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:
| $r_{ 1 }$ |
$=$ |
$ 3 + 25\cdot 31 + 4\cdot 31^{2} + 12\cdot 31^{3} + 18\cdot 31^{4} + 17\cdot 31^{5} + 6\cdot 31^{6} + 29\cdot 31^{7} + 5\cdot 31^{8} + 5\cdot 31^{9} + 9\cdot 31^{10} + 2\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a^{2} + 14 a + 19 + \left(27 a^{2} + 7 a + 28\right)\cdot 31 + \left(24 a^{2} + 20 a + 26\right)\cdot 31^{2} + \left(4 a^{2} + 21 a + 23\right)\cdot 31^{3} + \left(26 a^{2} + 4 a + 27\right)\cdot 31^{4} + \left(18 a^{2} + 17 a + 22\right)\cdot 31^{5} + \left(21 a^{2} + 22 a + 24\right)\cdot 31^{6} + \left(9 a^{2} + 2 a + 16\right)\cdot 31^{7} + \left(10 a^{2} + 25 a + 27\right)\cdot 31^{8} + \left(19 a^{2} + 28 a + 12\right)\cdot 31^{9} + \left(18 a^{2} + 16 a + 12\right)\cdot 31^{10} + \left(3 a^{2} + 12 a + 2\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a^{2} + 24 a + 27 + \left(16 a^{2} + 25 a\right)\cdot 31 + \left(25 a^{2} + 12 a + 17\right)\cdot 31^{2} + \left(13 a^{2} + 22 a + 19\right)\cdot 31^{3} + \left(24 a^{2} + 6 a + 26\right)\cdot 31^{4} + \left(4 a^{2} + 15 a + 23\right)\cdot 31^{5} + \left(28 a^{2} + 9 a + 18\right)\cdot 31^{6} + \left(2 a^{2} + 3 a + 22\right)\cdot 31^{7} + \left(28 a^{2} + 14 a + 18\right)\cdot 31^{8} + \left(16 a^{2} + 4 a + 21\right)\cdot 31^{9} + \left(12 a + 10\right)\cdot 31^{10} + \left(30 a^{2} + 18 a + 30\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 24\cdot 31 + 12\cdot 31^{2} + 8\cdot 31^{3} + 8\cdot 31^{5} + 16\cdot 31^{6} + 13\cdot 31^{7} + 6\cdot 31^{8} + 15\cdot 31^{9} + 2\cdot 31^{10} + 11\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a^{2} + 9 a + 10 + \left(25 a^{2} + 7 a + 27\right)\cdot 31 + \left(22 a^{2} + 27 a + 4\right)\cdot 31^{2} + \left(7 a^{2} + 8 a + 5\right)\cdot 31^{3} + \left(27 a^{2} + 18 a + 18\right)\cdot 31^{4} + \left(9 a^{2} + 4 a + 6\right)\cdot 31^{5} + \left(12 a^{2} + 23 a + 8\right)\cdot 31^{6} + \left(a^{2} + 11 a + 11\right)\cdot 31^{7} + \left(9 a^{2} + 14 a + 16\right)\cdot 31^{8} + \left(14 a^{2} + 19 a + 9\right)\cdot 31^{9} + \left(11 a^{2} + 21 a + 28\right)\cdot 31^{10} + \left(29 a^{2} + 15 a + 29\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a^{2} + 11 a + 24 + \left(14 a^{2} + 5 a + 19\right)\cdot 31 + \left(3 a^{2} + 9 a + 12\right)\cdot 31^{2} + \left(10 a^{2} + 22 a + 27\right)\cdot 31^{3} + \left(12 a^{2} + a + 28\right)\cdot 31^{4} + \left(18 a^{2} + 29 a + 1\right)\cdot 31^{5} + \left(17 a^{2} + 25 a + 22\right)\cdot 31^{6} + \left(14 a^{2} + 24 a + 9\right)\cdot 31^{7} + \left(2 a^{2} + 15 a + 22\right)\cdot 31^{8} + \left(24 a^{2} + 25 a + 5\right)\cdot 31^{9} + \left(4 a^{2} + 22 a + 3\right)\cdot 31^{10} + \left(21 a^{2} + 23 a + 14\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 15 + 12\cdot 31 + 13\cdot 31^{2} + 10\cdot 31^{3} + 12\cdot 31^{4} + 5\cdot 31^{5} + 8\cdot 31^{6} + 19\cdot 31^{7} + 18\cdot 31^{8} + 10\cdot 31^{9} + 19\cdot 31^{10} + 17\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 a^{2} + 8 a + 2 + \left(9 a^{2} + 16 a + 6\right)\cdot 31 + \left(14 a^{2} + 14 a + 30\right)\cdot 31^{2} + \left(18 a^{2} + 1\right)\cdot 31^{3} + \left(8 a^{2} + 8 a + 16\right)\cdot 31^{4} + \left(2 a^{2} + 9 a + 1\right)\cdot 31^{5} + \left(28 a^{2} + 16 a + 29\right)\cdot 31^{6} + \left(19 a^{2} + 16 a + 2\right)\cdot 31^{7} + \left(11 a^{2} + 22 a + 18\right)\cdot 31^{8} + \left(28 a^{2} + 13 a + 8\right)\cdot 31^{9} + \left(23 a + 21\right)\cdot 31^{10} + \left(29 a^{2} + 2 a + 29\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ a^{2} + 27 a + 11 + \left(30 a + 10\right)\cdot 31 + \left(2 a^{2} + 8 a + 1\right)\cdot 31^{2} + \left(7 a^{2} + 17 a + 15\right)\cdot 31^{3} + \left(25 a^{2} + 22 a + 6\right)\cdot 31^{4} + \left(7 a^{2} + 17 a + 5\right)\cdot 31^{5} + \left(16 a^{2} + 26 a + 21\right)\cdot 31^{6} + \left(13 a^{2} + 2 a + 29\right)\cdot 31^{7} + \left(a + 20\right)\cdot 31^{8} + \left(21 a^{2} + a + 3\right)\cdot 31^{9} + \left(25 a^{2} + 27 a + 17\right)\cdot 31^{10} + \left(10 a^{2} + 19 a + 17\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3)(4,6)(7,9)$ |
| $(1,4)(3,6)(5,8)$ |
| $(1,3,5)(2,7,9)(4,6,8)$ |
| $(1,4,7)(2,5,8)(3,6,9)$ |
| $(1,7,4)(3,6,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,4)(3,6)(5,8)$ | $0$ |
| $9$ | $2$ | $(1,3)(4,6)(7,9)$ | $2$ |
| $9$ | $2$ | $(1,6)(3,4)(5,8)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ | $-3$ |
| $6$ | $3$ | $(1,3,5)(2,7,9)(4,6,8)$ | $0$ |
| $6$ | $3$ | $(1,7,4)(2,5,8)$ | $0$ |
| $12$ | $3$ | $(1,6,8)(2,4,9)(3,5,7)$ | $0$ |
| $18$ | $6$ | $(1,6,5,4,3,8)(2,7,9)$ | $0$ |
| $18$ | $6$ | $(1,6,7,3,4,9)(2,5,8)$ | $-1$ |
| $18$ | $6$ | $(1,8,7,2,4,5)(3,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
