Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 44.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{3} + 6 x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a^{2} + a + 1 + \left(2 a^{2} + 5 a + 23\right)\cdot 37 + \left(23 a^{2} + 32 a + 15\right)\cdot 37^{2} + \left(29 a^{2} + 17 a + 22\right)\cdot 37^{3} + \left(21 a^{2} + 29 a + 15\right)\cdot 37^{4} + \left(19 a^{2} + 32 a + 29\right)\cdot 37^{5} + \left(8 a^{2} + 24 a + 6\right)\cdot 37^{6} + \left(16 a^{2} + 35 a + 28\right)\cdot 37^{7} + \left(29 a^{2} + 11 a + 21\right)\cdot 37^{8} + \left(2 a^{2} + 26 a + 21\right)\cdot 37^{9} + \left(20 a^{2} + 28\right)\cdot 37^{10} + \left(24 a^{2} + 17 a + 27\right)\cdot 37^{11} + \left(21 a^{2} + 5 a + 34\right)\cdot 37^{12} + \left(14 a^{2} + 14 a + 14\right)\cdot 37^{13} + \left(15 a^{2} + 19 a + 14\right)\cdot 37^{14} + \left(13 a^{2} + 33 a + 5\right)\cdot 37^{15} + \left(a^{2} + 28 a + 8\right)\cdot 37^{16} + \left(36 a + 12\right)\cdot 37^{17} + \left(5 a^{2} + 16 a + 3\right)\cdot 37^{18} + \left(16 a^{2} + 29 a + 15\right)\cdot 37^{19} + \left(3 a^{2} + 21 a + 24\right)\cdot 37^{20} + \left(10 a^{2} + 22 a + 15\right)\cdot 37^{21} + \left(19 a^{2} + 27 a + 6\right)\cdot 37^{22} + \left(12 a^{2} + 14 a + 23\right)\cdot 37^{23} + \left(2 a^{2} + 5\right)\cdot 37^{24} + \left(32 a^{2} + 12 a + 17\right)\cdot 37^{25} + \left(23 a + 9\right)\cdot 37^{26} + \left(4 a^{2} + 36 a + 28\right)\cdot 37^{27} + \left(15 a^{2} + 26 a + 33\right)\cdot 37^{28} + \left(16 a^{2} + 21\right)\cdot 37^{29} + \left(26 a^{2} + 14 a + 34\right)\cdot 37^{30} + \left(21 a^{2} + 5\right)\cdot 37^{31} + \left(26 a^{2} + 35 a + 36\right)\cdot 37^{32} + \left(31 a^{2} + 34 a + 14\right)\cdot 37^{33} + \left(3 a^{2} + 10 a + 27\right)\cdot 37^{34} + \left(35 a^{2} + 2 a + 2\right)\cdot 37^{35} + \left(a^{2} + 33 a + 9\right)\cdot 37^{36} + \left(28 a^{2} + 18 a + 10\right)\cdot 37^{37} + \left(29 a^{2} + 3 a + 30\right)\cdot 37^{38} + \left(4 a^{2} + 32 a + 30\right)\cdot 37^{39} + \left(4 a^{2} + 8\right)\cdot 37^{40} + \left(2 a^{2} + 10 a + 28\right)\cdot 37^{41} + \left(27 a^{2} + a + 29\right)\cdot 37^{42} + \left(17 a^{2} + 13 a + 24\right)\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a^{2} + 3 a + 35 + \left(13 a^{2} + 4 a + 28\right)\cdot 37 + \left(17 a^{2} + 3\right)\cdot 37^{2} + \left(11 a^{2} + 22 a + 33\right)\cdot 37^{3} + \left(3 a^{2} + 9 a + 2\right)\cdot 37^{4} + \left(6 a^{2} + 32 a + 5\right)\cdot 