Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 27 + 18\cdot 43 + \left(15 a + 15\right)\cdot 43^{2} + \left(30 a + 1\right)\cdot 43^{3} + \left(24 a + 28\right)\cdot 43^{4} + \left(38 a + 34\right)\cdot 43^{5} + \left(13 a + 32\right)\cdot 43^{6} + \left(16 a + 22\right)\cdot 43^{7} + \left(4 a + 37\right)\cdot 43^{8} + \left(26 a + 35\right)\cdot 43^{9} + \left(25 a + 21\right)\cdot 43^{10} +O\left(43^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 14\cdot 43 + 38\cdot 43^{2} + 41\cdot 43^{3} + 21\cdot 43^{4} + 6\cdot 43^{5} + 36\cdot 43^{6} + 9\cdot 43^{7} + 13\cdot 43^{8} + 37\cdot 43^{9} + 17\cdot 43^{10} +O\left(43^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 + 28\cdot 43 + 36\cdot 43^{2} + 34\cdot 43^{3} + 27\cdot 43^{4} + 22\cdot 43^{5} + 22\cdot 43^{6} + 32\cdot 43^{7} + 27\cdot 43^{8} + 18\cdot 43^{9} + 14\cdot 43^{10} +O\left(43^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 34 + \left(33 a + 25\right)\cdot 43 + \left(30 a + 27\right)\cdot 43^{2} + \left(26 a + 40\right)\cdot 43^{3} + \left(39 a + 7\right)\cdot 43^{4} + \left(33 a + 11\right)\cdot 43^{5} + \left(14 a + 24\right)\cdot 43^{6} + \left(33 a + 31\right)\cdot 43^{7} + \left(21 a + 39\right)\cdot 43^{8} + \left(9 a + 38\right)\cdot 43^{9} + \left(15 a + 23\right)\cdot 43^{10} +O\left(43^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 24 + \left(42 a + 22\right)\cdot 43 + \left(27 a + 29\right)\cdot 43^{2} + \left(12 a + 16\right)\cdot 43^{3} + \left(18 a + 22\right)\cdot 43^{4} + \left(4 a + 5\right)\cdot 43^{5} + \left(29 a + 8\right)\cdot 43^{6} + \left(26 a + 25\right)\cdot 43^{7} + \left(38 a + 25\right)\cdot 43^{8} + \left(16 a + 14\right)\cdot 43^{9} + \left(17 a + 21\right)\cdot 43^{10} +O\left(43^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 31 + \left(9 a + 19\right)\cdot 43 + \left(12 a + 24\right)\cdot 43^{2} + \left(16 a + 36\right)\cdot 43^{3} + \left(3 a + 20\right)\cdot 43^{4} + \left(9 a + 5\right)\cdot 43^{5} + \left(28 a + 5\right)\cdot 43^{6} + \left(9 a + 7\right)\cdot 43^{7} + \left(21 a + 28\right)\cdot 43^{8} + \left(33 a + 26\right)\cdot 43^{9} + \left(27 a + 29\right)\cdot 43^{10} +O\left(43^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(1,4)(5,6)$ |
| $(1,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-3$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $3$ | $2$ | $(1,5)(2,3)$ | $-1$ |
| $6$ | $2$ | $(1,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,2,5,3)$ | $-1$ |
| $6$ | $4$ | $(1,5)(2,6,3,4)$ | $1$ |
| $8$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
