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