37^{5} + \left(20 a^{2} + 9 a + 18\right)\cdot 37^{6} + \left(3 a^{2} + 31\right)\cdot 37^{7} + \left(14 a^{2} + 17 a + 30\right)\cdot 37^{8} + \left(7 a^{2} + 14 a + 27\right)\cdot 37^{9} + \left(2 a^{2} + 10 a + 22\right)\cdot 37^{10} + \left(12 a^{2} + 19 a + 10\right)\cdot 37^{11} + \left(5 a^{2} + 13 a + 18\right)\cdot 37^{12} + \left(35 a^{2} + 12\right)\cdot 37^{13} + \left(25 a^{2} + 33 a + 17\right)\cdot 37^{14} + \left(28 a^{2} + 30 a + 11\right)\cdot 37^{15} + \left(10 a^{2} + 12 a + 4\right)\cdot 37^{16} + \left(30 a^{2} + 2 a + 36\right)\cdot 37^{17} + \left(5 a^{2} + 23 a + 13\right)\cdot 37^{18} + \left(19 a^{2} + 12 a + 30\right)\cdot 37^{19} + \left(13 a^{2} + 24 a + 19\right)\cdot 37^{20} + \left(32 a^{2} + 2 a + 13\right)\cdot 37^{21} + \left(23 a^{2} + 27 a + 36\right)\cdot 37^{22} + \left(7 a^{2} + 13 a + 9\right)\cdot 37^{23} + \left(11 a^{2} + 30 a + 11\right)\cdot 37^{24} + \left(33 a^{2} + 32 a + 35\right)\cdot 37^{25} + \left(33 a^{2} + 16 a + 21\right)\cdot 37^{26} + \left(7 a^{2} + 33 a + 22\right)\cdot 37^{27} + \left(18 a^{2} + 28\right)\cdot 37^{28} + \left(22 a^{2} + 23 a + 6\right)\cdot 37^{29} + \left(6 a^{2} + 18 a + 27\right)\cdot 37^{30} + \left(25 a^{2} + 15 a + 30\right)\cdot 37^{31} + \left(27 a^{2} + 23\right)\cdot 37^{32} + \left(28 a^{2} + 16 a + 17\right)\cdot 37^{33} + \left(29 a^{2} + 13 a + 30\right)\cdot 37^{34} + \left(7 a^{2} + 25 a + 31\right)\cdot 37^{35} + \left(31 a^{2} + 20 a + 28\right)\cdot 37^{36} + \left(26 a + 32\right)\cdot 37^{37} + \left(36 a^{2} + 36 a + 36\right)\cdot 37^{38} + \left(24 a^{2} + 8 a + 3\right)\cdot 37^{39} + \left(31 a^{2} + 33 a + 34\right)\cdot 37^{40} + \left(22 a^{2} + 33 a + 16\right)\cdot 37^{41} + \left(30 a^{2} + 16 a + 36\right)\cdot 37^{42} + \left(25 a^{2} + 6 a + 29\right)\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 28\cdot 37 + 24\cdot 37^{2} + 32\cdot 37^{3} + 6\cdot 37^{4} + 15\cdot 37^{5} + 27\cdot 37^{6} + 13\cdot 37^{7} + 6\cdot 37^{8} + 19\cdot 37^{9} + 8\cdot 37^{10} + 33\cdot 37^{11} + 24\cdot 37^{12} + 7\cdot 37^{13} + 28\cdot 37^{14} + 7\cdot 37^{15} + 20\cdot 37^{16} + 14\cdot 37^{17} + 27\cdot 37^{18} + 30\cdot 37^{19} + 14\cdot 37^{20} + 25\cdot 37^{21} + 33\cdot 37^{22} + 37^{24} + 3\cdot 37^{25} + 5\cdot 37^{26} + 13\cdot 37^{27} + 10\cdot 37^{28} + 16\cdot 37^{29} + 29\cdot 37^{30} + 17\cdot 37^{31} + 5\cdot 37^{32} + 34\cdot 37^{33} + 9\cdot 37^{34} + 30\cdot 37^{35} + 19\cdot 37^{36} + 