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