Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 40 + \left(15 a + 42\right)\cdot 43 + \left(30 a + 10\right)\cdot 43^{2} + \left(24 a + 20\right)\cdot 43^{3} + \left(42 a + 12\right)\cdot 43^{4} + \left(14 a + 26\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 13 + \left(27 a + 42\right)\cdot 43 + \left(12 a + 25\right)\cdot 43^{2} + \left(18 a + 14\right)\cdot 43^{3} + 30\cdot 43^{4} + \left(28 a + 41\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 18\cdot 43 + 22\cdot 43^{2} + 7\cdot 43^{3} + 38\cdot 43^{4} + 22\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 36 + \left(42 a + 6\right)\cdot 43 + \left(26 a + 35\right)\cdot 43^{2} + \left(4 a + 18\right)\cdot 43^{3} + \left(35 a + 25\right)\cdot 43^{4} + \left(12 a + 37\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 40\cdot 43^{2} + 33\cdot 43^{3} + 32\cdot 43^{4} + 27\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 37 + 4\cdot 43 + \left(16 a + 20\right)\cdot 43^{2} + \left(38 a + 39\right)\cdot 43^{3} + \left(7 a + 12\right)\cdot 43^{4} + \left(30 a + 15\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 a + 36 + \left(25 a + 42\right)\cdot 43 + 40 a\cdot 43^{2} + \left(22 a + 6\right)\cdot 43^{3} + \left(26 a + 8\right)\cdot 43^{4} + \left(37 a + 16\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 31 a + 5 + \left(17 a + 13\right)\cdot 43 + \left(2 a + 16\right)\cdot 43^{2} + \left(20 a + 31\right)\cdot 43^{3} + \left(16 a + 11\right)\cdot 43^{4} + \left(5 a + 27\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,3)(2,8,5)$ |
| $(1,4,2,7)(3,6,5,8)$ |
| $(1,6,2,8)(3,7,5,4)$ |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,5)(2,3)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,3)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,4,8)(2,7,6)$ | $-1$ |
| $6$ | $4$ | $(1,4,2,7)(3,6,5,8)$ | $0$ |
| $8$ | $6$ | $(1,2)(3,6,7,5,8,4)$ | $1$ |
| $6$ | $8$ | $(1,8,5,4,2,6,3,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,6,5,7,2,8,3,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
