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