Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 19 + \left(2 a + 1\right)\cdot 43 + \left(6 a + 41\right)\cdot 43^{2} + \left(34 a + 28\right)\cdot 43^{3} + 32 a\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 9\cdot 43 + 20\cdot 43^{2} + 24\cdot 43^{3} + 18\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 5 + \left(24 a + 4\right)\cdot 43 + \left(2 a + 11\right)\cdot 43^{2} + \left(23 a + 11\right)\cdot 43^{3} + \left(41 a + 12\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 24 + \left(40 a + 41\right)\cdot 43 + \left(36 a + 1\right)\cdot 43^{2} + \left(8 a + 14\right)\cdot 43^{3} + \left(10 a + 42\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 33\cdot 43 + 22\cdot 43^{2} + 18\cdot 43^{3} + 24\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 38 + \left(18 a + 38\right)\cdot 43 + \left(40 a + 31\right)\cdot 43^{2} + \left(19 a + 31\right)\cdot 43^{3} + \left(a + 30\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,6,2)(3,5,4)$ |
| $(1,2)(3,6)(4,5)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
