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