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