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