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