Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 14\cdot 43 + 42\cdot 43^{2} + 21\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 28 + \left(39 a + 16\right)\cdot 43 + \left(23 a + 29\right)\cdot 43^{2} + \left(30 a + 39\right)\cdot 43^{3} + \left(11 a + 30\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 a + 25 + \left(35 a + 21\right)\cdot 43 + \left(36 a + 42\right)\cdot 43^{2} + \left(33 a + 22\right)\cdot 43^{3} + \left(21 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 28\cdot 43 + 21\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 15 + \left(3 a + 26\right)\cdot 43 + \left(19 a + 13\right)\cdot 43^{2} + \left(12 a + 3\right)\cdot 43^{3} + \left(31 a + 12\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 18 + \left(7 a + 21\right)\cdot 43 + 6 a\cdot 43^{2} + \left(9 a + 20\right)\cdot 43^{3} + \left(21 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(2,5)(3,6)$ |
| $(1,5,3)(2,6,4)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,4)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$0$ |
| $6$ |
$4$ |
$(2,6,5,3)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
