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