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:
| $r_{ 1 }$ |
$=$ |
$ 12 + 27\cdot 43 + 38\cdot 43^{2} + 26\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a + 30 + \left(41 a + 13\right)\cdot 43 + \left(26 a + 7\right)\cdot 43^{2} + \left(37 a + 16\right)\cdot 43^{3} + \left(40 a + 41\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 4 + \left(38 a + 41\right)\cdot 43 + \left(40 a + 41\right)\cdot 43^{2} + \left(10 a + 14\right)\cdot 43^{3} + \left(38 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 15\cdot 43 + 4\cdot 43^{2} + 16\cdot 43^{3} + 39\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 13 + \left(a + 29\right)\cdot 43 + \left(16 a + 35\right)\cdot 43^{2} + \left(5 a + 26\right)\cdot 43^{3} + \left(2 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 39 + \left(4 a + 1\right)\cdot 43 + \left(2 a + 1\right)\cdot 43^{2} + \left(32 a + 28\right)\cdot 43^{3} + \left(4 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,4)(2,5)$ |
| $(1,6,2)(3,5,4)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,4)(3,6)$ | $1$ |
| $8$ | $3$ | $(1,6,2)(3,5,4)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
