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