# Properties

 Label 9.729212883968.16t1294.a.a Dimension $9$ Group $S_4\wr C_2$ Conductor $729212883968$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $9$ Group: $S_4\wr C_2$ Conductor: $$729212883968$$$$\medspace = 2^{10} \cdot 19^{3} \cdot 47^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.2.94939073536.1 Galois orbit size: $1$ Smallest permutation container: 16T1294 Parity: odd Determinant: 1.3572.2t1.a.a Projective image: $S_4\wr C_2$ Projective field: Galois closure of 8.2.94939073536.1

## Defining polynomial

 $f(x)$ $=$ $x^{8} - 4 x^{7} + 2 x^{6} - 2 x^{5} + 37 x^{4} - 42 x^{3} - 62 x^{2} + 116 x - 44$.

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 106.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$

Roots:
 $r_{ 1 }$ $=$ $22 a + 18 + \left(a + 34\right)\cdot 37 + \left(24 a + 21\right)\cdot 37^{2} + 28\cdot 37^{3} + \left(17 a + 3\right)\cdot 37^{4} + \left(5 a + 27\right)\cdot 37^{5} + \left(23 a + 18\right)\cdot 37^{6} + \left(14 a + 5\right)\cdot 37^{7} + 34\cdot 37^{8} + \left(36 a + 5\right)\cdot 37^{9} + \left(29 a + 9\right)\cdot 37^{10} + \left(28 a + 10\right)\cdot 37^{11} + \left(36 a + 32\right)\cdot 37^{12} + 18 a\cdot 37^{13} + \left(6 a + 15\right)\cdot 37^{14} + \left(34 a + 26\right)\cdot 37^{15} + \left(2 a + 14\right)\cdot 37^{16} + \left(16 a + 23\right)\cdot 37^{17} + \left(7 a + 22\right)\cdot 37^{18} + \left(33 a + 17\right)\cdot 37^{19} + \left(a + 11\right)\cdot 37^{20} + \left(11 a + 31\right)\cdot 37^{21} + \left(7 a + 34\right)\cdot 37^{22} + \left(4 a + 20\right)\cdot 37^{23} + \left(30 a + 23\right)\cdot 37^{24} + \left(35 a + 6\right)\cdot 37^{25} + \left(a + 10\right)\cdot 37^{26} + \left(23 a + 35\right)\cdot 37^{27} + \left(15 a + 15\right)\cdot 37^{28} + \left(32 a + 30\right)\cdot 37^{29} + \left(34 a + 6\right)\cdot 37^{30} + \left(4 a + 8\right)\cdot 37^{31} + \left(6 a + 22\right)\cdot 37^{32} + \left(31 a + 19\right)\cdot 37^{33} + \left(3 a + 11\right)\cdot 37^{34} + \left(3 a + 19\right)\cdot 37^{35} + \left(4 a + 5\right)\cdot 37^{36} + \left(33 a + 18\right)\cdot 37^{37} + \left(23 a + 32\right)\cdot 37^{38} + 4 a\cdot 37^{39} + \left(2 a + 26\right)\cdot 37^{40} + \left(23 a + 31\right)\cdot 37^{41} + \left(15 a + 34\right)\cdot 37^{42} + \left(17 a + 2\right)\cdot 37^{43} + \left(9 a + 21\right)\cdot 37^{44} + \left(10 a + 33\right)\cdot 37^{45} + \left(9 a + 25\right)\cdot 37^{46} + \left(30 a + 23\right)\cdot 37^{47} + 26\cdot 37^{48} + \left(20 a + 16\right)\cdot 37^{49} + \left(20 a + 4\right)\cdot 37^{50} + \left(17 a + 31\right)\cdot 37^{51} + \left(30 a + 27\right)\cdot 37^{52} + 33\cdot 37^{53} + \left(12 a + 9\right)\cdot 37^{54} + 3 a\cdot 37^{55} + \left(18 a + 24\right)\cdot 37^{56} + \left(27 a + 30\right)\cdot 37^{57} + \left(21 a + 21\right)\cdot 37^{58} + \left(17 a + 25\right)\cdot 37^{59} + \left(4 a + 23\right)\cdot 37^{60} + \left(14 a + 17\right)\cdot 37^{61} + \left(5 a + 8\right)\cdot 37^{62} + \left(22 a + 19\right)\cdot 37^{63} + \left(22 a + 13\right)\cdot 37^{64} + \left(2 a + 25\right)\cdot 37^{65} + \left(35 a + 13\right)\cdot 37^{66} + \left(25 a + 18\right)\cdot 37^{67} + \left(35 a + 26\right)\cdot 37^{68} + \left(9 a + 3\right)\cdot 37^{69} + \left(16 a + 24\right)\cdot 37^{70} + \left(2 a + 15\right)\cdot 37^{71} + \left(5 a + 21\right)\cdot 37^{72} + \left(20 a + 6\right)\cdot 37^{73} + \left(18 a + 18\right)\cdot 37^{74} + \left(32 a + 26\right)\cdot 37^{75} + \left(11 a + 18\right)\cdot 37^{76} + \left(28 a + 13\right)\cdot 37^{77} + \left(26 a + 16\right)\cdot 37^{78} + \left(7 