↑

Properties

Label 12.2e10_7e6_11e11.18t218.1c1
Dimension 12
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $ 2^{10} \cdot 7^{6} \cdot 11^{11}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$12$
Group:$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor:$34372231921370663936= 2^{10} \cdot 7^{6} \cdot 11^{11} $
Artin number field: Splitting field of $f= x^{9} - 3 x^{8} + 4 x^{7} - 12 x^{6} + 26 x^{5} - 34 x^{4} + 44 x^{3} - 44 x^{2} + 33 x - 11 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 18T218
Parity: Odd
Determinant: 1.11.2t1.1c1

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(3,4,8)$
$(1,5,6)$
$(1,3,5,4)(6,8)$
$(1,2,3)(4,5,7)(6,9,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$12$
$27$$2$$(1,5)(2,7)$$0$
$54$$2$$(1,2)(3,4)(5,7)(6,9)$$2$
$6$$3$$(2,7,9)$$0$
$8$$3$$(1,5,6)(2,7,9)(3,4,8)$$3$
$12$$3$$(1,5,6)(2,7,9)$$-3$
$72$$3$$(1,2,3)(4,5,7)(6,9,8)$$0$
$54$$4$$(1,2,5,7)(6,9)$$0$
$54$$6$$(1,5)(2,9,7)(3,4)$$0$
$108$$6$$(1,2,6,9,5,7)(3,4)$$-1$
$72$$9$$(1,2,4,5,7,8,6,9,3)$$0$
$72$$9$$(1,2,8,6,9,4,5,7,3)$$0$
$54$$12$$(1,3,5,4)(2,7,9)(6,8)$$0$
$54$$12$$(1,3,5,4)(2,9,7)(6,8)$$0$
The blue line marks the conjugacy class containing complex conjugation.