Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 5\cdot 43 + 37\cdot 43^{2} + 31\cdot 43^{3} + 21\cdot 43^{4} + 5\cdot 43^{5} + 9\cdot 43^{6} + 14\cdot 43^{7} + 17\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 16 + \left(12 a + 21\right)\cdot 43 + \left(22 a + 16\right)\cdot 43^{2} + \left(2 a + 31\right)\cdot 43^{3} + \left(31 a + 28\right)\cdot 43^{4} + \left(13 a + 8\right)\cdot 43^{5} + \left(17 a + 41\right)\cdot 43^{6} + \left(18 a + 20\right)\cdot 43^{7} + \left(24 a + 18\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 37\cdot 43 + 5\cdot 43^{2} + 11\cdot 43^{3} + 21\cdot 43^{4} + 37\cdot 43^{5} + 33\cdot 43^{6} + 28\cdot 43^{7} + 25\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 4 + \left(2 a + 38\right)\cdot 43 + \left(41 a + 23\right)\cdot 43^{2} + \left(37 a + 1\right)\cdot 43^{3} + \left(4 a + 38\right)\cdot 43^{4} + \left(15 a + 37\right)\cdot 43^{5} + \left(33 a + 33\right)\cdot 43^{6} + \left(23 a + 4\right)\cdot 43^{7} + \left(a + 11\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 28 + \left(30 a + 21\right)\cdot 43 + \left(20 a + 26\right)\cdot 43^{2} + \left(40 a + 11\right)\cdot 43^{3} + \left(11 a + 14\right)\cdot 43^{4} + \left(29 a + 34\right)\cdot 43^{5} + \left(25 a + 1\right)\cdot 43^{6} + \left(24 a + 22\right)\cdot 43^{7} + \left(18 a + 24\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 40 + \left(40 a + 4\right)\cdot 43 + \left(a + 19\right)\cdot 43^{2} + \left(5 a + 41\right)\cdot 43^{3} + \left(38 a + 4\right)\cdot 43^{4} + \left(27 a + 5\right)\cdot 43^{5} + \left(9 a + 9\right)\cdot 43^{6} + \left(19 a + 38\right)\cdot 43^{7} + \left(41 a + 31\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(1,3)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(2,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,3,4)$ | $-1$ |
| $6$ | $4$ | $(1,3)(2,6,5,4)$ | $1$ |
| $8$ | $6$ | $(1,6,5,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