a + 7\right)\cdot 37^{79} + 2 a\cdot 37^{80} + \left(8 a + 31\right)\cdot 37^{81} + \left(34 a + 20\right)\cdot 37^{82} + \left(22 a + 25\right)\cdot 37^{83} + \left(36 a + 10\right)\cdot 37^{84} + \left(18 a + 34\right)\cdot 37^{85} + \left(16 a + 18\right)\cdot 37^{86} + \left(13 a + 26\right)\cdot 37^{87} + \left(33 a + 32\right)\cdot 37^{88} + \left(21 a + 25\right)\cdot 37^{89} + \left(25 a + 20\right)\cdot 37^{90} + \left(2 a + 12\right)\cdot 37^{91} + \left(29 a + 36\right)\cdot 37^{92} + 35 a\cdot 37^{93} + \left(6 a + 2\right)\cdot 37^{94} + \left(6 a + 36\right)\cdot 37^{95} + \left(5 a + 17\right)\cdot 37^{96} + \left(14 a + 10\right)\cdot 37^{97} + \left(24 a + 17\right)\cdot 37^{98} + \left(36 a + 9\right)\cdot 37^{99} + \left(2 a + 22\right)\cdot 37^{100} + \left(22 a + 32\right)\cdot 37^{101} + \left(29 a + 20\right)\cdot 37^{102} + \left(9 a + 29\right)\cdot 37^{103} + \left(28 a + 21\right)\cdot 37^{104} + \left(33 a + 32\right)\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 2 }$ $=$ $8 a + 34 + \left(17 a + 16\right)\cdot 37 + \left(29 a + 28\right)\cdot 37^{2} + \left(32 a + 23\right)\cdot 37^{3} + \left(10 a + 12\right)\cdot 37^{4} + 6 a\cdot 37^{5} + \left(30 a + 10\right)\cdot 37^{6} + \left(19 a + 26\right)\cdot 37^{7} + \left(2 a + 32\right)\cdot 37^{8} + \left(6 a + 3\right)\cdot 37^{9} + \left(8 a + 28\right)\cdot 37^{10} + \left(28 a + 5\right)\cdot 37^{11} + \left(35 a + 36\right)\cdot 37^{12} + \left(9 a + 32\right)\cdot 37^{13} + \left(21 a + 17\right)\cdot 37^{14} + \left(11 a + 25\right)\cdot 37^{15} + \left(6 a + 26\right)\cdot 37^{16} + \left(15 a + 29\right)\cdot 37^{17} + \left(10 a + 12\right)\cdot 37^{18} + \left(16 a + 3\right)\cdot 37^{19} + \left(9 a + 9\right)\cdot 37^{20} + \left(2 a + 3\right)\cdot 37^{21} + \left(28 a + 12\right)\cdot 37^{22} + \left(6 a + 30\right)\cdot 37^{23} + \left(29 a + 29\right)\cdot 37^{24} + \left(31 a + 35\right)\cdot 37^{25} + \left(15 a + 24\right)\cdot 37^{26} + \left(2 a + 33\right)\cdot 37^{27} + \left(3 a + 14\right)\cdot 37^{28} + \left(32 a + 16\right)\cdot 37^{29} + \left(3 a + 3\right)\cdot 37^{30} + \left(6 a + 26\right)\cdot 37^{31} + \left(25 a + 31\right)\cdot 37^{32} + \left(26 a + 9\right)\cdot 37^{33} + \left(19 a + 7\right)\cdot 37^{34} + 22\cdot 37^{35} + \left(14 a + 15\right)\cdot 37^{36} + \left(35 a + 20\right)\cdot 37^{37} + \left(4 a + 36\right)\cdot 37^{38} + \left(33 a + 11\right)\cdot 37^{39} + \left(29 a + 21\right)\cdot 37^{40} + \left(3 a + 4\right)\cdot 37^{41} + \left(20 a + 18\right)\cdot 37^{42} + \left(34 a + 3\right)\cdot 37^{43} + \left(17 a + 24\right)\cdot 37^{44} + \left(17 a + 35\right)\cdot 37^{45} + \left(32 a + 33\right)\cdot 37^{46} + \left(15 a + 15\right)\cdot 37^{47} + \left(22 a + 5\right)\cdot 37^{48} + \left(29 a + 25\right)\cdot 37^{49} + 7 a\cdot 37^{50} + \left(16 a + 8\right)\cdot 37^{51} + \left(24 a + 27\right)\cdot 37^{52} + \left(6 a + 15\right)\cdot 37^{53} + \left(9 a + 25\right)\cdot 37^{54} + \left(31 a + 33\right)\cdot 37^{55} + \left(13 a + 21\right)\cdot 37^{56} + \left(18 a + 4\right)\cdot 37^{57} + \left(27 a + 32\right)\cdot 37^{58} + \left(33 a + 25\right)\cdot 37^{59} + \left(16 a + 14\right)\cdot 37^{60} + \left(24 a + 8\right)\cdot 37^{61} + \left(16 a + 22\right)\cdot 37^{62} + \left(2 a + 16\right)\cdot 37^{63} + \left(3 a + 21\right)\cdot 37^{64} + \left(3 a + 31\right)\cdot 37^{65} + \left(2 a + 25\right)\cdot 37^{66} + \left(12 a + 16\right)\cdot 37^{67} + \left(32 a + 4\right)\cdot 37^{68} + \left(23 a + 18\right)\cdot 