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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 27 + \left(3 a + 36\right)\cdot 43 + \left(22 a + 8\right)\cdot 43^{2} + \left(14 a + 39\right)\cdot 43^{3} + \left(9 a + 38\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 18\cdot 43 + 14\cdot 43^{2} + 6\cdot 43^{3} + 2\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 37 + \left(35 a + 13\right)\cdot 43 + \left(9 a + 27\right)\cdot 43^{2} + \left(37 a + 4\right)\cdot 43^{3} + \left(24 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 33 + \left(7 a + 10\right)\cdot 43 + \left(33 a + 1\right)\cdot 43^{2} + \left(5 a + 32\right)\cdot 43^{3} + \left(18 a + 35\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 24\cdot 43 + 6\cdot 43^{2} + 15\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 42 + \left(39 a + 24\right)\cdot 43 + \left(20 a + 27\right)\cdot 43^{2} + \left(28 a + 31\right)\cdot 43^{3} + \left(33 a + 33\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,5)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(2,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,2,5,3,6,4)$ | $1$ |
| $12$ | $6$ | $(1,5,6)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
