 29^{52} + 11\cdot 29^{53} + 13\cdot 29^{54} + 16\cdot 29^{55} + 14\cdot 29^{56} + 9\cdot 29^{57} + 13\cdot 29^{59} + 8\cdot 29^{60} + 3\cdot 29^{61} + 25\cdot 29^{62} + 11\cdot 29^{63} + 28\cdot 29^{64} + 9\cdot 29^{65} + 15\cdot 29^{66} + 12\cdot 29^{67} + 9\cdot 29^{68} + 22\cdot 29^{69} + 28\cdot 29^{70} + 25\cdot 29^{71} + 18\cdot 29^{72} + 4\cdot 29^{73} + 15\cdot 29^{74} + 12\cdot 29^{75} + 26\cdot 29^{76} + 17\cdot 29^{77} + 20\cdot 29^{78} + 29^{79} + 19\cdot 29^{80} + 17\cdot 29^{82} + 9\cdot 29^{83} + 23\cdot 29^{84} + 15\cdot 29^{85} + 23\cdot 29^{86} + 12\cdot 29^{87} + 13\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 4 }$ $=$ $7 a^{2} + 12 a + 16 + \left(27 a^{2} + 27 a + 15\right)\cdot 29 + \left(10 a + 2\right)\cdot 29^{2} + \left(23 a^{2} + 20 a\right)\cdot 29^{3} + \left(14 a^{2} + 27 a + 14\right)\cdot 29^{4} + \left(8 a^{2} + 8 a + 7\right)\cdot 29^{5} + \left(7 a^{2} + 27 a + 19\right)\cdot 29^{6} + \left(25 a^{2} + 22 a + 25\right)\cdot 29^{7} + \left(21 a^{2} + 21 a + 12\right)\cdot 29^{8} + \left(11 a^{2} + 3 a + 6\right)\cdot 29^{9} + \left(8 a^{2} + 24 a + 9\right)\cdot 29^{10} + \left(15 a^{2} + 20 a + 23\right)\cdot 29^{11} + \left(8 a^{2} + 19 a + 8\right)\cdot 29^{12} + \left(22 a^{2} + 4 a + 18\right)\cdot 29^{13} + \left(20 a^{2} + 8\right)\cdot 29^{14} + \left(2 a^{2} + 6 a + 11\right)\cdot 29^{15} + \left(11 a^{2} + 26 a + 16\right)\cdot 29^{16} + \left(12 a^{2} + 25 a + 28\right)\cdot 29^{17} + \left(21 a^{2} + 15 a + 12\right)\cdot 29^{18} + \left(3 a^{2} + 5 a + 17\right)\cdot 29^{19} + \left(4 a^{2} + 23 a + 27\right)\cdot 29^{20} + 17 a\cdot 29^{21} + \left(21 a^{2} + 19 a + 26\right)\cdot 29^{22} + \left(25 a^{2} + 25 a + 3\right)\cdot 29^{23} + \left(23 a^{2} + 10 a + 14\right)\cdot 29^{24} + \left(5 a^{2} + 17 a + 28\right)\cdot 29^{25} + \left(19 a^{2} + 24 a + 6\right)\cdot 29^{26} + \left(3 a^{2} + 17 a + 28\right)\cdot 29^{27} + \left(19 a^{2} + 28 a + 25\right)\cdot 29^{28} + \left(5 a^{2} + 27 a + 25\right)\cdot 29^{29} + \left(10 a^{2} + 14 a + 6\right)\cdot 29^{30} + \left(18 a^{2} + 26 a\right)\cdot 29^{31} + \left(15 a^{2} + 5 a + 17\right)\cdot 29^{32} + \left(13 a^{2} + 9 a + 25\right)\cdot 29^{33} + \left(a^{2} + 16 a + 15\right)\cdot 29^{34} + \left(13 a^{2} + 11 a + 22\right)\cdot 29^{35} + \left(24 a^{2} + 22 a + 5\right)\cdot 29^{36} + \left(25 a^{2} + 15 a + 26\right)\cdot 29^{37} + \left(3 a^{2} + 6\right)\cdot 29^{38} + \left(23 a^{2} + 19 a + 10\right)\cdot 29^{39} + \left(3 a^{2} + 16 a + 18\right)\cdot 29^{40} + \left(15 a^{2} + 21 a + 28\right)\cdot 29^{41} + \left(11 a^{2} + 12 a + 24\right)\cdot 29^{42} + \left(16 a^{2} + 21 a + 20\right)\cdot 29^{43} + \left(26 a^{2} + 12 a + 21\right)\cdot 29^{44} + \left(25 a^{2} + 13 a + 5\right)\cdot 29^{45} + \left(9 a^{2} + 4 a + 9\right)\cdot 29^{46} + \left(18 a^{2} + 7 a + 24\right)\cdot 29^{47} + \left(2 a^{2} + 3 a + 9\right)\cdot 29^{48} + \left(19 a^{2} + 13 a + 18\right)\cdot 29^{49} + \left(2 a^{2} + 8 a + 18\right)\cdot 29^{50} + \left(7 a^{2} + 5 a + 18\right)\cdot 29^{51} + \left(22 a^{2} + 24 a + 22\right)\cdot 29^{52} + \left(11 a^{2} + 15 a + 10\right)\cdot 29^{53} + \left(8 a^{2} + 22 a + 21\right)\cdot 29^{54} + \left(25 a^{2} + 19 a + 26\right)\cdot 29^{55} + \left(7 a^{2} + 24 a + 25\right)\cdot 29^{56} + \left(22 a^{2} + 12 a + 1\right)\cdot 29^{57} + \left(23 a^{2} + 8 a + 15\right)\cdot 29^{58} + \left(26 a^{2} + 15 a + 9\right)\cdot 29^{59} + \left(4 a^{2} + 26 a\right)\cdot 29^{60} + \left(6 a + 18\right)\cdot 29^{61} + \left(14 a^{2} + 26 a + 20\right)\cdot 29^{62} + \left(19 a^{2} + 5 a + 23\right)\cdot 29^{63} + \left(9 a^{2} + 7 a + 10\right)\cdot 29^{64} + \left(18 a + 15\right)\cdot 29^{65} + \left(2 a^{2} + 14 a + 16\right)\cdot 29^{66} + \left(27 a^{2} + 7 a + 20\right)\cdot 29^{67} + \left(a^{2} + 23 a + 1\right)\cdot 29^{68} + \left(12 a^{2} + 16 a\right)\cdot 29^{69} + \left(17 a^{2} + 6 a + 23\right)\cdot 29^{70} + \left(16 a^{2} + 15 a + 11\right)\cdot 29^{71} + \left(6 a^{2} + 14\right)\cdot 29^{72} + \left(15 a^{2} + 19 a + 12\right)\cdot 29^{73} + \left(8 a^{2} + 19 a + 10\right)\cdot 29^{74} + \left(13 a^{2} + 28 a + 12\right)\cdot 29^{75} + \left(6 a^{2} + 16 a + 25\right)\cdot 29^{76} + \left(2 a^{2} + 20 a + 25\right)\cdot 29^{77} + \left(23 a^{2} + 27 a + 22\right)\cdot 29^{78} + \left(5 a^{2} + 23 a + 15\right)\cdot 29^{79} + \left(5 a^{2} + 13 a + 15\right)\cdot 29^{80} + \left(a^{2} + 25 a + 12\right)\cdot 29^{81} + \left(a^{2} + 8 a + 2\right)\cdot 29^{82} + \left(27 a^{2} + 28 a + 16\right)\cdot 29^{83} + \left(23 a^{2} + 4 a + 22\right)\cdot 29^{84} + \left(5 a^{2} + 2 a + 25\right)\cdot 29^{85} + \left(a^{2} + 13 a + 17\right)\cdot 29^{86} + \left(20 a^{2} + 28 a + 19\right)\cdot 