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 + 2 + \left(40 a + 10\right)\cdot 43 + \left(14 a + 31\right)\cdot 43^{2} + \left(36 a + 8\right)\cdot 43^{3} + \left(22 a + 27\right)\cdot 43^{4} + \left(9 a + 37\right)\cdot 43^{5} + \left(30 a + 15\right)\cdot 43^{6} + \left(18 a + 6\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 11 + \left(40 a + 31\right)\cdot 43 + \left(20 a + 24\right)\cdot 43^{2} + \left(28 a + 7\right)\cdot 43^{3} + \left(24 a + 17\right)\cdot 43^{4} + \left(42 a + 4\right)\cdot 43^{5} + \left(23 a + 42\right)\cdot 43^{6} + \left(33 a + 4\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 28\cdot 43 + 26\cdot 43^{2} + 5\cdot 43^{3} + 34\cdot 43^{4} + 3\cdot 43^{5} + 43^{6} + 3\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 12\cdot 43 + 36\cdot 43^{2} + 18\cdot 43^{3} + 23\cdot 43^{4} + 36\cdot 43^{5} + 9\cdot 43^{6} + 19\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 37 + \left(2 a + 15\right)\cdot 43 + \left(28 a + 5\right)\cdot 43^{2} + \left(6 a + 30\right)\cdot 43^{3} + \left(20 a + 13\right)\cdot 43^{4} + \left(33 a + 24\right)\cdot 43^{5} + \left(12 a + 36\right)\cdot 43^{6} + \left(24 a + 37\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 10 + \left(2 a + 30\right)\cdot 43 + \left(22 a + 4\right)\cdot 43^{2} + \left(14 a + 15\right)\cdot 43^{3} + \left(18 a + 13\right)\cdot 43^{4} + 22\cdot 43^{5} + \left(19 a + 23\right)\cdot 43^{6} + \left(9 a + 14\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2)(5,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)(2,5)$ |
$-1$ |
| $3$ |
$2$ |
$(2,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,6,2)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6)(2,4,5,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,3,2,6,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