37^{69} + \left(11 a + 29\right)\cdot 37^{70} + \left(18 a + 30\right)\cdot 37^{71} + \left(22 a + 7\right)\cdot 37^{72} + \left(14 a + 30\right)\cdot 37^{73} + \left(21 a + 11\right)\cdot 37^{74} + \left(14 a + 10\right)\cdot 37^{75} + \left(3 a + 11\right)\cdot 37^{76} + \left(16 a + 16\right)\cdot 37^{77} + \left(30 a + 2\right)\cdot 37^{78} + \left(3 a + 35\right)\cdot 37^{79} + \left(35 a + 4\right)\cdot 37^{80} + \left(18 a + 26\right)\cdot 37^{81} + \left(5 a + 5\right)\cdot 37^{82} + \left(25 a + 9\right)\cdot 37^{83} + \left(8 a + 15\right)\cdot 37^{84} + \left(14 a + 14\right)\cdot 37^{85} + \left(21 a + 14\right)\cdot 37^{86} + \left(15 a + 8\right)\cdot 37^{87} + \left(28 a + 6\right)\cdot 37^{88} + \left(24 a + 4\right)\cdot 37^{89} + \left(20 a + 21\right)\cdot 37^{90} + \left(27 a + 5\right)\cdot 37^{91} + \left(14 a + 2\right)\cdot 37^{92} + \left(8 a + 25\right)\cdot 37^{93} + \left(31 a + 17\right)\cdot 37^{94} + \left(35 a + 28\right)\cdot 37^{95} + \left(26 a + 12\right)\cdot 37^{96} + \left(8 a + 15\right)\cdot 37^{97} + \left(11 a + 15\right)\cdot 37^{98} + \left(9 a + 27\right)\cdot 37^{99} + 12\cdot 37^{100} + \left(7 a + 3\right)\cdot 37^{101} + \left(28 a + 26\right)\cdot 37^{102} + \left(26 a + 18\right)\cdot 37^{103} + \left(33 a + 20\right)\cdot 37^{104} + 29 a\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 3 }$ $=$ $15 a + 32 + \left(35 a + 18\right)\cdot 37 + \left(12 a + 5\right)\cdot 37^{2} + \left(36 a + 7\right)\cdot 37^{3} + \left(19 a + 34\right)\cdot 37^{4} + \left(31 a + 31\right)\cdot 37^{5} + \left(13 a + 31\right)\cdot 37^{6} + \left(22 a + 3\right)\cdot 37^{7} + \left(36 a + 21\right)\cdot 37^{8} + 37^{9} + \left(7 a + 19\right)\cdot 37^{10} + \left(8 a + 21\right)\cdot 37^{11} + 2\cdot 37^{12} + \left(18 a + 3\right)\cdot 37^{13} + \left(30 a + 22\right)\cdot 37^{14} + \left(2 a + 8\right)\cdot 37^{15} + \left(34 a + 29\right)\cdot 37^{16} + \left(20 a + 10\right)\cdot 37^{17} + \left(29 a + 36\right)\cdot 37^{18} + \left(3 a + 31\right)\cdot 37^{19} + \left(35 a + 22\right)\cdot 37^{20} + \left(25 a + 36\right)\cdot 37^{21} + \left(29 a + 15\right)\cdot 37^{22} + \left(32 a + 30\right)\cdot 37^{23} + \left(6 a + 28\right)\cdot 37^{24} + \left(a + 8\right)\cdot 37^{25} + \left(35 a + 19\right)\cdot 37^{26} + \left(13 a + 14\right)\cdot 37^{27} + \left(21 a + 18\right)\cdot 37^{28} + \left(4 a + 33\right)\cdot 37^{29} + \left(2 a + 2\right)\cdot 37^{30} + \left(32 a + 30\right)\cdot 37^{31} + \left(30 a + 4\right)\cdot 37^{32} + \left(5 a + 27\right)\cdot 37^{33} + \left(33 a + 32\right)\cdot 37^{34} + \left(33 a + 27\right)\cdot 37^{35} + \left(32 a + 18\right)\cdot 37^{36} + \left(3 a + 35\right)\cdot 37^{37} + \left(13 a + 20\right)\cdot 37^{38} + \left(32 a + 32\right)\cdot 37^{39} + \left(34 a + 29\right)\cdot 37^{40} + \left(13 a + 10\right)\cdot 37^{41} + 21 a\cdot 37^{42} + \left(19 a + 20\right)\cdot 37^{43} + \left(27 a + 4\right)\cdot 37^{44} + \left(26 a + 28\right)\cdot 37^{45} + \left(27 a + 15\right)\cdot 37^{46} + \left(6 a + 24\right)\cdot 37^{47} + \left(36 a + 36\right)\cdot 37^{48} + \left(16 a + 21\right)\cdot 37^{49} + \left(16 a + 29\right)\cdot 37^{50} + \left(19 a + 6\right)\cdot 37^{51} + \left(6 a + 21\right)\cdot 37^{52} + \left(36 a + 6\right)\cdot 37^{53} + \left(24 a + 20\right)\cdot 37^{54} + \left(33 a + 1\right)\cdot 37^{55} + \left(18 a + 19\right)\cdot 37^{56} + \left(9 a + 11\right)\cdot 37^{57} + \left(15 a + 7\right)\cdot 37^{58} + 19 a\cdot 37^{59} + \left(32 a + 24\right)\cdot 37^{60} + \left(22 a + 32\right)\cdot 