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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 5 + \left(15 a + 20\right)\cdot 23 + 20 a\cdot 23^{2} + \left(20 a + 3\right)\cdot 23^{3} + \left(15 a + 9\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 15 + \left(15 a + 1\right)\cdot 23 + \left(20 a + 15\right)\cdot 23^{2} + \left(20 a + 8\right)\cdot 23^{3} + \left(15 a + 22\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 17 + \left(7 a + 21\right)\cdot 23 + \left(2 a + 3\right)\cdot 23^{2} + \left(2 a + 1\right)\cdot 23^{3} + \left(7 a + 20\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 4 + \left(7 a + 3\right)\cdot 23 + \left(2 a + 18\right)\cdot 23^{2} + \left(2 a + 6\right)\cdot 23^{3} + \left(7 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 9 + 7 a\cdot 23 + \left(2 a + 17\right)\cdot 23^{2} + 2 a\cdot 23^{3} + \left(7 a + 9\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 20 + \left(15 a + 21\right)\cdot 23 + \left(20 a + 13\right)\cdot 23^{2} + \left(20 a + 2\right)\cdot 23^{3} + \left(15 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(1,3)(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,6,2)(3,5,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,4,6,3,2,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,2,3,6,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
