Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 43 + 25\cdot 43^{2} + 26\cdot 43^{4} + 37\cdot 43^{5} + 23\cdot 43^{6} + 22\cdot 43^{7} + 5\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 2 + \left(27 a + 6\right)\cdot 43 + \left(6 a + 32\right)\cdot 43^{2} + \left(21 a + 35\right)\cdot 43^{3} + \left(18 a + 22\right)\cdot 43^{4} + \left(19 a + 42\right)\cdot 43^{5} + \left(19 a + 42\right)\cdot 43^{6} + \left(41 a + 31\right)\cdot 43^{7} + \left(14 a + 34\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 12 + \left(30 a + 16\right)\cdot 43 + \left(12 a + 30\right)\cdot 43^{2} + \left(4 a + 25\right)\cdot 43^{3} + \left(9 a + 40\right)\cdot 43^{4} + \left(34 a + 8\right)\cdot 43^{5} + \left(38 a + 19\right)\cdot 43^{6} + \left(22 a + 29\right)\cdot 43^{7} + \left(24 a + 20\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 42 + \left(15 a + 36\right)\cdot 43 + \left(36 a + 10\right)\cdot 43^{2} + \left(21 a + 7\right)\cdot 43^{3} + \left(24 a + 20\right)\cdot 43^{4} + 23 a\cdot 43^{5} + 23 a\cdot 43^{6} + \left(a + 11\right)\cdot 43^{7} + \left(28 a + 8\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 41\cdot 43 + 17\cdot 43^{2} + 42\cdot 43^{3} + 16\cdot 43^{4} + 5\cdot 43^{5} + 19\cdot 43^{6} + 20\cdot 43^{7} + 37\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 32 + \left(12 a + 26\right)\cdot 43 + \left(30 a + 12\right)\cdot 43^{2} + \left(38 a + 17\right)\cdot 43^{3} + \left(33 a + 2\right)\cdot 43^{4} + \left(8 a + 34\right)\cdot 43^{5} + \left(4 a + 23\right)\cdot 43^{6} + \left(20 a + 13\right)\cdot 43^{7} + \left(18 a + 22\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,5)$ |
| $(3,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,5)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(2,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,6,4,5,3,2)$ |
$0$ |
| $4$ |
$6$ |
$(1,2,3,5,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
