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:
| $r_{ 1 }$ |
$=$ |
$ 42 a + 11 + \left(15 a + 40\right)\cdot 43 + \left(13 a + 16\right)\cdot 43^{2} + \left(39 a + 38\right)\cdot 43^{3} + \left(32 a + 21\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 28\cdot 43 + 11\cdot 43^{2} + 22\cdot 43^{3} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 31\cdot 43 + 11\cdot 43^{2} + 26\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 18 + \left(11 a + 33\right)\cdot 43 + \left(2 a + 41\right)\cdot 43^{2} + \left(28 a + 18\right)\cdot 43^{3} + 40 a\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 39 + \left(31 a + 23\right)\cdot 43 + \left(40 a + 32\right)\cdot 43^{2} + \left(14 a + 1\right)\cdot 43^{3} + \left(2 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 10 + \left(27 a + 14\right)\cdot 43 + \left(29 a + 14\right)\cdot 43^{2} + \left(3 a + 21\right)\cdot 43^{3} + \left(10 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,4)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,3,6)(2,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
