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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 20 + \left(4 a + 30\right)\cdot 43 + \left(29 a + 26\right)\cdot 43^{2} + 9\cdot 43^{3} + \left(31 a + 28\right)\cdot 43^{4} + \left(33 a + 27\right)\cdot 43^{5} + \left(5 a + 7\right)\cdot 43^{6} + \left(3 a + 30\right)\cdot 43^{7} + 17 a\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 19\cdot 43 + 10\cdot 43^{2} + 11\cdot 43^{3} + 3\cdot 43^{4} + 42\cdot 43^{5} + 5\cdot 43^{6} + 18\cdot 43^{7} + 27\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 42 + \left(38 a + 12\right)\cdot 43 + \left(13 a + 8\right)\cdot 43^{2} + \left(42 a + 24\right)\cdot 43^{3} + \left(11 a + 15\right)\cdot 43^{4} + \left(9 a + 30\right)\cdot 43^{5} + \left(37 a + 22\right)\cdot 43^{6} + \left(39 a + 27\right)\cdot 43^{7} + \left(25 a + 14\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 23 + \left(38 a + 12\right)\cdot 43 + \left(13 a + 16\right)\cdot 43^{2} + \left(42 a + 33\right)\cdot 43^{3} + \left(11 a + 14\right)\cdot 43^{4} + \left(9 a + 15\right)\cdot 43^{5} + \left(37 a + 35\right)\cdot 43^{6} + \left(39 a + 12\right)\cdot 43^{7} + \left(25 a + 42\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 23\cdot 43 + 32\cdot 43^{2} + 31\cdot 43^{3} + 39\cdot 43^{4} + 37\cdot 43^{6} + 24\cdot 43^{7} + 15\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 1 + \left(4 a + 30\right)\cdot 43 + \left(29 a + 34\right)\cdot 43^{2} + 18\cdot 43^{3} + \left(31 a + 27\right)\cdot 43^{4} + \left(33 a + 12\right)\cdot 43^{5} + \left(5 a + 20\right)\cdot 43^{6} + \left(3 a + 15\right)\cdot 43^{7} + \left(17 a + 28\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,6,5)(2,4,3)$ |
| $(1,6)(3,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,6,5)(2,4,3)$ | $0$ |
| $6$ | $4$ | $(1,2,4,5)(3,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