29^{87} + \left(9 a^{2} + 16 a + 18\right)\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 5 }$ $=$ $26 a^{2} + 8 a + 22 + \left(26 a^{2} + 28 a + 24\right)\cdot 29 + \left(27 a^{2} + 2 a + 28\right)\cdot 29^{2} + \left(28 a^{2} + 13 a + 7\right)\cdot 29^{3} + \left(7 a^{2} + 12 a + 24\right)\cdot 29^{4} + \left(28 a^{2} + 11 a + 4\right)\cdot 29^{5} + \left(7 a^{2} + 22 a + 20\right)\cdot 29^{6} + \left(6 a^{2} + 4 a + 19\right)\cdot 29^{7} + \left(5 a^{2} + 3 a + 19\right)\cdot 29^{8} + 25 a\cdot 29^{9} + \left(23 a^{2} + 28 a + 19\right)\cdot 29^{10} + \left(6 a^{2} + 21 a + 21\right)\cdot 29^{11} + \left(4 a^{2} + 7 a + 12\right)\cdot 29^{12} + \left(7 a^{2} + 26 a + 17\right)\cdot 29^{13} + \left(2 a^{2} + 23 a + 22\right)\cdot 29^{14} + \left(5 a^{2} + 3 a + 4\right)\cdot 29^{15} + \left(28 a^{2} + 22 a + 10\right)\cdot 29^{16} + \left(14 a^{2} + 5 a + 22\right)\cdot 29^{17} + \left(14 a + 23\right)\cdot 29^{18} + \left(3 a^{2} + 19 a + 6\right)\cdot 29^{19} + \left(17 a^{2} + 6\right)\cdot 29^{20} + \left(24 a^{2} + 17 a + 14\right)\cdot 29^{21} + \left(25 a + 18\right)\cdot 29^{22} + \left(27 a^{2} + 14 a + 5\right)\cdot 29^{23} + \left(3 a^{2} + 14 a + 26\right)\cdot 29^{24} + \left(13 a^{2} + 2 a + 18\right)\cdot 29^{25} + \left(12 a^{2} + 6 a + 7\right)\cdot 29^{26} + \left(23 a^{2} + 9 a + 6\right)\cdot 29^{27} + \left(11 a + 11\right)\cdot 29^{28} + \left(12 a^{2} + 3 a + 5\right)\cdot 29^{29} + \left(21 a^{2} + 13 a + 12\right)\cdot 29^{30} + \left(27 a^{2} + 17 a + 22\right)\cdot 29^{31} + \left(12 a + 16\right)\cdot 29^{32} + \left(11 a^{2} + 10 a + 12\right)\cdot 29^{33} + \left(10 a^{2} + 9 a + 8\right)\cdot 29^{34} + \left(11 a^{2} + 14 a + 20\right)\cdot 29^{35} + \left(13 a^{2} + 4 a\right)\cdot 29^{36} + \left(6 a^{2} + 18 a + 10\right)\cdot 29^{37} + \left(13 a^{2} + 12 a + 19\right)\cdot 29^{38} + \left(25 a^{2} + 16 a + 3\right)\cdot 29^{39} + \left(11 a^{2} + 27 a\right)\cdot 29^{40} + \left(a^{2} + 21 a + 20\right)\cdot 29^{41} + \left(3 a^{2} + 8 a + 13\right)\cdot 29^{42} + \left(24 a^{2} + 2 a + 21\right)\cdot 29^{43} + \left(4 a + 6\right)\cdot 29^{44} + \left(27 a^{2} + 16 a + 7\right)\cdot 29^{45} + \left(2 a^{2} + 4 a + 19\right)\cdot 29^{46} + \left(17 a^{2} + 24 a + 22\right)\cdot 29^{47} + \left(21 a^{2} + 23 a + 15\right)\cdot 29^{48} + \left(3 a^{2} + 9 a + 7\right)\cdot 