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