30\cdot 37^{37} + 26\cdot 37^{38} + 5\cdot 37^{39} + 26\cdot 37^{40} + 37^{42} + 10\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a^{2} + 15 a + 15 + \left(17 a^{2} + 24 a + 7\right)\cdot 37 + \left(26 a^{2} + 18 a + 29\right)\cdot 37^{2} + \left(16 a^{2} + 27 a + 7\right)\cdot 37^{3} + \left(31 a^{2} + 30 a + 17\right)\cdot 37^{4} + \left(5 a^{2} + 35 a + 11\right)\cdot 37^{5} + \left(24 a^{2} + 28 a + 32\right)\cdot 37^{6} + \left(29 a^{2} + 36 a + 7\right)\cdot 37^{7} + \left(16 a^{2} + 29 a + 8\right)\cdot 37^{8} + \left(2 a^{2} + 23 a + 20\right)\cdot 37^{9} + \left(25 a^{2} + 11\right)\cdot 37^{10} + \left(4 a^{2} + 21 a + 22\right)\cdot 37^{11} + \left(3 a^{2} + 16 a + 34\right)\cdot 37^{12} + \left(26 a + 30\right)\cdot 37^{13} + \left(34 a^{2} + 13 a + 14\right)\cdot 37^{14} + \left(36 a^{2} + 5 a + 25\right)\cdot 37^{15} + \left(17 a^{2} + 21 a\right)\cdot 37^{16} + \left(13 a^{2} + 28 a + 29\right)\cdot 37^{17} + \left(15 a^{2} + 17 a + 7\right)\cdot 37^{18} + \left(12 a^{2} + 29 a\right)\cdot 37^{19} + \left(10 a^{2} + 36 a + 15\right)\cdot 37^{20} + \left(16 a^{2} + 15 a + 3\right)\cdot 37^{21} + \left(35 a^{2} + 12 a + 34\right)\cdot 37^{22} + \left(31 a^{2} + 22 a + 26\right)\cdot 37^{23} + \left(10 a^{2} + 30 a + 2\right)\cdot 37^{24} + \left(20 a^{2} + 29 a + 7\right)\cdot 37^{25} + \left(5 a^{2} + 14 a + 28\right)\cdot 37^{26} + \left(21 a^{2} + 6 a + 22\right)\cdot 37^{27} + \left(13 a^{2} + 7 a + 27\right)\cdot 37^{28} + \left(31 a^{2} + 9 a + 7\right)\cdot 37^{29} + \left(27 a^{2} + 25 a + 3\right)\cdot 37^{30} + \left(26 a^{2} + 26\right)\cdot 37^{31} + \left(16 a^{2} + 14 a + 33\right)\cdot 37^{32} + \left(28 a^{2} + 4 a + 1\right)\cdot 37^{33} + \left(22 a^{2} + 27 a + 29\right)\cdot 37^{34} + \left(23 a^{2} + 33 a + 30\right)\cdot 37^{35} + \left(23 a^{2} + 4 a + 21\right)\cdot 37^{36} + \left(32 a^{2} + 22 a + 28\right)\cdot 37^{37} + \left(12 a^{2} + 36 a + 36\right)\cdot 37^{38} + \left(5 a^{2} + 11 a + 32\right)\cdot 37^{39} + \left(35 a^{2} + 14 a + 21\right)\cdot 37^{40} + \left(5 a + 23\right)\cdot 37^{41} + \left(35 a^{2} + 17 a + 24\right)\cdot 37^{42} + \left(33 a^{2} + 6 a + 15\right)\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a^{2} + 21 a + 35 + \left(16 a^{2} + 7 a + 3\right)\cdot 37 + \left(24 a^{2} + 23 a + 21\right)\cdot 37^{2} + \left(27 a^{2} + 28 a + 14\right)\cdot 37^{3} + \left(20 a^{2} + 13 a + 11\right)\cdot 37^{4} + \left(11 a^{2} + 5 a + 