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:
| $r_{ 1 }$ |
$=$ |
$ 3 + 35\cdot 43 + 7\cdot 43^{2} + 4\cdot 43^{3} + 42\cdot 43^{4} + 37\cdot 43^{5} + 8\cdot 43^{6} + 9\cdot 43^{7} + 30\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 28 + \left(29 a + 13\right)\cdot 43 + \left(a + 11\right)\cdot 43^{2} + \left(42 a + 24\right)\cdot 43^{3} + \left(5 a + 25\right)\cdot 43^{4} + \left(37 a + 33\right)\cdot 43^{5} + 32 a\cdot 43^{6} + \left(20 a + 31\right)\cdot 43^{7} + \left(5 a + 6\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 32\cdot 43 + 33\cdot 43^{2} + 15\cdot 43^{3} + 35\cdot 43^{4} + 15\cdot 43^{5} + 18\cdot 43^{6} + 23\cdot 43^{7} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 36 + \left(4 a + 2\right)\cdot 43 + \left(40 a + 7\right)\cdot 43^{2} + \left(17 a + 21\right)\cdot 43^{3} + \left(30 a + 33\right)\cdot 43^{4} + \left(8 a + 20\right)\cdot 43^{5} + \left(11 a + 29\right)\cdot 43^{6} + \left(41 a + 29\right)\cdot 43^{7} + \left(22 a + 37\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 25 + \left(13 a + 3\right)\cdot 43 + \left(41 a + 26\right)\cdot 43^{2} + 21\cdot 43^{3} + \left(37 a + 32\right)\cdot 43^{4} + \left(5 a + 21\right)\cdot 43^{5} + \left(10 a + 39\right)\cdot 43^{6} + \left(22 a + 18\right)\cdot 43^{7} + \left(37 a + 34\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 2 + \left(38 a + 41\right)\cdot 43 + \left(2 a + 42\right)\cdot 43^{2} + \left(25 a + 41\right)\cdot 43^{3} + \left(12 a + 2\right)\cdot 43^{4} + \left(34 a + 42\right)\cdot 43^{5} + \left(31 a + 31\right)\cdot 43^{6} + \left(a + 16\right)\cdot 43^{7} + \left(20 a + 19\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,5)$ |
| $(1,2,4)(3,6,5)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)(4,5)$ |
$-1$ |
| $3$ |
$2$ |
$(4,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(3,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,3,4)$ |
$-1$ |
| $6$ |
$4$ |
$(1,3)(2,4,6,5)$ |
$1$ |
| $8$ |
$6$ |
$(1,2,4,3,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
