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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 14\cdot 23 + 14\cdot 23^{2} + 5\cdot 23^{3} + 9\cdot 23^{4} + 5\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + \left(13 a + 12\right)\cdot 23 + \left(7 a + 3\right)\cdot 23^{2} + \left(2 a + 1\right)\cdot 23^{3} + \left(13 a + 20\right)\cdot 23^{4} + \left(15 a + 12\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 9 + 9 a\cdot 23 + \left(15 a + 5\right)\cdot 23^{2} + \left(20 a + 21\right)\cdot 23^{3} + \left(9 a + 20\right)\cdot 23^{4} + \left(7 a + 7\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 22 + \left(5 a + 20\right)\cdot 23 + \left(11 a + 2\right)\cdot 23^{2} + \left(20 a + 10\right)\cdot 23^{3} + \left(2 a + 8\right)\cdot 23^{4} + \left(13 a + 2\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 17 + \left(17 a + 22\right)\cdot 23 + \left(11 a + 19\right)\cdot 23^{2} + \left(2 a + 16\right)\cdot 23^{3} + \left(20 a + 16\right)\cdot 23^{4} + \left(9 a + 2\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 + 21\cdot 23 + 22\cdot 23^{2} + 13\cdot 23^{3} + 16\cdot 23^{4} + 14\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4,3,5,2)$ |
| $(2,3)(4,5)$ |
| $(2,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,4,5)(2,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,6,3)$ | $-2$ |
| $2$ | $3$ | $(1,4,5)(2,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)$ | $1$ |
| $6$ | $6$ | $(1,6,4,3,5,2)$ | $0$ |
| $6$ | $6$ | $(1,2,4,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