37^{61} + \left(31 a + 15\right)\cdot 37^{62} + \left(14 a + 28\right)\cdot 37^{63} + \left(14 a + 7\right)\cdot 37^{64} + \left(34 a + 13\right)\cdot 37^{65} + \left(a + 3\right)\cdot 37^{66} + \left(11 a + 13\right)\cdot 37^{67} + \left(a + 32\right)\cdot 37^{68} + \left(27 a + 7\right)\cdot 37^{69} + \left(20 a + 5\right)\cdot 37^{70} + \left(34 a + 9\right)\cdot 37^{71} + \left(31 a + 2\right)\cdot 37^{72} + \left(16 a + 8\right)\cdot 37^{73} + \left(18 a + 35\right)\cdot 37^{74} + \left(4 a + 26\right)\cdot 37^{75} + \left(25 a + 33\right)\cdot 37^{76} + \left(8 a + 3\right)\cdot 37^{77} + \left(10 a + 21\right)\cdot 37^{78} + \left(29 a + 11\right)\cdot 37^{79} + \left(34 a + 1\right)\cdot 37^{80} + \left(28 a + 24\right)\cdot 37^{81} + \left(2 a + 1\right)\cdot 37^{82} + \left(14 a + 9\right)\cdot 37^{83} + 23\cdot 37^{84} + \left(18 a + 36\right)\cdot 37^{85} + \left(20 a + 28\right)\cdot 37^{86} + \left(23 a + 26\right)\cdot 37^{87} + \left(3 a + 4\right)\cdot 37^{88} + \left(15 a + 6\right)\cdot 37^{89} + \left(11 a + 27\right)\cdot 37^{90} + \left(34 a + 34\right)\cdot 37^{91} + \left(7 a + 1\right)\cdot 37^{92} + \left(a + 4\right)\cdot 37^{93} + \left(30 a + 31\right)\cdot 37^{94} + \left(30 a + 16\right)\cdot 37^{95} + \left(31 a + 32\right)\cdot 37^{96} + \left(22 a + 24\right)\cdot 37^{97} + \left(12 a + 26\right)\cdot 37^{98} + 20\cdot 37^{99} + \left(34 a + 34\right)\cdot 37^{100} + \left(14 a + 6\right)\cdot 37^{101} + \left(7 a + 6\right)\cdot 37^{102} + \left(27 a + 2\right)\cdot 37^{103} + \left(8 a + 14\right)\cdot 37^{104} + \left(3 a + 28\right)\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 4 }$ $=$ $a + 14 + \left(23 a + 19\right)\cdot 37 + \left(17 a + 31\right)\cdot 37^{2} + \left(a + 6\right)\cdot 37^{3} + \left(34 a + 12\right)\cdot 37^{4} + \left(34 a + 29\right)\cdot 37^{5} + \left(3 a + 24\right)\cdot 37^{6} + \left(24 a + 26\right)\cdot 37^{7} + \left(7 a + 22\right)\cdot 37^{8} + \left(33 a + 7\right)\cdot 37^{9} + \left(3 a + 17\right)\cdot 37^{10} + \left(4 a + 26\right)\cdot 37^{11} + \left(18 a + 35\right)\cdot 37^{12} + \left(12 a + 36\right)\cdot 37^{13} + \left(19 a + 33\right)\cdot 37^{14} + \left(29 a + 7\right)\cdot 37^{15} + \left(23 a + 17\right)\cdot 37^{16} + \left(21 a + 25\right)\cdot 37^{17} + \left(25 a + 9\right)\cdot 37^{18} + \left(17 a + 36\right)\cdot 37^{19} + \left(6 a + 6\right)\cdot 37^{20} + \left(25 a + 27\right)\cdot 37^{21} + \left(31 a + 20\right)\cdot 37^{22} + \left(4 a + 8\right)\cdot 37^{23} + \left(28 a + 1\right)\cdot 37^{24} + \left(13 a + 36\right)\cdot 37^{25} + \left(25 a + 8\right)\cdot 37^{26} + \left(21 a + 12\right)\cdot 37^{27} + \left(25 a + 22\right)\cdot 37^{28} + \left(26 a + 18\right)\cdot 37^{29} + \left(5 a + 10\right)\cdot 37^{30} + \left(12 a + 33\right)\cdot 37^{31} + \left(29 a + 15\right)\cdot 37^{32} + 21 a\cdot 37^{33} + \left(31 a + 20\right)\cdot 37^{34} + \left(27 a + 11\right)\cdot 37^{35} + \left(17 a + 22\right)\cdot 37^{36} + \left(30 a + 11\right)\cdot 37^{37} + \left(13 a + 13\right)\cdot 37^{38} + \left(32 a + 35\right)\cdot 37^{39} + \left(35 a + 36\right)\cdot 37^{40} + \left(19 a + 15\right)\cdot 37^{41} + \left(3 a + 30\right)\cdot 37^{42} + \left(21 a + 31\right)\cdot 37^{43} + \left(33 a + 10\right)\cdot 37^{44} + \left(15 a + 9\right)\cdot 37^{45} + \left(4 a + 35\right)\cdot 37^{46} + \left(26 a + 20\right)\cdot 37^{47} + \left(8 a + 21\right)\cdot 37^{48} + \left(25 a + 10\right)\cdot 37^{49} + \left(9 a + 4\right)\cdot 37^{50} + \left(26 a + 13\right)\cdot 