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