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