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:
| $r_{ 1 }$ |
$=$ |
$ 23 + 8\cdot 43 + 27\cdot 43^{2} + 14\cdot 43^{3} + 29\cdot 43^{4} + 35\cdot 43^{5} + 43^{6} + 37\cdot 43^{7} + 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 1 + \left(a + 34\right)\cdot 43 + \left(15 a + 5\right)\cdot 43^{2} + \left(22 a + 33\right)\cdot 43^{3} + \left(2 a + 2\right)\cdot 43^{4} + \left(31 a + 11\right)\cdot 43^{5} + \left(31 a + 6\right)\cdot 43^{6} + \left(7 a + 40\right)\cdot 43^{7} + \left(a + 17\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 24 + \left(41 a + 12\right)\cdot 43 + \left(27 a + 19\right)\cdot 43^{2} + \left(20 a + 40\right)\cdot 43^{3} + \left(40 a + 25\right)\cdot 43^{4} + \left(11 a + 39\right)\cdot 43^{5} + \left(11 a + 6\right)\cdot 43^{6} + \left(35 a + 16\right)\cdot 43^{7} + \left(41 a + 11\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 34\cdot 43 + 15\cdot 43^{2} + 28\cdot 43^{3} + 13\cdot 43^{4} + 7\cdot 43^{5} + 41\cdot 43^{6} + 5\cdot 43^{7} + 41\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 42 + \left(41 a + 8\right)\cdot 43 + \left(27 a + 37\right)\cdot 43^{2} + \left(20 a + 9\right)\cdot 43^{3} + \left(40 a + 40\right)\cdot 43^{4} + \left(11 a + 31\right)\cdot 43^{5} + \left(11 a + 36\right)\cdot 43^{6} + \left(35 a + 2\right)\cdot 43^{7} + \left(41 a + 25\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 19 + \left(a + 30\right)\cdot 43 + \left(15 a + 23\right)\cdot 43^{2} + \left(22 a + 2\right)\cdot 43^{3} + \left(2 a + 17\right)\cdot 43^{4} + \left(31 a + 3\right)\cdot 43^{5} + \left(31 a + 36\right)\cdot 43^{6} + \left(7 a + 26\right)\cdot 43^{7} + \left(a + 31\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,4,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6,4,3)(2,5)$ |
$1$ |
| $8$ |
$6$ |
$(1,6,5,4,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
