Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 34 + \left(27 a + 22\right)\cdot 43 + \left(a + 32\right)\cdot 43^{2} + \left(31 a + 21\right)\cdot 43^{3} + \left(23 a + 20\right)\cdot 43^{4} + \left(12 a + 17\right)\cdot 43^{5} + \left(12 a + 40\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 4 + \left(30 a + 2\right)\cdot 43 + \left(31 a + 21\right)\cdot 43^{2} + \left(22 a + 4\right)\cdot 43^{3} + \left(42 a + 33\right)\cdot 43^{4} + \left(41 a + 21\right)\cdot 43^{5} + \left(16 a + 12\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 16 + \left(15 a + 25\right)\cdot 43 + \left(41 a + 6\right)\cdot 43^{2} + \left(11 a + 8\right)\cdot 43^{3} + \left(19 a + 13\right)\cdot 43^{4} + \left(30 a + 6\right)\cdot 43^{5} + \left(30 a + 40\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 5\cdot 43 + 23\cdot 43^{2} + 16\cdot 43^{3} + 33\cdot 43^{4} + 14\cdot 43^{5} + 14\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 9 + \left(15 a + 20\right)\cdot 43 + \left(41 a + 10\right)\cdot 43^{2} + \left(11 a + 21\right)\cdot 43^{3} + \left(19 a + 22\right)\cdot 43^{4} + \left(30 a + 25\right)\cdot 43^{5} + \left(30 a + 2\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 39 + \left(12 a + 40\right)\cdot 43 + \left(11 a + 21\right)\cdot 43^{2} + \left(20 a + 38\right)\cdot 43^{3} + 9\cdot 43^{4} + \left(a + 21\right)\cdot 43^{5} + \left(26 a + 30\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 a + 27 + \left(27 a + 17\right)\cdot 43 + \left(a + 36\right)\cdot 43^{2} + \left(31 a + 34\right)\cdot 43^{3} + \left(23 a + 29\right)\cdot 43^{4} + \left(12 a + 36\right)\cdot 43^{5} + \left(12 a + 2\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 29 + 37\cdot 43 + 19\cdot 43^{2} + 26\cdot 43^{3} + 9\cdot 43^{4} + 28\cdot 43^{5} + 28\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6)(2,5,3)$ |
| $(1,5)(2,7)(3,6)$ |
| $(1,8,5,4)(2,7,6,3)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,7,5,3)(2,4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,5)(2,7)(3,6)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(2,4,7)(3,6,8)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,5,3)(2,4,6,8)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,5)(2,3,4,6,7,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,2,4,3,5,6,8,7)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,6,4,7,5,2,8,3)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
