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:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 5 + \left(36 a + 9\right)\cdot 43 + \left(20 a + 42\right)\cdot 43^{2} + \left(36 a + 17\right)\cdot 43^{3} + \left(33 a + 3\right)\cdot 43^{4} + \left(8 a + 21\right)\cdot 43^{5} + \left(12 a + 2\right)\cdot 43^{6} + \left(11 a + 31\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 33 + \left(36 a + 14\right)\cdot 43 + \left(20 a + 23\right)\cdot 43^{2} + \left(36 a + 7\right)\cdot 43^{3} + \left(33 a + 24\right)\cdot 43^{4} + \left(8 a + 26\right)\cdot 43^{5} + \left(12 a + 37\right)\cdot 43^{6} + \left(11 a + 42\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 40 + \left(6 a + 10\right)\cdot 43 + \left(22 a + 26\right)\cdot 43^{2} + \left(6 a + 33\right)\cdot 43^{3} + 9 a\cdot 43^{4} + \left(34 a + 39\right)\cdot 43^{5} + \left(30 a + 5\right)\cdot 43^{6} + \left(31 a + 30\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 20\cdot 43 + 5\cdot 43^{2} + 18\cdot 43^{3} + 28\cdot 43^{4} + 43^{5} + 17\cdot 43^{6} + 40\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 25 + \left(6 a + 16\right)\cdot 43 + \left(22 a + 7\right)\cdot 43^{2} + \left(6 a + 23\right)\cdot 43^{3} + \left(9 a + 21\right)\cdot 43^{4} + \left(34 a + 1\right)\cdot 43^{5} + \left(30 a + 41\right)\cdot 43^{6} + \left(31 a + 41\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 14\cdot 43 + 24\cdot 43^{2} + 28\cdot 43^{3} + 7\cdot 43^{4} + 39\cdot 43^{5} + 24\cdot 43^{6} + 28\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(3,6)(4,5)$ |
| $(1,3)(2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,3)(2,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,3)(2,4,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,4,3,2,6,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
