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