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