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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 14 + \left(18 a + 6\right)\cdot 43 + \left(29 a + 15\right)\cdot 43^{2} + \left(9 a + 36\right)\cdot 43^{3} + \left(27 a + 13\right)\cdot 43^{4} + \left(25 a + 42\right)\cdot 43^{5} + \left(20 a + 17\right)\cdot 43^{6} + \left(21 a + 1\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 37\cdot 43 + 43^{2} + 33\cdot 43^{3} + 40\cdot 43^{4} + 2\cdot 43^{5} + 12\cdot 43^{6} + 39\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 3 + \left(18 a + 1\right)\cdot 43 + \left(29 a + 17\right)\cdot 43^{2} + \left(9 a + 26\right)\cdot 43^{3} + \left(27 a + 11\right)\cdot 43^{4} + \left(25 a + 2\right)\cdot 43^{5} + \left(20 a + 30\right)\cdot 43^{6} + \left(21 a + 40\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 29 + \left(24 a + 36\right)\cdot 43 + \left(13 a + 27\right)\cdot 43^{2} + \left(33 a + 6\right)\cdot 43^{3} + \left(15 a + 29\right)\cdot 43^{4} + 17 a\cdot 43^{5} + \left(22 a + 25\right)\cdot 43^{6} + \left(21 a + 41\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 + 5\cdot 43 + 41\cdot 43^{2} + 9\cdot 43^{3} + 2\cdot 43^{4} + 40\cdot 43^{5} + 30\cdot 43^{6} + 3\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 40 + \left(24 a + 41\right)\cdot 43 + \left(13 a + 25\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + \left(15 a + 31\right)\cdot 43^{4} + \left(17 a + 40\right)\cdot 43^{5} + \left(22 a + 12\right)\cdot 43^{6} + \left(21 a + 2\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(2,6)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,2,4,6,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
