Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 16 + \left(14 a + 20\right)\cdot 43 + \left(22 a + 17\right)\cdot 43^{2} + \left(9 a + 6\right)\cdot 43^{3} + \left(37 a + 29\right)\cdot 43^{4} + \left(8 a + 35\right)\cdot 43^{5} + \left(27 a + 33\right)\cdot 43^{6} + \left(14 a + 27\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 11 + \left(26 a + 19\right)\cdot 43 + \left(10 a + 29\right)\cdot 43^{2} + \left(4 a + 24\right)\cdot 43^{3} + \left(29 a + 30\right)\cdot 43^{4} + \left(12 a + 29\right)\cdot 43^{5} + \left(5 a + 3\right)\cdot 43^{6} + \left(10 a + 19\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 33 + \left(16 a + 23\right)\cdot 43 + \left(32 a + 13\right)\cdot 43^{2} + \left(38 a + 18\right)\cdot 43^{3} + \left(13 a + 12\right)\cdot 43^{4} + \left(30 a + 13\right)\cdot 43^{5} + \left(37 a + 39\right)\cdot 43^{6} + \left(32 a + 23\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 16\cdot 43 + 24\cdot 43^{2} + 22\cdot 43^{3} + 17\cdot 43^{4} + 24\cdot 43^{5} + 28\cdot 43^{6} + 10\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 28 + \left(28 a + 22\right)\cdot 43 + \left(20 a + 25\right)\cdot 43^{2} + \left(33 a + 36\right)\cdot 43^{3} + \left(5 a + 13\right)\cdot 43^{4} + \left(34 a + 7\right)\cdot 43^{5} + \left(15 a + 9\right)\cdot 43^{6} + \left(28 a + 15\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 26\cdot 43 + 18\cdot 43^{2} + 20\cdot 43^{3} + 25\cdot 43^{4} + 18\cdot 43^{5} + 14\cdot 43^{6} + 32\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(1,5)$ |
| $(2,3)$ |
| $(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,5)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(2,3)$ |
$-1$ |
| $4$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,6,3,5,4,2)$ |
$0$ |
| $4$ |
$6$ |
$(1,2,4,5,3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