29^{49} + \left(6 a^{2} + a + 23\right)\cdot 29^{50} + \left(11 a^{2} + 7 a + 4\right)\cdot 29^{51} + \left(11 a^{2} + 25 a + 8\right)\cdot 29^{52} + \left(17 a^{2} + 2 a + 18\right)\cdot 29^{53} + \left(9 a^{2} + 6 a + 3\right)\cdot 29^{54} + \left(6 a^{2} + 4 a + 11\right)\cdot 29^{55} + \left(28 a^{2} + 25 a + 14\right)\cdot 29^{56} + \left(4 a^{2} + 14 a + 17\right)\cdot 29^{57} + \left(28 a^{2} + 4 a + 1\right)\cdot 29^{58} + \left(27 a^{2} + 12 a + 11\right)\cdot 29^{59} + \left(10 a^{2} + 10 a + 8\right)\cdot 29^{60} + \left(22 a^{2} + 9 a + 28\right)\cdot 29^{61} + \left(5 a^{2} + 3 a + 28\right)\cdot 29^{62} + \left(28 a^{2} + 7 a + 15\right)\cdot 29^{63} + \left(28 a^{2} + 27 a + 7\right)\cdot 29^{64} + \left(11 a^{2} + 7 a + 21\right)\cdot 29^{65} + \left(5 a^{2} + 8 a + 1\right)\cdot 29^{66} + \left(15 a^{2} + 24\right)\cdot 29^{67} + \left(9 a^{2} + 9 a + 11\right)\cdot 29^{68} + \left(3 a^{2} + 21 a + 17\right)\cdot 29^{69} + \left(27 a^{2} + 16\right)\cdot 29^{70} + \left(4 a^{2} + 27 a + 15\right)\cdot 29^{71} + \left(26 a^{2} + 9 a + 11\right)\cdot 29^{72} + \left(a^{2} + 19 a + 4\right)\cdot 29^{73} + \left(20 a^{2} + 24 a + 16\right)\cdot 29^{74} + \left(14 a^{2} + 22 a + 4\right)\cdot 29^{75} + \left(22 a^{2} + 15 a + 8\right)\cdot 29^{76} + \left(13 a^{2} + 17 a + 12\right)\cdot 29^{77} + \left(a^{2} + 3 a + 13\right)\cdot 29^{78} + \left(10 a^{2} + 19 a + 21\right)\cdot 29^{79} + \left(15 a^{2} + 23 a + 9\right)\cdot 29^{80} + \left(4 a^{2} + 28 a + 7\right)\cdot 29^{81} + \left(6 a^{2} + 12 a + 9\right)\cdot 29^{82} + \left(3 a^{2} + 9 a + 13\right)\cdot 29^{83} + \left(6 a^{2} + a + 8\right)\cdot 29^{84} + \left(15 a^{2} + 21 a + 9\right)\cdot 29^{85} + \left(2 a^{2} + 18 a\right)\cdot 29^{86} + \left(4 a^{2} + 17 a + 8\right)\cdot 29^{87} + \left(19 a^{2} + 20 a + 2\right)\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 6 }$ $=$ $13 a^{2} + 21 a + 15 + \left(8 a^{2} + 10 a + 9\right)\cdot 29 + \left(9 a^{2} + 28 a + 12\right)\cdot 29^{2} + \left(11 a^{2} + 10 a + 14\right)\cdot 29^{3} + \left(20 a^{2} + 19 a + 20\right)\cdot 29^{4} + \left(24 a^{2} + 16 a + 10\right)\cdot 29^{5} + \left(22 a^{2} + 10 a + 4\right)\cdot 29^{6} + \left(4 a^{2} + 4 a + 1\right)\cdot 29^{7} + \left(27 a^{2} + 25 a + 20\right)\cdot 29^{8} + \left(18 a^{2} + a + 16\right)\cdot 29^{9} + \left(21 a^{2} + 20 a + 24\right)\cdot 