34\right)\cdot 37^{5} + \left(4 a^{2} + 20 a + 26\right)\cdot 37^{6} + \left(28 a^{2} + a + 1\right)\cdot 37^{7} + \left(27 a^{2} + 32 a + 15\right)\cdot 37^{8} + \left(31 a^{2} + 23 a + 26\right)\cdot 37^{9} + \left(28 a^{2} + 35 a + 26\right)\cdot 37^{10} + \left(7 a^{2} + 35 a + 34\right)\cdot 37^{11} + \left(12 a^{2} + 14 a + 33\right)\cdot 37^{12} + \left(22 a^{2} + 33 a + 8\right)\cdot 37^{13} + \left(24 a^{2} + 3 a + 14\right)\cdot 37^{14} + \left(23 a^{2} + 35 a + 9\right)\cdot 37^{15} + \left(17 a^{2} + 23 a + 36\right)\cdot 37^{16} + \left(23 a^{2} + 8 a + 31\right)\cdot 37^{17} + \left(16 a^{2} + 2 a + 12\right)\cdot 37^{18} + \left(8 a^{2} + 15 a + 21\right)\cdot 37^{19} + \left(23 a^{2} + 15 a + 29\right)\cdot 37^{20} + \left(10 a^{2} + 35 a + 17\right)\cdot 37^{21} + \left(19 a^{2} + 33 a + 6\right)\cdot 37^{22} + \left(29 a^{2} + 36 a + 17\right)\cdot 37^{23} + \left(23 a^{2} + 5 a + 17\right)\cdot 37^{24} + \left(21 a^{2} + 32 a + 12\right)\cdot 37^{25} + \left(30 a^{2} + 35 a + 17\right)\cdot 37^{26} + \left(11 a^{2} + 30 a + 22\right)\cdot 37^{27} + \left(8 a^{2} + 2 a + 6\right)\cdot 37^{28} + \left(26 a^{2} + 27 a + 24\right)\cdot 37^{29} + \left(19 a^{2} + 34 a + 7\right)\cdot 37^{30} + \left(25 a^{2} + 35 a + 21\right)\cdot 37^{31} + \left(30 a^{2} + 24 a + 15\right)\cdot 37^{32} + \left(13 a^{2} + 34 a + 17\right)\cdot 37^{33} + \left(10 a^{2} + 35 a + 16\right)\cdot 37^{34} + \left(15 a^{2} + 34\right)\cdot 37^{35} + \left(11 a^{2} + 36 a + 9\right)\cdot 37^{36} + \left(13 a^{2} + 32 a + 25\right)\cdot 37^{37} + \left(31 a^{2} + 33 a + 36\right)\cdot 37^{38} + \left(26 a^{2} + 29 a + 7\right)\cdot 37^{39} + \left(34 a^{2} + 21 a + 20\right)\cdot 37^{40} + \left(33 a^{2} + 21 a + 7\right)\cdot 37^{41} + \left(11 a^{2} + 18 a + 6\right)\cdot 37^{42} + \left(22 a^{2} + 17 a + 6\right)\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 6\cdot 37 + 32\cdot 37^{2} + 34\cdot 37^{3} + 16\cdot 37^{4} + 4\cdot 37^{5} + 20\cdot 37^{6} + 7\cdot 37^{7} + 25\cdot 37^{8} + 28\cdot 37^{9} + 30\cdot 37^{10} + 31\cdot 37^{11} + 28\cdot 37^{12} + 25\cdot 37^{13} + 2\cdot 37^{14} + 3\cdot 37^{15} + 14\cdot 37^{16} + 19\cdot 37^{17} + 13\cdot 37^{18} + 34\cdot 37^{19} + 18\cdot 37^{20} + 26\cdot 37^{21} + 22\cdot 37^{22} + 30\cdot 37^{23} + 36\cdot 37^{24} + 31\cdot 37^{25} + 21\cdot 37^{26} + 14\cdot 37^{27} + 17\cdot 37^{28} + 31\cdot 37^{29} + 33\cdot 37^{30} + 27\cdot 37^{31} + 21\cdot 37^{32} + 37^{33} + 