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