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