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:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 14 + 39 a\cdot 43 + \left(20 a + 13\right)\cdot 43^{2} + \left(24 a + 32\right)\cdot 43^{3} + 22 a\cdot 43^{4} + \left(22 a + 5\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 8\cdot 43 + 3\cdot 43^{2} + 9\cdot 43^{3} + 23\cdot 43^{4} + 22\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 39 + \left(17 a + 37\right)\cdot 43 + \left(17 a + 18\right)\cdot 43^{2} + \left(6 a + 24\right)\cdot 43^{3} + \left(29 a + 23\right)\cdot 43^{4} + \left(a + 11\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 6 + \left(3 a + 5\right)\cdot 43 + \left(22 a + 37\right)\cdot 43^{2} + \left(18 a + 35\right)\cdot 43^{3} + \left(20 a + 41\right)\cdot 43^{4} + \left(20 a + 4\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 2\cdot 43 + 38\cdot 43^{2} + 13\cdot 43^{3} + 36\cdot 43^{4} + 14\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 20 + \left(25 a + 31\right)\cdot 43 + \left(25 a + 18\right)\cdot 43^{2} + \left(36 a + 13\right)\cdot 43^{3} + \left(13 a + 3\right)\cdot 43^{4} + \left(41 a + 27\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(2,5)$ |
| $(3,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
| $4$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
