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 + 28 + \left(3 a + 27\right)\cdot 43 + \left(35 a + 26\right)\cdot 43^{2} + \left(7 a + 7\right)\cdot 43^{3} + \left(27 a + 19\right)\cdot 43^{4} + 5 a\cdot 43^{5} + \left(39 a + 33\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 12\cdot 43 + 35\cdot 43^{2} + 8\cdot 43^{3} + 9\cdot 43^{4} + 41\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 31 + \left(34 a + 36\right)\cdot 43 + \left(19 a + 6\right)\cdot 43^{2} + \left(28 a + 23\right)\cdot 43^{3} + \left(5 a + 6\right)\cdot 43^{4} + \left(34 a + 38\right)\cdot 43^{5} + \left(22 a + 20\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 41 + \left(8 a + 17\right)\cdot 43 + \left(23 a + 35\right)\cdot 43^{2} + \left(14 a + 31\right)\cdot 43^{3} + \left(37 a + 26\right)\cdot 43^{4} + \left(8 a + 23\right)\cdot 43^{5} + \left(20 a + 9\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 + 31\cdot 43 + 38\cdot 43^{2} + 43^{3} + 31\cdot 43^{4} + 28\cdot 43^{5} + 42\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 7 + \left(13 a + 23\right)\cdot 43 + \left(34 a + 39\right)\cdot 43^{2} + \left(22 a + 21\right)\cdot 43^{3} + \left(26 a + 18\right)\cdot 43^{4} + 41 a\cdot 43^{5} + \left(32 a + 5\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 23 a + 27 + \left(29 a + 16\right)\cdot 43 + \left(8 a + 17\right)\cdot 43^{2} + \left(20 a + 10\right)\cdot 43^{3} + \left(16 a + 22\right)\cdot 43^{4} + \left(a + 15\right)\cdot 43^{5} + \left(10 a + 39\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 18 a + 10 + \left(39 a + 6\right)\cdot 43 + \left(7 a + 15\right)\cdot 43^{2} + \left(35 a + 23\right)\cdot 43^{3} + \left(15 a + 38\right)\cdot 43^{4} + \left(37 a + 21\right)\cdot 43^{5} + \left(3 a + 23\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,5)(6,8)$ |
| $(1,7,6,8)(2,3,5,4)$ |
| $(1,2,6,5)(3,7,4,8)$ |
| $(1,6)(2,5)(3,4)(7,8)$ |
| $(1,8,2)(5,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-4$ |
| $12$ | $2$ | $(1,7)(2,5)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,5,4)(2,3,6)$ | $1$ |
| $6$ | $4$ | $(1,7,6,8)(2,3,5,4)$ | $0$ |
| $8$ | $6$ | $(1,7,2,6,8,5)(3,4)$ | $-1$ |
| $6$ | $8$ | $(1,4,8,5,6,3,7,2)$ | $0$ |
| $6$ | $8$ | $(1,3,8,2,6,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