29^{10} + \left(16 a^{2} + 5 a + 18\right)\cdot 29^{11} + \left(17 a^{2} + 7 a + 21\right)\cdot 29^{12} + \left(22 a^{2} + 14 a + 16\right)\cdot 29^{13} + \left(22 a^{2} + 17 a + 23\right)\cdot 29^{14} + \left(10 a^{2} + 9 a + 19\right)\cdot 29^{15} + \left(24 a + 22\right)\cdot 29^{16} + \left(27 a^{2} + 25 a + 23\right)\cdot 29^{17} + \left(7 a^{2} + 13 a + 27\right)\cdot 29^{18} + \left(16 a^{2} + 13 a + 11\right)\cdot 29^{19} + \left(17 a + 14\right)\cdot 29^{20} + \left(25 a^{2} + 19 a + 27\right)\cdot 29^{21} + \left(16 a^{2} + 25 a + 12\right)\cdot 29^{22} + \left(19 a^{2} + 28 a + 26\right)\cdot 29^{23} + \left(11 a^{2} + 6 a + 6\right)\cdot 29^{24} + \left(26 a^{2} + 10 a + 25\right)\cdot 29^{25} + \left(21 a^{2} + 5 a + 21\right)\cdot 29^{26} + \left(27 a^{2} + 10 a + 12\right)\cdot 29^{27} + \left(28 a^{2} + 18 a + 13\right)\cdot 29^{28} + \left(3 a^{2} + 28 a + 21\right)\cdot 29^{29} + \left(11 a^{2} + 12\right)\cdot 29^{30} + \left(15 a^{2} + 23 a + 20\right)\cdot 29^{31} + \left(11 a + 28\right)\cdot 29^{32} + \left(15 a^{2} + 2 a + 6\right)\cdot 29^{33} + \left(19 a^{2} + 24 a + 8\right)\cdot 29^{34} + \left(3 a^{2} + 15 a + 21\right)\cdot 29^{35} + \left(10 a^{2} + 28 a + 5\right)\cdot 29^{36} + \left(19 a^{2} + 26 a + 22\right)\cdot 29^{37} + \left(20 a^{2} + 18\right)\cdot 29^{38} + \left(a^{2} + 2 a + 26\right)\cdot 29^{39} + \left(4 a^{2} + 28 a + 20\right)\cdot 29^{40} + \left(27 a^{2} + 20 a + 20\right)\cdot 29^{41} + \left(5 a^{2} + 14 a + 19\right)\cdot 29^{42} + \left(17 a^{2} + 21 a + 21\right)\cdot 29^{43} + \left(12 a^{2} + 5 a + 19\right)\cdot 29^{44} + \left(3 a^{2} + 14 a + 23\right)\cdot 29^{45} + \left(12 a^{2} + 22 a + 11\right)\cdot 29^{46} + \left(11 a^{2} + 16 a + 23\right)\cdot 29^{47} + \left(25 a^{2} + 26 a + 15\right)\cdot 29^{48} + \left(22 a^{2} + 2 a + 5\right)\cdot 29^{49} + \left(2 a^{2} + 16 a + 19\right)\cdot 29^{50} + \left(23 a^{2} + 17 a + 4\right)\cdot 29^{51} + \left(19 a^{2} + 21\right)\cdot 29^{52} + \left(8 a^{2} + 14 a + 7\right)\cdot 29^{53} + \left(6 a^{2} + 24 a + 23\right)\cdot 29^{54} + \left(23 a^{2} + 5 a + 15\right)\cdot 29^{55} + \left(12 a^{2} + 21 a + 2\right)\cdot 29^{56} + \left(16 a^{2} + 3 a + 9\right)\cdot 29^{57} + \left(4 a^{2} + 6 a + 25\right)\cdot 29^{58} + \left(19 a^{2} + 27 a + 1\right)\cdot 29^{59} + \left(9 a^{2} + 26 a + 10\right)\cdot 