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