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