29^{60} + \left(25 a^{2} + 2 a + 13\right)\cdot 29^{61} + \left(27 a^{2} + 12 a + 9\right)\cdot 29^{62} + \left(14 a^{2} + 15 a + 6\right)\cdot 29^{63} + \left(a^{2} + 17 a + 2\right)\cdot 29^{64} + \left(14 a^{2} + 7 a + 25\right)\cdot 29^{65} + \left(13 a^{2} + 13 a + 12\right)\cdot 29^{66} + \left(a^{2} + 27 a + 7\right)\cdot 29^{67} + \left(5 a^{2} + 10 a + 13\right)\cdot 29^{68} + \left(27 a + 2\right)\cdot 29^{69} + \left(27 a^{2} + 15 a + 7\right)\cdot 29^{70} + \left(8 a^{2} + 11 a + 3\right)\cdot 29^{71} + \left(21 a^{2} + 8 a + 22\right)\cdot 29^{72} + \left(19 a^{2} + 8 a + 24\right)\cdot 29^{73} + \left(5 a^{2} + 3 a + 21\right)\cdot 29^{74} + \left(4 a^{2} + 16 a + 20\right)\cdot 29^{75} + \left(3 a^{2} + 9 a + 14\right)\cdot 29^{76} + \left(14 a^{2} + 13 a + 22\right)\cdot 29^{77} + \left(25 a^{2} + 10 a + 7\right)\cdot 29^{78} + \left(3 a^{2} + 14\right)\cdot 29^{79} + \left(22 a^{2} + 22 a + 13\right)\cdot 29^{80} + \left(24 a^{2} + 8 a + 13\right)\cdot 29^{81} + \left(9 a^{2} + 19 a + 7\right)\cdot 29^{82} + \left(27 a^{2} + 26 a + 4\right)\cdot 29^{83} + \left(12 a^{2} + 22 a + 19\right)\cdot 29^{84} + \left(13 a^{2} + 13 a + 12\right)\cdot 29^{85} + \left(3 a^{2} + 10 a + 6\right)\cdot 29^{86} + \left(12 a^{2} + 6 a + 2\right)\cdot 29^{87} + \left(22 a^{2} + 6 a + 6\right)\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 7 }$ $=$ $10 + 4\cdot 29 + 25\cdot 29^{2} + 4\cdot 29^{3} + 17\cdot 29^{4} + 11\cdot 29^{5} + 24\cdot 29^{7} + 19\cdot 29^{8} + 27\cdot 29^{9} + 5\cdot 29^{10} + 20\cdot 29^{11} + 7\cdot 29^{12} + 5\cdot 29^{13} + 28\cdot 29^{14} + 5\cdot 29^{15} + 24\cdot 29^{16} + 21\cdot 29^{17} + 17\cdot 29^{18} + 20\cdot 29^{19} + 20\cdot 29^{20} + 26\cdot 29^{21} + 5\cdot 29^{22} + 4\cdot 29^{23} + 24\cdot 29^{24} + 24\cdot 29^{25} + 26\cdot 29^{26} + 16\cdot 29^{27} + 27\cdot 29^{28} + 2\cdot 29^{29} + 20\cdot 29^{30} + 14\cdot 29^{31} + 11\cdot 29^{32} + 6\cdot 29^{33} + 16\cdot 29^{34} + 13\cdot 29^{35} + 22\cdot 29^{36} + 24\cdot 29^{37} + 23\cdot 29^{38} + 3\cdot 29^{39} + 18\cdot 29^{40} + 3\cdot 29^{41} + 3\cdot 29^{43} + 12\cdot 29^{44} + 28\cdot 29^{45} + 11\cdot 29^{46} + 10\cdot 29^{48} + 21\cdot 29^{49} + 12\cdot 29^{50} + 29^{51} + 21\cdot 29^{52} + 14\cdot 29^{53} + 27\cdot 29^{54} + 20\cdot 29^{55} + 11\cdot 29^{56} + 25\cdot 29^{57} + 20\cdot 29^{58} + 20\cdot 29^{59} + 18\cdot 