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