Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{3} + 2 x + 18 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 a^{2} + 9 a + 6 + \left(2 a^{2} + 14 a\right)\cdot 23 + \left(19 a^{2} + 21 a + 15\right)\cdot 23^{2} + \left(17 a^{2} + 8 a + 18\right)\cdot 23^{3} + \left(12 a^{2} + 13 a + 19\right)\cdot 23^{4} + \left(19 a^{2} + 22 a + 19\right)\cdot 23^{5} + \left(18 a^{2} + 13 a + 13\right)\cdot 23^{6} + \left(15 a + 17\right)\cdot 23^{7} + \left(20 a^{2} + 8 a + 20\right)\cdot 23^{8} + \left(10 a^{2} + 16 a + 22\right)\cdot 23^{9} + \left(16 a^{2} + 17 a + 9\right)\cdot 23^{10} + \left(11 a + 16\right)\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 8 a + 15 + \left(4 a^{2} + 7 a + 10\right)\cdot 23 + \left(5 a^{2} + 10 a + 19\right)\cdot 23^{2} + \left(17 a^{2} + 14 a + 17\right)\cdot 23^{3} + \left(18 a^{2} + 19 a + 4\right)\cdot 23^{4} + \left(11 a^{2} + 19 a + 17\right)\cdot 23^{5} + \left(7 a^{2} + 22 a + 21\right)\cdot 23^{6} + \left(10 a^{2} + 11 a + 14\right)\cdot 23^{7} + \left(4 a^{2} + 20 a + 7\right)\cdot 23^{8} + \left(14 a^{2} + 8 a + 4\right)\cdot 23^{9} + \left(7 a^{2} + 9 a + 21\right)\cdot 23^{10} + \left(21 a^{2} + 10 a + 20\right)\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a^{2} + 14 a + 8 + \left(17 a^{2} + 9 a + 12\right)\cdot 23 + 4 a^{2}23^{2} + \left(9 a^{2} + 14 a + 11\right)\cdot 23^{3} + \left(3 a^{2} + 13 a + 13\right)\cdot 23^{4} + \left(13 a^{2} + 21 a + 21\right)\cdot 23^{5} + \left(5 a^{2} + 18 a + 22\right)\cdot 23^{6} + \left(12 a^{2} + 15 a + 17\right)\cdot 23^{7} + \left(7 a^{2} + 6 a + 10\right)\cdot 23^{8} + \left(3 a^{2} + 15 a + 21\right)\cdot 23^{9} + \left(17 a^{2} + 9 a + 16\right)\cdot 23^{10} + \left(12 a^{2} + 9 a + 12\right)\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 12\cdot 23 + 4\cdot 23^{2} + 8\cdot 23^{3} + 3\cdot 23^{4} + 14\cdot 23^{5} + 20\cdot 23^{6} + 10\cdot 23^{7} + 7\cdot 23^{8} + 19\cdot 23^{9} + 8\cdot 23^{10} + 12\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 + 18\cdot 23 + 6\cdot 23^{2} + 11\cdot 23^{3} + 10\cdot 23^{4} + 22\cdot 23^{5} + 20\cdot 23^{6} + 9\cdot 23^{7} + 18\cdot 23^{8} + 7\cdot 23^{9} + 4\cdot 23^{10} +O\left(23^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 8\cdot 23 + 14\cdot 23^{2} + 22\cdot 23^{3} + 19\cdot 23^{4} + 15\cdot 23^{5} + 14\cdot 23^{6} + 16\cdot 23^{7} + 12\cdot 23^{8} + 11\cdot 23^{9} + 17\cdot 23^{10} + 22\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 20 a^{2} + 13 a + 5 + \left(22 a^{2} + 20 a + 12\right)\cdot 23 + \left(8 a^{2} + 22 a + 21\right)\cdot 23^{2} + \left(20 a^{2} + 2 a + 2\right)\cdot 23^{3} + \left(9 a^{2} + 4 a + 22\right)\cdot 23^{4} + \left(3 a^{2} + 18 a\right)\cdot 23^{5} + \left(8 a^{2} + 11\right)\cdot 23^{6} + \left(17 a^{2} + a + 9\right)\cdot 23^{7} + \left(8 a + 9\right)\cdot 23^{8} + \left(a^{2} + 5 a + 18\right)\cdot 23^{9} + \left(17 a^{2} + 13 a + 16\right)\cdot 23^{10} + \left(15 a^{2} + 13 a + 16\right)\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 a^{2} + 19 a + 14 + \left(5 a^{2} + 15 a + 12\right)\cdot 23 + \left(9 a^{2} + 22 a + 6\right)\cdot 23^{2} + \left(16 a^{2} + 5 a + 5\right)\cdot 23^{3} + \left(9 a^{2} + 5 a + 14\right)\cdot 23^{4} + \left(6 a^{2} + 6 a + 12\right)\cdot 23^{5} + \left(9 a^{2} + 3 a + 12\right)\cdot 23^{6} + \left(16 a^{2} + 6 a\right)\cdot 23^{7} + \left(14 a^{2} + 8 a + 5\right)\cdot 23^{8} + \left(18 a^{2} + 2 a + 11\right)\cdot 23^{9} + \left(11 a^{2} + 17\right)\cdot 23^{10} + \left(17 a^{2} + 3\right)\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 8 a^{2} + 6 a + 15 + \left(16 a^{2} + a + 3\right)\cdot 23 + \left(21 a^{2} + 14 a + 3\right)\cdot 23^{2} + \left(10 a^{2} + 22 a + 17\right)\cdot 23^{3} + \left(14 a^{2} + 12 a + 6\right)\cdot 23^{4} + \left(14 a^{2} + 3 a + 13\right)\cdot 23^{5} + \left(19 a^{2} + 9 a + 22\right)\cdot 23^{6} + \left(11 a^{2} + 18 a + 16\right)\cdot 23^{7} + \left(21 a^{2} + 16 a + 22\right)\cdot 23^{8} + \left(20 a^{2} + 20 a + 20\right)\cdot 23^{9} + \left(21 a^{2} + 18 a + 1\right)\cdot 23^{10} + 9\cdot 23^{11} +O\left(23^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,9)(3,8,7)$ |
| $(1,2,9)(4,6,5)$ |
| $(1,6)(2,5)(3,7)(4,9)$ |
| $(1,8,6,2,3,4,9,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,6)(2,5)(3,7)(4,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,9)(3,7,8)(4,5,6)$ |
$-3$ |
| $3$ |
$3$ |
$(1,2,9)(3,8,7)$ |
$0$ |
| $3$ |
$3$ |
$(1,9,2)(3,7,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,2,5,9,4)(7,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,4,9,5,2,6)(7,8)$ |
$0$ |
| $6$ |
$9$ |
$(1,8,6,2,3,4,9,7,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,8,5,2,3,6,9,7,4)$ |
$0$ |
| $6$ |
$9$ |
$(1,5,3,9,4,8,2,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