29^{60} + 4\cdot 29^{61} + 23\cdot 29^{62} + 6\cdot 29^{63} + 6\cdot 29^{64} + 13\cdot 29^{65} + 16\cdot 29^{66} + 17\cdot 29^{67} + 2\cdot 29^{68} + 19\cdot 29^{69} + 2\cdot 29^{71} + 12\cdot 29^{72} + 23\cdot 29^{73} + 2\cdot 29^{74} + 16\cdot 29^{75} + 7\cdot 29^{76} + 18\cdot 29^{77} + 23\cdot 29^{78} + 4\cdot 29^{79} + 3\cdot 29^{80} + 25\cdot 29^{81} + 25\cdot 29^{82} + 29^{83} + 28\cdot 29^{84} + 3\cdot 29^{85} + 9\cdot 29^{86} + 21\cdot 29^{87} + 11\cdot 29^{88} +O\left(29^{ 89 }\right)$ $r_{ 8 }$ $=$ $22 a^{2} + 18 a + 27 + \left(12 a^{2} + 10 a + 24\right)\cdot 29 + \left(23 a^{2} + 9 a + 11\right)\cdot 29^{2} + \left(7 a^{2} + 19 a + 19\right)\cdot 29^{3} + \left(27 a^{2} + 18 a\right)\cdot 29^{4} + \left(19 a^{2} + 22 a + 14\right)\cdot 29^{5} + \left(10 a^{2} + 7\right)\cdot 29^{6} + \left(5 a^{2} + 19 a + 21\right)\cdot 29^{7} + \left(20 a^{2} + 6 a + 10\right)\cdot 29^{8} + \left(18 a^{2} + 16 a + 16\right)\cdot 29^{9} + \left(2 a^{2} + 25 a + 18\right)\cdot 29^{10} + \left(4 a^{2} + 7 a + 11\right)\cdot 29^{11} + \left(6 a^{2} + 3 a + 6\right)\cdot 29^{12} + \left(7 a^{2} + 9 a + 25\right)\cdot 29^{13} + \left(25 a^{2} + a + 26\right)\cdot 29^{14} + 3 a^{2}29^{15} + \left(7 a^{2} + 14 a + 22\right)\cdot 29^{16} + \left(15 a^{2} + 9 a + 17\right)\cdot 29^{17} + \left(20 a^{2} + 28 a + 15\right)\cdot 29^{18} + \left(13 a^{2} + 2 a + 8\right)\cdot 29^{19} + \left(19 a^{2} + a + 20\right)\cdot 29^{20} + \left(23 a^{2} + 15 a + 25\right)\cdot 29^{21} + \left(11 a^{2} + 19 a + 15\right)\cdot 29^{22} + \left(28 a^{2} + 12 a + 28\right)\cdot 29^{23} + \left(24 a^{2} + 7 a + 14\right)\cdot 29^{24} + \left(28 a^{2} + 25 a + 28\right)\cdot 29^{25} + \left(17 a^{2} + 26 a + 6\right)\cdot 29^{26} + \left(22 a^{2} + 23 a + 25\right)\cdot 29^{27} + \left(24 a^{2} + 26 a + 7\right)\cdot 29^{28} + \left(27 a^{2} + 5 a + 24\right)\cdot 29^{29} + \left(13 a^{2} + 17 a + 6\right)\cdot 29^{30} + \left(27 a^{2} + 26 a + 17\right)\cdot 29^{31} + \left(17 a^{2} + 5 a + 3\right)\cdot 29^{32} + \left(9 a^{2} + 8 a + 19\right)\cdot 29^{33} + \left(13 a^{2} + 19 a + 9\right)\cdot 29^{34} + \left(21 a^{2} + 18 a + 6\right)\cdot 29^{35} + \left(8 a^{2} + 2 a + 23\right)\cdot 29^{36} + \left(11 a^{2} + 4 a + 1\right)\cdot 29^{37} + \left(23 a^{2} + 21 a + 3\right)\cdot 29^{38} + \left(26 a^{2} + 26 a + 2\right)\cdot 29^{39} + \left(24 a^{2} + 