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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 15 + \left(30 a + 24\right)\cdot 43 + \left(9 a + 23\right)\cdot 43^{2} + \left(18 a + 22\right)\cdot 43^{3} + \left(40 a + 28\right)\cdot 43^{4} + \left(27 a + 2\right)\cdot 43^{5} + \left(16 a + 28\right)\cdot 43^{6} + \left(5 a + 9\right)\cdot 43^{7} + \left(32 a + 36\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 3 + \left(12 a + 24\right)\cdot 43 + \left(33 a + 2\right)\cdot 43^{2} + \left(24 a + 31\right)\cdot 43^{3} + \left(2 a + 7\right)\cdot 43^{4} + \left(15 a + 33\right)\cdot 43^{5} + \left(26 a + 16\right)\cdot 43^{6} + \left(37 a + 41\right)\cdot 43^{7} + \left(10 a + 19\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 29\cdot 43 + 41\cdot 43^{2} + 2\cdot 43^{3} + 39\cdot 43^{4} + 26\cdot 43^{5} + 11\cdot 43^{6} + 20\cdot 43^{7} + 14\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 28 + \left(12 a + 18\right)\cdot 43 + \left(33 a + 19\right)\cdot 43^{2} + \left(24 a + 20\right)\cdot 43^{3} + \left(2 a + 14\right)\cdot 43^{4} + \left(15 a + 40\right)\cdot 43^{5} + \left(26 a + 14\right)\cdot 43^{6} + \left(37 a + 33\right)\cdot 43^{7} + \left(10 a + 6\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 40 + \left(30 a + 18\right)\cdot 43 + \left(9 a + 40\right)\cdot 43^{2} + \left(18 a + 11\right)\cdot 43^{3} + \left(40 a + 35\right)\cdot 43^{4} + \left(27 a + 9\right)\cdot 43^{5} + \left(16 a + 26\right)\cdot 43^{6} + \left(5 a + 1\right)\cdot 43^{7} + \left(32 a + 23\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 13\cdot 43 + 43^{2} + 40\cdot 43^{3} + 3\cdot 43^{4} + 16\cdot 43^{5} + 31\cdot 43^{6} + 22\cdot 43^{7} + 28\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $1$ |
| $8$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
