Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 15 + \left(5 a + 31\right)\cdot 43 + \left(6 a + 5\right)\cdot 43^{2} + \left(27 a + 27\right)\cdot 43^{3} + \left(16 a + 6\right)\cdot 43^{4} + \left(26 a + 2\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 24 + \left(34 a + 31\right)\cdot 43 + \left(31 a + 6\right)\cdot 43^{2} + \left(32 a + 34\right)\cdot 43^{3} + \left(33 a + 36\right)\cdot 43^{4} + \left(36 a + 9\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 33 + \left(8 a + 33\right)\cdot 43 + \left(15 a + 6\right)\cdot 43^{2} + \left(20 a + 17\right)\cdot 43^{3} + \left(a + 30\right)\cdot 43^{4} + \left(41 a + 35\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 6\cdot 43 + 12\cdot 43^{2} + 24\cdot 43^{3} + 18\cdot 43^{4} + 3\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 25\cdot 43 + 31\cdot 43^{2} + 6\cdot 43^{3} + 34\cdot 43^{4} + 20\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 29 + \left(34 a + 3\right)\cdot 43 + \left(27 a + 13\right)\cdot 43^{2} + \left(22 a + 22\right)\cdot 43^{3} + \left(41 a + 11\right)\cdot 43^{4} + \left(a + 32\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 a + 41 + \left(37 a + 10\right)\cdot 43 + \left(36 a + 6\right)\cdot 43^{2} + \left(15 a + 5\right)\cdot 43^{3} + \left(26 a + 39\right)\cdot 43^{4} + \left(16 a + 11\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 6 a + 18 + \left(8 a + 29\right)\cdot 43 + \left(11 a + 3\right)\cdot 43^{2} + \left(10 a + 35\right)\cdot 43^{3} + \left(9 a + 37\right)\cdot 43^{4} + \left(6 a + 12\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,6)(3,8)(4,5)$ |
| $(1,7)(2,4)(5,6)$ |
| $(1,8,7,3)(2,4,6,5)$ |
| $(1,5,2)(4,6,7)$ |
| $(1,4,7,5)(2,3,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,6)(3,8)(4,5)$ |
$-4$ |
| $12$ |
$2$ |
$(1,7)(2,4)(5,6)$ |
$0$ |
| $8$ |
$3$ |
$(2,4,8)(3,6,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,4,7,5)(2,3,6,8)$ |
$0$ |
| $8$ |
$6$ |
$(1,7)(2,3,4,6,8,5)$ |
$-1$ |
| $6$ |
$8$ |
$(1,2,3,5,7,6,8,4)$ |
$0$ |
| $6$ |
$8$ |
$(1,6,3,4,7,2,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