37^{51} + \left(34 a + 5\right)\cdot 37^{52} + \left(8 a + 14\right)\cdot 37^{53} + \left(5 a + 36\right)\cdot 37^{54} + \left(16 a + 20\right)\cdot 37^{55} + \left(8 a + 23\right)\cdot 37^{56} + \left(14 a + 3\right)\cdot 37^{57} + \left(18 a + 3\right)\cdot 37^{58} + \left(3 a + 31\right)\cdot 37^{59} + \left(23 a + 20\right)\cdot 37^{60} + \left(30 a + 36\right)\cdot 37^{61} + 9\cdot 37^{62} + \left(14 a + 35\right)\cdot 37^{63} + \left(26 a + 22\right)\cdot 37^{64} + \left(17 a + 18\right)\cdot 37^{65} + \left(5 a + 34\right)\cdot 37^{66} + \left(36 a + 7\right)\cdot 37^{67} + \left(29 a + 5\right)\cdot 37^{68} + \left(24 a + 25\right)\cdot 37^{69} + \left(5 a + 18\right)\cdot 37^{70} + \left(19 a + 29\right)\cdot 37^{71} + \left(6 a + 2\right)\cdot 37^{72} + \left(35 a + 9\right)\cdot 37^{73} + \left(24 a + 34\right)\cdot 37^{74} + \left(10 a + 1\right)\cdot 37^{75} + \left(12 a + 6\right)\cdot 37^{76} + 20 a\cdot 37^{77} + \left(31 a + 15\right)\cdot 37^{78} + \left(4 a + 17\right)\cdot 37^{79} + \left(34 a + 19\right)\cdot 37^{80} + \left(29 a + 15\right)\cdot 37^{81} + \left(12 a + 25\right)\cdot 37^{82} + \left(17 a + 4\right)\cdot 37^{83} + \left(3 a + 22\right)\cdot 37^{84} + \left(29 a + 19\right)\cdot 37^{85} + 7\cdot 37^{86} + \left(4 a + 21\right)\cdot 37^{87} + \left(32 a + 25\right)\cdot 37^{88} + \left(7 a + 20\right)\cdot 37^{89} + \left(6 a + 10\right)\cdot 37^{90} + \left(16 a + 3\right)\cdot 37^{91} + 8 a\cdot 37^{92} + \left(19 a + 7\right)\cdot 37^{93} + \left(6 a + 29\right)\cdot 37^{94} + \left(7 a + 13\right)\cdot 37^{95} + \left(2 a + 22\right)\cdot 37^{96} + \left(32 a + 32\right)\cdot 37^{97} + \left(12 a + 24\right)\cdot 37^{98} + \left(21 a + 5\right)\cdot 37^{99} + \left(22 a + 8\right)\cdot 37^{100} + \left(25 a + 20\right)\cdot 37^{101} + \left(3 a + 35\right)\cdot 37^{102} + 9\cdot 37^{103} + \left(6 a + 7\right)\cdot 37^{104} + \left(20 a + 5\right)\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 5 }$ $=$ $23 + 11\cdot 37 + 25\cdot 37^{2} + 4\cdot 37^{3} + 9\cdot 37^{4} + 34\cdot 37^{6} + 22\cdot 37^{7} + 23\cdot 37^{8} + 29\cdot 37^{9} + 37^{10} + 5\cdot 37^{11} + 37^{12} + 32\cdot 37^{13} + 25\cdot 37^{14} + 12\cdot 37^{15} + 8\cdot 37^{16} + 29\cdot 37^{17} + 17\cdot 37^{18} + 12\cdot 37^{19} + 37^{20} + 19\cdot 37^{21} + 34\cdot 37^{22} + 24\cdot 37^{23} + 21\cdot 37^{24} + 28\cdot 37^{25} + 32\cdot 37^{26} + 30\cdot 37^{27} + 21\cdot 37^{28} + 14\cdot 37^{29} + 2\cdot 37^{30} + 26\cdot 37^{31} + 24\cdot 37^{32} + 5\cdot 37^{33} + 28\cdot 37^{34} + 15\cdot 37^{35} + 16\cdot 37^{37} + 14\cdot 37^{38} + 13\cdot 37^{39} + 33\cdot 37^{40} + 6\cdot 37^{41} + 11\cdot 37^{42} + 16\cdot 37^{43} + 24\cdot 37^{44} + 9\cdot 37^{46} + 18\cdot 37^{47} + 17\cdot 37^{48} + 17\cdot 37^{49} + 21\cdot 37^{50} + 22\cdot 37^{51} + 27\cdot 37^{52} + 7\cdot 37^{53} + 11\cdot 37^{54} + 12\cdot 37^{55} + 20\cdot 37^{56} + 6\cdot 37^{57} + 20\cdot 37^{58} + 29\cdot 37^{59} + 2\cdot 37^{60} + 30\cdot 37^{61} + 31\cdot 37^{62} + 12\cdot 37^{63} + 11\cdot 37^{64} + 5\cdot 37^{65} + 16\cdot 37^{66} + 3\cdot 37^{67} + 22\cdot 37^{68} + 27\cdot 37^{69} + 10\cdot 37^{70} + 27\cdot 37^{71} + 6\cdot 37^{72} + 30\cdot 37^{73} + 25\cdot 37^{74} + 2\cdot 37^{75} + 27\cdot 37^{76} + 35\cdot 37^{77} + 29\cdot 37^{78} + 31\cdot 37^{79} + 15\cdot 37^{80} + 34\cdot 37^{81} + 10\cdot 37^{82} + 30\cdot 37^{83} + 12\cdot 37^{84} + 11\cdot 37^{85} + 12\cdot 37^{86} + 29\cdot 37^{87} + 4\cdot 37^{88} + 33\cdot 37^{89} + 37^{90} + 20\cdot 37^{91} + 23\cdot 37^{92} + 16\cdot 37^{93} + 5\cdot 