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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 21 + \left(14 a + 20\right)\cdot 23 + \left(6 a + 14\right)\cdot 23^{2} + \left(18 a + 4\right)\cdot 23^{3} + \left(21 a + 16\right)\cdot 23^{4} + \left(a + 9\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 7 + \left(17 a + 22\right)\cdot 23 + \left(7 a + 16\right)\cdot 23^{2} + \left(21 a + 6\right)\cdot 23^{3} + \left(14 a + 1\right)\cdot 23^{4} + \left(9 a + 13\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 15 + \left(7 a + 16\right)\cdot 23 + 7 a\cdot 23^{2} + \left(22 a + 18\right)\cdot 23^{3} + \left(a + 20\right)\cdot 23^{4} + \left(12 a + 18\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 15 + \left(8 a + 7\right)\cdot 23 + \left(16 a + 13\right)\cdot 23^{2} + \left(4 a + 11\right)\cdot 23^{3} + \left(a + 18\right)\cdot 23^{4} + \left(21 a + 14\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 21 + \left(5 a + 3\right)\cdot 23 + \left(15 a + 15\right)\cdot 23^{2} + \left(a + 18\right)\cdot 23^{3} + \left(8 a + 9\right)\cdot 23^{4} + \left(13 a + 17\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 14 + \left(15 a + 20\right)\cdot 23 + \left(15 a + 7\right)\cdot 23^{2} + 9\cdot 23^{3} + \left(21 a + 2\right)\cdot 23^{4} + \left(10 a + 18\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3,4)$ |
| $(1,3,5,4,6,2)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(2,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
| $3$ | $6$ | $(1,2,6,4,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
