# Properties

 Label 9.592704000000.12t165.a.a Dimension 9 Group $(A_4\wr C_2):C_2$ Conductor $2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $9$ Group: $(A_4\wr C_2):C_2$ Conductor: $592704000000= 2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}$ Artin number field: Splitting field of 8.4.1244678400.2 defined by $f= x^{8} - 4 x^{7} + 8 x^{5} + 9 x^{4} - 10 x^{3} - 10 x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 12T165 Parity: Even Determinant: 1.21.2t1.a.a Projective image: $(A_4\wr C_2):C_2$ Projective field: Galois closure of 8.4.1244678400.2

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $9$ $6$ $2$ $(2,7)(4,8)$ $-3$ $9$ $2$ $(1,5)(2,7)(3,6)(4,8)$ $1$ $12$ $2$ $(1,8)(2,3)(4,5)(6,7)$ $3$ $12$ $2$ $(1,2)(3,4)(5,7)(6,8)$ $3$ $36$ $2$ $(3,5)(7,8)$ $1$ $16$ $3$ $(1,6,3)$ $0$ $32$ $3$ $(1,6,3)(2,8,7)$ $0$ $32$ $3$ $(1,6,3)(2,8,4)$ $0$ $36$ $4$ $(1,8,5,4)(2,6,7,3)$ $-1$ $36$ $4$ $(1,6,5,3)(2,4,7,8)$ $1$ $36$ $4$ $(1,2,5,7)(3,4,6,8)$ $-1$ $72$ $4$ $(2,8,7,4)(5,6)$ $-1$ $48$ $6$ $(1,6,3)(2,7)(4,8)$ $0$ $96$ $6$ $(1,7,6,2,3,8)(4,5)$ $0$ $96$ $6$ $(1,8,6,4,3,2)(5,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.