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:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 37 + \left(37 a + 23\right)\cdot 43 + \left(36 a + 15\right)\cdot 43^{2} + \left(20 a + 29\right)\cdot 43^{3} + \left(34 a + 3\right)\cdot 43^{4} + \left(28 a + 18\right)\cdot 43^{5} + \left(26 a + 8\right)\cdot 43^{6} + \left(12 a + 24\right)\cdot 43^{7} + \left(18 a + 21\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 29\cdot 43 + 12\cdot 43^{3} + 21\cdot 43^{4} + 20\cdot 43^{5} + 19\cdot 43^{6} + 16\cdot 43^{7} + 19\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 9\cdot 43 + 9\cdot 43^{2} + 32\cdot 43^{3} + 37\cdot 43^{4} + 5\cdot 43^{5} + 12\cdot 43^{6} + 35\cdot 43^{7} + 31\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 28 + \left(27 a + 1\right)\cdot 43 + \left(42 a + 15\right)\cdot 43^{2} + \left(21 a + 31\right)\cdot 43^{3} + \left(12 a + 7\right)\cdot 43^{4} + \left(19 a + 11\right)\cdot 43^{5} + \left(32 a + 13\right)\cdot 43^{6} + \left(8 a + 33\right)\cdot 43^{7} + \left(16 a + 10\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 4 + \left(5 a + 8\right)\cdot 43 + \left(6 a + 15\right)\cdot 43^{2} + \left(22 a + 13\right)\cdot 43^{3} + \left(8 a + 17\right)\cdot 43^{4} + \left(14 a + 12\right)\cdot 43^{5} + \left(16 a + 6\right)\cdot 43^{6} + \left(30 a + 10\right)\cdot 43^{7} + \left(24 a + 27\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 1 + \left(15 a + 13\right)\cdot 43 + 30\cdot 43^{2} + \left(21 a + 10\right)\cdot 43^{3} + \left(30 a + 41\right)\cdot 43^{4} + \left(23 a + 17\right)\cdot 43^{5} + \left(10 a + 26\right)\cdot 43^{6} + \left(34 a + 9\right)\cdot 43^{7} + \left(26 a + 18\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,5,2)(3,4,6)$ |
| $(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(2,6,3,5)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
| $8$ | $6$ | $(1,2,6,4,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
