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