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