35\cdot 37^{34} + 10\cdot 37^{35} + 5\cdot 37^{36} + 2\cdot 37^{37} + 6\cdot 37^{38} + 25\cdot 37^{39} + 15\cdot 37^{40} + 15\cdot 37^{41} + 11\cdot 37^{42} + 15\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 a^{2} + 34 a + 4 + \left(25 a^{2} + 18 a + 5\right)\cdot 37 + \left(20 a^{2} + 31 a + 17\right)\cdot 37^{2} + \left(23 a^{2} + 19 a + 7\right)\cdot 37^{3} + \left(19 a^{2} + 27 a + 31\right)\cdot 37^{4} + \left(13 a^{2} + 18 a + 34\right)\cdot 37^{5} + \left(27 a^{2} + 13 a + 9\right)\cdot 37^{6} + \left(33 a^{2} + 28 a + 4\right)\cdot 37^{7} + \left(28 a^{2} + 18 a + 16\right)\cdot 37^{8} + \left(2 a^{2} + 18 a + 9\right)\cdot 37^{9} + \left(12 a^{2} + 5 a + 25\right)\cdot 37^{10} + \left(14 a^{2} + 21 a + 19\right)\cdot 37^{11} + \left(8 a^{2} + 36 a + 30\right)\cdot 37^{12} + \left(20 a^{2} + 11 a + 26\right)\cdot 37^{13} + \left(30 a^{2} + 5 a + 35\right)\cdot 37^{14} + \left(3 a^{2} + 10 a + 22\right)\cdot 37^{15} + \left(21 a^{2} + 4 a + 8\right)\cdot 37^{16} + \left(7 a^{2} + 35 a + 19\right)\cdot 37^{17} + \left(2 a^{2} + 26 a + 36\right)\cdot 37^{18} + \left(9 a^{2} + 8 a + 26\right)\cdot 37^{19} + \left(27 a^{2} + 12 a\right)\cdot 37^{20} + \left(10 a^{2} + 27 a + 1\right)\cdot 37^{21} + \left(19 a^{2} + 6 a + 18\right)\cdot 37^{22} + \left(28 a^{2} + 32 a + 19\right)\cdot 37^{23} + \left(22 a + 6\right)\cdot 37^{24} + \left(15 a^{2} + 29 a + 36\right)\cdot 37^{25} + \left(10 a^{2} + 31 a + 1\right)\cdot 37^{26} + \left(21 a^{2} + 23 a + 2\right)\cdot 37^{27} + \left(13 a + 32\right)\cdot 37^{28} + \left(30 a + 27\right)\cdot 37^{29} + \left(25 a^{2} + 4 a + 26\right)\cdot 37^{30} + \left(11 a^{2} + 33 a + 13\right)\cdot 37^{31} + \left(21 a^{2} + 22 a + 35\right)\cdot 37^{32} + \left(32 a^{2} + 3 a + 32\right)\cdot 37^{33} + \left(32 a^{2} + 21 a + 5\right)\cdot 37^{34} + \left(34 a^{2} + 9 a + 29\right)\cdot 37^{35} + \left(22 a^{2} + 31 a + 32\right)\cdot 37^{36} + \left(7 a^{2} + 32 a + 22\right)\cdot 37^{37} + \left(17 a^{2} + 25 a + 35\right)\cdot 37^{38} + \left(4 a^{2} + 34 a + 32\right)\cdot 37^{39} + \left(17 a^{2} + 6 a + 12\right)\cdot 37^{40} + \left(9 a^{2} + 28 a\right)\cdot 37^{41} + \left(20 a^{2} + 28 a + 32\right)\cdot 37^{42} + \left(23 a^{2} + 5 a + 20\right)\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a^{2} + 5 + \left(35 a^{2} + 14 a + 7\right)\cdot 37 + \left(35 a^{2} + 5 a + 4\right)\cdot 37^{2} + \left(a^{2} + 32 a + 32\right)\cdot 37^{3} + \left(14 