37^{94} + 17\cdot 37^{95} + 9\cdot 37^{96} + 14\cdot 37^{97} + 5\cdot 37^{98} + 30\cdot 37^{99} + 10\cdot 37^{100} + 32\cdot 37^{101} + 8\cdot 37^{102} + 10\cdot 37^{103} + 31\cdot 37^{104} + 6\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 6 }$ $=$ $21 + 6\cdot 37 + 12\cdot 37^{2} + 30\cdot 37^{3} + 16\cdot 37^{4} + 20\cdot 37^{5} + 9\cdot 37^{6} + 16\cdot 37^{7} + 35\cdot 37^{8} + 14\cdot 37^{9} + 18\cdot 37^{10} + 3\cdot 37^{11} + 7\cdot 37^{12} + 10\cdot 37^{13} + 26\cdot 37^{14} + 20\cdot 37^{15} + 2\cdot 37^{16} + 5\cdot 37^{17} + 30\cdot 37^{18} + 17\cdot 37^{19} + 13\cdot 37^{20} + 17\cdot 37^{21} + 7\cdot 37^{22} + 7\cdot 37^{23} + 16\cdot 37^{24} + 20\cdot 37^{25} + 9\cdot 37^{26} + 31\cdot 37^{27} + 15\cdot 37^{29} + 17\cdot 37^{30} + 12\cdot 37^{31} + 23\cdot 37^{32} + 9\cdot 37^{33} + 12\cdot 37^{34} + 29\cdot 37^{35} + 22\cdot 37^{36} + 4\cdot 37^{37} + 8\cdot 37^{38} + 22\cdot 37^{39} + 3\cdot 37^{40} + 28\cdot 37^{41} + 7\cdot 37^{42} + 24\cdot 37^{43} + 25\cdot 37^{44} + 24\cdot 37^{45} + 29\cdot 37^{46} + 24\cdot 37^{47} + 4\cdot 37^{48} + 17\cdot 37^{49} + 30\cdot 37^{50} + 3\cdot 37^{51} + 34\cdot 37^{52} + 36\cdot 37^{53} + 14\cdot 37^{54} + 34\cdot 37^{55} + 25\cdot 37^{56} + 11\cdot 37^{57} + 25\cdot 37^{58} + 23\cdot 37^{59} + 14\cdot 37^{60} + 19\cdot 37^{61} + 12\cdot 37^{62} + 9\cdot 37^{63} + 36\cdot 37^{64} + 23\cdot 37^{65} + 21\cdot 37^{66} + 26\cdot 37^{67} + 31\cdot 37^{68} + 28\cdot 37^{70} + 27\cdot 37^{71} + 17\cdot 37^{72} + 2\cdot 37^{73} + 26\cdot 37^{74} + 12\cdot 37^{75} + 33\cdot 37^{76} + 5\cdot 37^{77} + 19\cdot 37^{78} + 19\cdot 37^{79} + 35\cdot 37^{80} + 33\cdot 37^{81} + 27\cdot 37^{82} + 14\cdot 37^{83} + 20\cdot 37^{84} + 21\cdot 37^{85} + 35\cdot 37^{86} + 23\cdot 37^{87} + 4\cdot 37^{88} + 34\cdot 37^{90} + 25\cdot 37^{91} + 32\cdot 37^{92} + 11\cdot 37^{93} + 3\cdot 37^{94} + 7\cdot 37^{95} + 18\cdot 37^{96} + 16\cdot 37^{97} + 36\cdot 37^{98} + 33\cdot 37^{99} + 14\cdot 37^{100} + 32\cdot 37^{101} + 4\cdot 37^{102} + 10\cdot 37^{103} + 4\cdot 37^{104} + 19\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 7 }$ $=$ $36 a + 18 + \left(13 a + 36\right)\cdot 37 + \left(19 a + 4\right)\cdot 37^{2} + \left(35 a + 32\right)\cdot 37^{3} + \left(2 a + 35\right)\cdot 37^{4} + \left(2 a + 23\right)\cdot 37^{5} + \left(33 a + 5\right)\cdot 37^{6} + \left(12 a + 8\right)\cdot 37^{7} + \left(29 a + 29\right)\cdot 37^{8} + \left(3 a + 21\right)\cdot 37^{9} + \left(33 a + 36\right)\cdot 37^{10} + \left(32 a + 1\right)\cdot 37^{11} + \left(18 a + 30\right)\cdot 37^{12} + \left(24 a + 31\right)\cdot 37^{13} + \left(17 a + 24\right)\cdot 37^{14} + \left(7 a + 32\right)\cdot 37^{15} + \left(13 a + 8\right)\cdot 37^{16} + \left(15 a + 14\right)\cdot 37^{17} + \left(11 a + 16\right)\cdot 37^{18} + \left(19 a + 7\right)\cdot 37^{19} + \left(30 a + 15\right)\cdot 37^{20} + \left(11 a + 10\right)\cdot 37^{21} + \left(5 a + 11\right)\cdot 37^{22} + \left(32 a + 33\right)\cdot 37^{23} + \left(8 a + 34\right)\cdot 37^{24} + \left(23 a + 25\right)\cdot 37^{25} + \left(11 a + 22\right)\cdot 37^{26} + \left(15 a + 36\right)\cdot 37^{27} + \left(11 a + 28\right)\cdot 37^{28} + \left(10 a + 25\right)\cdot 37^{29} + \left(31 a + 6\right)\cdot 37^{30} + \left(24 a + 2\right)\cdot 37^{31} + \left(7 a + 10\right)\cdot 37^{32} + \left(15 a + 21\right)\cdot 37^{33} + \left(5 a + 13\right)\cdot 37^{34} + \left(9 a + 17\right)\cdot 37^{35} + \left(19 a + 28\right)\cdot 37^{36} + \left(6 a + 4\right)\cdot 37^{37} + \left(23 a + 1\right)\cdot 37^{38} + \left(4 a + 3\right)\cdot 