28 a + 10\right)\cdot 29^{40} + \left(28 a^{2} + 24 a + 13\right)\cdot 29^{41} + \left(8 a^{2} + 8 a + 4\right)\cdot 29^{42} + \left(20 a^{2} + 22 a + 16\right)\cdot 29^{43} + \left(20 a^{2} + 23 a + 1\right)\cdot 29^{44} + \left(12 a^{2} + 11 a + 7\right)\cdot 29^{45} + \left(22 a^{2} + 26 a + 6\right)\cdot 29^{46} + \left(22 a^{2} + 13 a + 19\right)\cdot 29^{47} + \left(11 a^{2} + 16 a + 26\right)\cdot 29^{48} + \left(24 a^{2} + 17 a + 26\right)\cdot 29^{49} + \left(17 a^{2} + 10 a + 19\right)\cdot 29^{50} + 3\cdot 29^{51} + \left(6 a^{2} + 26 a + 22\right)\cdot 29^{52} + \left(21 a^{2} + 10 a + 14\right)\cdot 29^{53} + \left(7 a^{2} + 3 a + 15\right)\cdot 29^{54} + \left(27 a^{2} + 27 a + 11\right)\cdot 29^{55} + \left(18 a^{2} + a + 20\right)\cdot 29^{56} + \left(23 a^{2} + 4 a + 18\right)\cdot 29^{57} + \left(2 a^{2} + 2 a + 3\right)\cdot 29^{58} + \left(13 a^{2} + 11 a + 13\right)\cdot 29^{59} + \left(3 a^{2} + 18 a + 11\right)\cdot 29^{60} + \left(12 a^{2} + 15 a + 5\right)\cdot 29^{61} + \left(24 a^{2} + 22 a + 24\right)\cdot 29^{62} + \left(23 a^{2} + 20 a + 27\right)\cdot 29^{63} + \left(14 a^{2} + 15 a + 19\right)\cdot 29^{64} + \left(11 a^{2} + 13 a + 21\right)\cdot 29^{65} + \left(18 a^{2} + 6 a + 9\right)\cdot 29^{66} + \left(14 a^{2} + 25 a + 15\right)\cdot 29^{67} + \left(28 a^{2} + 24 a + 15\right)\cdot 29^{68} + \left(9 a^{2} + 6 a + 15\right)\cdot 29^{69} + \left(25 a^{2} + 15 a + 14\right)\cdot 29^{70} + \left(28 a^{2} + 12 a + 10\right)\cdot 29^{71} + \left(20 a^{2} + 12 a + 2\right)\cdot 29^{72} + \left(23 a^{2} + 9 a + 1\right)\cdot 29^{73} + \left(19 a^{2} + 22 a + 2\right)\cdot 29^{74} + \left(14 a^{2} + 15 a + 25\right)\cdot 29^{75} + \left(24 a^{2} + 18 a + 23\right)\cdot 29^{76} + \left(16 a^{2} + 23 a + 6\right)\cdot 29^{77} + \left(9 a^{2} + 25 a + 25\right)\cdot 29^{78} + \left(10 a^{2} + 5 a + 22\right)\cdot 29^{79} + \left(8 a^{2} + 10 a + 4\right)\cdot 29^{80} + \left(a^{2} + 6 a + 11\right)\cdot 29^{81} + \left(25 a^{2} + 10 a + 8\right)\cdot 29^{82} + \left(27 a^{2} + a + 24\right)\cdot 29^{83} + \left(15 a^{2} + 15 a + 3\right)\cdot 29^{84} + \left(21 a^{2} + 4 a + 4\right)\cdot 29^{85} + \left(25 a + 22\right)\cdot 29^{86} + \left(20 a^{2} + 27 a + 12\right)\cdot 29^{87} + \left(19 a^{2} + 18 a + 2\right)\cdot 29^{88} +O\left(29^{ 89 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.