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