a^{2} + 36 a + 8\right)\cdot 37^{4} + \left(17 a^{2} + 22 a + 13\right)\cdot 37^{5} + \left(26 a^{2} + 13 a + 6\right)\cdot 37^{6} + \left(36 a^{2} + 8 a + 16\right)\cdot 37^{7} + \left(30 a^{2} + a + 24\right)\cdot 37^{8} + \left(26 a^{2} + 4 a + 31\right)\cdot 37^{9} + \left(22 a^{2} + 21 a + 30\right)\cdot 37^{10} + \left(10 a^{2} + 33 a + 4\right)\cdot 37^{11} + \left(23 a^{2} + 23 a + 16\right)\cdot 37^{12} + \left(18 a^{2} + 24 a + 20\right)\cdot 37^{13} + \left(17 a^{2} + 35 a + 20\right)\cdot 37^{14} + \left(4 a^{2} + 32 a + 25\right)\cdot 37^{15} + \left(5 a^{2} + 19 a + 18\right)\cdot 37^{16} + \left(36 a^{2} + 36 a + 22\right)\cdot 37^{17} + \left(28 a^{2} + 23 a + 32\right)\cdot 37^{18} + \left(8 a^{2} + 15 a + 25\right)\cdot 37^{19} + \left(33 a^{2} + 24\right)\cdot 37^{20} + \left(30 a^{2} + 7 a + 7\right)\cdot 37^{21} + \left(30 a^{2} + 3 a + 27\right)\cdot 37^{22} + \left(28 a + 19\right)\cdot 37^{23} + \left(25 a^{2} + 20 a + 29\right)\cdot 37^{24} + \left(25 a^{2} + 11 a + 4\right)\cdot 37^{25} + \left(29 a^{2} + 25 a + 5\right)\cdot 37^{26} + \left(7 a^{2} + 16 a + 22\right)\cdot 37^{27} + \left(18 a^{2} + 22 a + 28\right)\cdot 37^{28} + \left(14 a^{2} + 20 a + 11\right)\cdot 37^{29} + \left(5 a^{2} + 13 a + 22\right)\cdot 37^{30} + \left(25 a + 4\right)\cdot 37^{31} + \left(25 a^{2} + 13 a + 13\right)\cdot 37^{32} + \left(12 a^{2} + 17 a + 27\right)\cdot 37^{33} + \left(11 a^{2} + 2 a + 30\right)\cdot 37^{34} + \left(31 a^{2} + 2 a + 14\right)\cdot 37^{35} + \left(19 a^{2} + 22 a + 20\right)\cdot 37^{36} + \left(28 a^{2} + 14 a + 32\right)\cdot 37^{37} + \left(20 a^{2} + 11 a + 12\right)\cdot 37^{38} + \left(7 a^{2} + 30 a + 8\right)\cdot 37^{39} + \left(25 a^{2} + 33 a + 8\right)\cdot 37^{40} + \left(4 a^{2} + 11 a + 18\right)\cdot 37^{41} + \left(23 a^{2} + 28 a + 6\right)\cdot 37^{42} + \left(24 a^{2} + 24 a + 25\right)\cdot 37^{43} +O\left(37^{ 44 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,4,5,6,7,2)$ |
| $(1,6)(2,8)(3,7)(4,5)$ |
| $(1,4)(2,3)(5,6)(7,8)$ |
| $(3,6,5)(4,8,7)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $7$ |
| $7$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $-1$ |
| $28$ | $3$ | $(3,6,5)(4,8,7)$ | $1$ |
| $28$ | $3$ | $(3,5,6)(4,7,8)$ | $1$ |
| $28$ | $6$ | $(1,3,5,7,6,2)(4,8)$ | $-1$ |
| $28$ | $6$ | $(1,2,6,7,5,3)(4,8)$ | $-1$ |
| $24$ | $7$ | $(1,8,4,5,6,7,2)$ | $0$ |
| $24$ | $7$ | $(1,5,2,4,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
