Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 27 + \left(41 a + 39\right)\cdot 43 + \left(41 a + 20\right)\cdot 43^{2} + \left(5 a + 32\right)\cdot 43^{3} + \left(18 a + 27\right)\cdot 43^{4} + \left(6 a + 27\right)\cdot 43^{5} + \left(29 a + 31\right)\cdot 43^{6} + \left(28 a + 29\right)\cdot 43^{7} + \left(42 a + 36\right)\cdot 43^{8} + \left(21 a + 13\right)\cdot 43^{9} +O\left(43^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 3\cdot 43 + 15\cdot 43^{3} + 41\cdot 43^{4} + 42\cdot 43^{5} + 34\cdot 43^{6} + 23\cdot 43^{7} + 32\cdot 43^{8} +O\left(43^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 1 + \left(41 a + 20\right)\cdot 43 + \left(41 a + 21\right)\cdot 43^{2} + \left(5 a + 3\right)\cdot 43^{3} + \left(18 a + 3\right)\cdot 43^{4} + \left(6 a + 27\right)\cdot 43^{5} + \left(29 a + 31\right)\cdot 43^{6} + \left(28 a + 13\right)\cdot 43^{7} + \left(42 a + 35\right)\cdot 43^{8} + \left(21 a + 6\right)\cdot 43^{9} +O\left(43^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 16 + \left(a + 3\right)\cdot 43 + \left(a + 22\right)\cdot 43^{2} + \left(37 a + 10\right)\cdot 43^{3} + \left(24 a + 15\right)\cdot 43^{4} + \left(36 a + 15\right)\cdot 43^{5} + \left(13 a + 11\right)\cdot 43^{6} + \left(14 a + 13\right)\cdot 43^{7} + 6\cdot 43^{8} + \left(21 a + 29\right)\cdot 43^{9} +O\left(43^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 39\cdot 43 + 42\cdot 43^{2} + 27\cdot 43^{3} + 43^{4} + 8\cdot 43^{6} + 19\cdot 43^{7} + 10\cdot 43^{8} + 42\cdot 43^{9} +O\left(43^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 42 + \left(a + 22\right)\cdot 43 + \left(a + 21\right)\cdot 43^{2} + \left(37 a + 39\right)\cdot 43^{3} + \left(24 a + 39\right)\cdot 43^{4} + \left(36 a + 15\right)\cdot 43^{5} + \left(13 a + 11\right)\cdot 43^{6} + \left(14 a + 29\right)\cdot 43^{7} + 7\cdot 43^{8} + \left(21 a + 36\right)\cdot 43^{9} +O\left(43^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(1,3)(4,6)$ |
| $(1,6)(3,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(2,5)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,3)(4,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,4)(2,6,5,3)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
