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