37^{39} + a\cdot 37^{40} + \left(17 a + 23\right)\cdot 37^{41} + \left(33 a + 24\right)\cdot 37^{42} + \left(15 a + 1\right)\cdot 37^{43} + \left(3 a + 13\right)\cdot 37^{44} + \left(21 a + 2\right)\cdot 37^{45} + 32 a\cdot 37^{46} + \left(10 a + 10\right)\cdot 37^{47} + \left(28 a + 30\right)\cdot 37^{48} + \left(11 a + 28\right)\cdot 37^{49} + \left(27 a + 17\right)\cdot 37^{50} + \left(10 a + 34\right)\cdot 37^{51} + \left(2 a + 6\right)\cdot 37^{52} + \left(28 a + 15\right)\cdot 37^{53} + \left(31 a + 11\right)\cdot 37^{54} + \left(20 a + 6\right)\cdot 37^{55} + \left(28 a + 4\right)\cdot 37^{56} + \left(22 a + 15\right)\cdot 37^{57} + \left(18 a + 25\right)\cdot 37^{58} + \left(33 a + 26\right)\cdot 37^{59} + \left(13 a + 35\right)\cdot 37^{60} + \left(6 a + 24\right)\cdot 37^{61} + \left(36 a + 19\right)\cdot 37^{62} + \left(22 a + 16\right)\cdot 37^{63} + \left(10 a + 3\right)\cdot 37^{64} + \left(19 a + 26\right)\cdot 37^{65} + \left(31 a + 1\right)\cdot 37^{66} + 36\cdot 37^{67} + \left(7 a + 14\right)\cdot 37^{68} + \left(12 a + 20\right)\cdot 37^{69} + \left(31 a + 16\right)\cdot 37^{70} + \left(17 a + 26\right)\cdot 37^{71} + \left(30 a + 9\right)\cdot 37^{72} + \left(a + 32\right)\cdot 37^{73} + \left(12 a + 24\right)\cdot 37^{74} + \left(26 a + 19\right)\cdot 37^{75} + \left(24 a + 7\right)\cdot 37^{76} + \left(16 a + 32\right)\cdot 37^{77} + \left(5 a + 9\right)\cdot 37^{78} + \left(32 a + 5\right)\cdot 37^{79} + \left(2 a + 3\right)\cdot 37^{80} + \left(7 a + 27\right)\cdot 37^{81} + \left(24 a + 9\right)\cdot 37^{82} + \left(19 a + 24\right)\cdot 37^{83} + \left(33 a + 18\right)\cdot 37^{84} + \left(7 a + 21\right)\cdot 37^{85} + \left(36 a + 18\right)\cdot 37^{86} + \left(32 a + 36\right)\cdot 37^{87} + \left(4 a + 1\right)\cdot 37^{88} + \left(29 a + 20\right)\cdot 37^{89} + \left(30 a + 27\right)\cdot 37^{90} + \left(20 a + 24\right)\cdot 37^{91} + \left(28 a + 17\right)\cdot 37^{92} + \left(17 a + 1\right)\cdot 37^{93} + \left(30 a + 36\right)\cdot 37^{94} + \left(29 a + 35\right)\cdot 37^{95} + \left(34 a + 23\right)\cdot 37^{96} + \left(4 a + 10\right)\cdot 37^{97} + \left(24 a + 7\right)\cdot 37^{98} + \left(15 a + 4\right)\cdot 37^{99} + \left(14 a + 3\right)\cdot 37^{100} + \left(11 a + 26\right)\cdot 37^{101} + \left(33 a + 24\right)\cdot 37^{102} + \left(36 a + 6\right)\cdot 37^{103} + \left(30 a + 31\right)\cdot 37^{104} + \left(16 a + 5\right)\cdot 37^{105} +O\left(37^{ 106 }\right)$ $r_{ 8 }$ $=$ $29 a + 29 + \left(19 a + 3\right)\cdot 37 + \left(7 a + 18\right)\cdot 37^{2} + \left(4 a + 14\right)\cdot 37^{3} + \left(26 a + 23\right)\cdot 37^{4} + \left(30 a + 14\right)\cdot 37^{5} + \left(6 a + 13\right)\cdot 37^{6} + \left(17 a + 1\right)\cdot 37^{7} + \left(34 a + 23\right)\cdot 37^{8} + \left(30 a + 25\right)\cdot 37^{9} + \left(28 a + 17\right)\cdot 37^{10} + \left(8 a + 36\right)\cdot 37^{11} + \left(a + 2\right)\cdot 37^{12} + 27 a\cdot 37^{13} + \left(15 a + 19\right)\cdot 37^{14} + \left(25 a + 13\right)\cdot 37^{15} + \left(30 a + 3\right)\cdot 37^{16} + \left(21 a + 10\right)\cdot 37^{17} + \left(26 a + 2\right)\cdot 37^{18} + \left(20 a + 21\right)\cdot 37^{19} + \left(27 a + 30\right)\cdot 37^{20} + \left(34 a + 2\right)\cdot 37^{21} + \left(8 a + 11\right)\cdot 37^{22} + \left(30 a + 29\right)\cdot 37^{23} + \left(7 a + 28\right)\cdot 37^{24} + \left(5 a + 22\right)\cdot 37^{25} + \left(21 a + 19\right)\cdot 37^{26} + \left(34 a + 27\right)\cdot 37^{27} + \left(33 a + 24\right)\cdot 37^{28} + \left(4 a + 30\right)\cdot 37^{29} + \left(33 a + 23\right)\cdot 37^{30} + \left(30 a + 9\right)\cdot 37^{31} + \left(11 a + 15\right)\cdot 37^{32} + \left(10 a + 17\right)\cdot 37^{33} + \left(17 a + 22\right)\cdot 37^{34} + \left(36 a + 4\right)\cdot 37^{35} + \left(22 a + 34\right)\cdot 37^{36} + \left(a + 36\right)\cdot 37^{37} + \left(32 a + 20\right)\cdot 37^{38} + \left(3 a + 28\right)\cdot 37^{39} + \left(7 a + 33\right)\cdot 37^{40} + \left(33 a + 26\right)\cdot 37^{41} + \left(16 a + 20\right)\cdot 37^{42} + \left(2 a + 10\right)\cdot 37^{43} + \left(19 a + 24\right)\cdot 37^{44} + \left(19 a + 13\right)\cdot 37^{45} + \left(4 a + 35\right)\cdot 37^{46} + \left(21 a + 9\right)\cdot 37^{47} + \left(14 a + 5\right)\cdot 37^{48} + \left(7 a + 10\right)\cdot 37^{49} + \left(29 a + 2\right)\cdot 37^{50} + \left(20 a + 28\right)\cdot 37^{51} + \left(12 a + 34\right)\cdot 37^{52} + \left(30 a + 17\right)\cdot 37^{53} + \left(27 a + 18\right)\cdot 37^{54} + \left(5 a + 1\right)\cdot 37^{55} + \left(23 a + 9\right)\cdot 37^{56} + \left(18 a + 27\right)\cdot 37^{57} + \left(9 a + 12\right)\cdot 37^{58} + \left(3 a + 22\right)\cdot 37^{59} + \left(20 a + 11\right)\cdot 37^{60} + \left(12 a + 15\right)\cdot 37^{61} + \left(20 a + 27\right)\cdot 37^{62} + \left(34 a + 9\right)\cdot 37^{63} + \left(33 a + 31\right)\cdot 37^{64} + \left(33 a + 3\right)\cdot 37^{65} + \left(34 a + 31\right)\cdot 37^{66} + \left(24 a + 25\right)\cdot 37^{67} + \left(4 a + 10\right)\cdot 37^{68} + \left(13 a + 7\right)\cdot 37^{69} + \left(25 a + 15\right)\cdot 37^{70} + \left(18 a + 18\right)\cdot 37^{71} + \left(14 a + 5\right)\cdot 37^{72} + \left(22 a + 29\right)\cdot 37^{73} + \left(15 a + 8\right)\cdot 37^{74} + \left(22 a + 10\right)\cdot 37^{75} + \left(33 a + 10\right)\cdot 37^{76} + \left(20 a + 3\right)\cdot 37^{77} + \left(6 a + 34\right)\cdot 37^{78} + \left(33 a + 19\right)\cdot 37^{79} + \left(a + 30\right)\cdot 37^{80} + \left(18 a + 29\right)\cdot 37^{81} + \left(31 a + 8\right)\cdot 37^{82} + \left(11 a + 30\right)\cdot 37^{83} + \left(28 a + 24\right)\cdot 37^{84} + \left(22 a + 25\right)\cdot 37^{85} + \left(15 a + 11\right)\cdot 37^{86} + \left(21 a + 12\right)\cdot 37^{87} + \left(8 a + 30\right)\cdot 37^{88} + 12 a\cdot 37^{89} + \left(16 a + 5\right)\cdot 37^{90} + \left(9 a + 21\right)\cdot 37^{91} + \left(22 a + 33\right)\cdot 37^{92} + \left(28 a + 6\right)\cdot 37^{93} + \left(5 a + 23\right)\cdot 37^{94} + \left(a + 29\right)\cdot 37^{95} + \left(10 a + 10\right)\cdot 37^{96} + \left(28 a + 23\right)\cdot 37^{97} + \left(25 a + 14\right)\cdot 37^{98} + \left(27 a + 16\right)\cdot 37^{99} + \left(36 a + 4\right)\cdot 37^{100} + \left(29 a + 31\right)\cdot 37^{101} + \left(8 a + 20\right)\cdot 37^{102} + \left(10 a + 23\right)\cdot 37^{103} + \left(3 a + 17\right)\cdot 37^{104} + \left(7 a + 12\right)\cdot 37^{105} +O\left(37^{ 106 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $9$ $6$ $2$ $(1,3)(2,8)$ $-3$ $9$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $1$ $12$ $2$ $(1,2)$ $3$ $24$ $2$ $(1,4)(2,5)(3,6)(7,8)$ $3$ $36$ $2$ $(1,2)(4,5)$ $1$ $36$ $2$ $(1,2)(4,6)(5,7)$ $-1$ $16$ $3$ $(1,3,8)$ $0$ $64$ $3$ $(1,3,8)(4,6,7)$ $0$ $12$ $4$ $(1,2,3,8)$ $-3$ $36$ $4$ $(1,2,3,8)(4,5,6,7)$ $1$ $36$ $4$ $(1,3)(2,8)(4,5,6,7)$ $1$ $72$ $4$ $(1,6,3,4)(2,7,8,5)$ $-1$ $72$ $4$ $(1,2)(4,5,6,7)$ $-1$ $144$ $4$ $(1,5,2,4)(3,6)(7,8)$ $1$ $48$ $6$ $(1,8,3)(4,6)(5,7)$ $0$ $96$ $6$ $(1,2)(5,7,6)$ $0$ $192$ $6$ $(1,6,3,7,8,4)(2,5)$ $0$ $144$ $8$ $(1,5,2,6,3,7,8,4)$ $-1$ $96$ $12$ $(1,3,8)(4,5,6,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.