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(15 a + 19\right)\cdot 23 + \left(15 a + 9\right)\cdot 23^{2} + \left(8 a + 12\right)\cdot 23^{3} + \left(16 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 7 + \left(15 a + 8\right)\cdot 23 + \left(15 a + 7\right)\cdot 23^{2} + \left(8 a + 20\right)\cdot 23^{3} + \left(16 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 8 + \left(15 a + 21\right)\cdot 23 + \left(15 a + 4\right)\cdot 23^{2} + \left(8 a + 22\right)\cdot 23^{3} + \left(16 a + 17\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 5 + \left(7 a + 19\right)\cdot 23 + \left(7 a + 20\right)\cdot 23^{2} + 14 a\cdot 23^{3} + \left(6 a + 19\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 10 + \left(7 a + 17\right)\cdot 23 + \left(7 a + 2\right)\cdot 23^{2} + \left(14 a + 14\right)\cdot 23^{3} + \left(6 a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 4 + \left(7 a + 6\right)\cdot 23 + 7 a\cdot 23^{2} + \left(14 a + 22\right)\cdot 23^{3} + \left(6 a + 2\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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,3,5,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,2,5,3,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
