Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 5 + \left(9 a + 12\right)\cdot 23 + \left(5 a + 6\right)\cdot 23^{2} + \left(21 a + 14\right)\cdot 23^{3} + \left(9 a + 18\right)\cdot 23^{4} + \left(19 a + 17\right)\cdot 23^{5} + \left(4 a + 12\right)\cdot 23^{6} + \left(3 a + 17\right)\cdot 23^{7} + \left(8 a + 6\right)\cdot 23^{8} + 20\cdot 23^{9} + 17\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 20 + \left(17 a + 4\right)\cdot 23 + \left(10 a + 3\right)\cdot 23^{2} + \left(15 a + 13\right)\cdot 23^{3} + \left(6 a + 5\right)\cdot 23^{4} + \left(14 a + 10\right)\cdot 23^{5} + \left(2 a + 19\right)\cdot 23^{6} + \left(7 a + 3\right)\cdot 23^{7} + \left(18 a + 11\right)\cdot 23^{8} + \left(14 a + 10\right)\cdot 23^{9} + \left(19 a + 21\right)\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 6 + \left(17 a + 16\right)\cdot 23 + \left(10 a + 15\right)\cdot 23^{2} + \left(15 a + 12\right)\cdot 23^{3} + \left(6 a + 19\right)\cdot 23^{4} + \left(14 a + 13\right)\cdot 23^{5} + \left(2 a + 12\right)\cdot 23^{6} + \left(7 a + 7\right)\cdot 23^{7} + \left(18 a + 5\right)\cdot 23^{8} + \left(14 a + 1\right)\cdot 23^{9} + 19 a\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 11 + \left(9 a + 6\right)\cdot 23 + \left(5 a + 15\right)\cdot 23^{2} + \left(21 a + 17\right)\cdot 23^{3} + \left(9 a + 5\right)\cdot 23^{4} + \left(19 a + 22\right)\cdot 23^{5} + \left(4 a + 19\right)\cdot 23^{6} + \left(3 a + 3\right)\cdot 23^{7} + \left(8 a + 3\right)\cdot 23^{8} + 10\cdot 23^{9} + 5\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 18 + \left(13 a + 10\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(a + 8\right)\cdot 23^{3} + \left(13 a + 4\right)\cdot 23^{4} + \left(3 a + 5\right)\cdot 23^{5} + \left(18 a + 10\right)\cdot 23^{6} + \left(19 a + 5\right)\cdot 23^{7} + \left(14 a + 16\right)\cdot 23^{8} + \left(22 a + 2\right)\cdot 23^{9} + \left(22 a + 5\right)\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 3 + \left(5 a + 18\right)\cdot 23 + \left(12 a + 19\right)\cdot 23^{2} + \left(7 a + 9\right)\cdot 23^{3} + \left(16 a + 17\right)\cdot 23^{4} + \left(8 a + 12\right)\cdot 23^{5} + \left(20 a + 3\right)\cdot 23^{6} + \left(15 a + 19\right)\cdot 23^{7} + \left(4 a + 11\right)\cdot 23^{8} + \left(8 a + 12\right)\cdot 23^{9} + \left(3 a + 1\right)\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 17 + \left(5 a + 6\right)\cdot 23 + \left(12 a + 7\right)\cdot 23^{2} + \left(7 a + 10\right)\cdot 23^{3} + \left(16 a + 3\right)\cdot 23^{4} + \left(8 a + 9\right)\cdot 23^{5} + \left(20 a + 10\right)\cdot 23^{6} + \left(15 a + 15\right)\cdot 23^{7} + \left(4 a + 17\right)\cdot 23^{8} + \left(8 a + 21\right)\cdot 23^{9} + \left(3 a + 22\right)\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 8 a + 12 + \left(13 a + 16\right)\cdot 23 + \left(17 a + 7\right)\cdot 23^{2} + \left(a + 5\right)\cdot 23^{3} + \left(13 a + 17\right)\cdot 23^{4} + 3 a\cdot 23^{5} + \left(18 a + 3\right)\cdot 23^{6} + \left(19 a + 19\right)\cdot 23^{7} + \left(14 a + 19\right)\cdot 23^{8} + \left(22 a + 12\right)\cdot 23^{9} + \left(22 a + 17\right)\cdot 23^{10} +O\left(23^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,5)(3,7)$ |
| $(1,3,5,7)$ |
| $(1,7,5,3)(2,8,6,4)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(3,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,8)(4,7)(5,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,3,5,7)(2,8,6,4)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,7,5,3)(2,4,6,8)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,5,3)(2,8,6,4)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,5,7)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,7,5,3)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,7,5,3)(2,6)(4,8)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,3,5,7)(2,6)(4,8)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $4$ |
$4$ |
$(1,2,5,6)(3,8,7,4)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,2,3,8,5,6,7,4)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,8,7,2,5,4,3,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
