Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 33 + \left(37 a + 39\right)\cdot 43 + \left(40 a + 11\right)\cdot 43^{2} + \left(16 a + 14\right)\cdot 43^{3} + \left(40 a + 34\right)\cdot 43^{4} + \left(3 a + 12\right)\cdot 43^{5} + \left(9 a + 6\right)\cdot 43^{6} + \left(24 a + 3\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 9 + \left(10 a + 28\right)\cdot 43 + \left(28 a + 30\right)\cdot 43^{2} + \left(8 a + 29\right)\cdot 43^{3} + \left(28 a + 9\right)\cdot 43^{4} + \left(12 a + 33\right)\cdot 43^{5} + \left(16 a + 38\right)\cdot 43^{6} + \left(35 a + 23\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 20 + \left(5 a + 4\right)\cdot 43 + \left(2 a + 15\right)\cdot 43^{2} + \left(26 a + 33\right)\cdot 43^{3} + \left(2 a + 14\right)\cdot 43^{4} + \left(39 a + 19\right)\cdot 43^{5} + \left(33 a + 11\right)\cdot 43^{6} + \left(18 a + 18\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 19 + \left(32 a + 28\right)\cdot 43 + \left(14 a + 5\right)\cdot 43^{2} + \left(34 a + 10\right)\cdot 43^{3} + \left(14 a + 29\right)\cdot 43^{4} + \left(30 a + 17\right)\cdot 43^{5} + \left(26 a + 42\right)\cdot 43^{6} + \left(7 a + 42\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 33\cdot 43 + 5\cdot 43^{2} + 13\cdot 43^{3} + 32\cdot 43^{4} + 21\cdot 43^{5} + 33\cdot 43^{6} + 2\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 37\cdot 43 + 16\cdot 43^{2} + 28\cdot 43^{3} + 8\cdot 43^{4} + 24\cdot 43^{5} + 39\cdot 43^{6} + 37\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(3,5)(4,6)$ |
| $(1,3,5)(2,4,6)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(3,6,4,5)$ | $-1$ |
| $6$ | $4$ | $(1,6,2,5)(3,4)$ | $1$ |
| $8$ | $6$ | $(1,3,6,2,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
