Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 17 + \left(36 a + 8\right)\cdot 43 + \left(12 a + 11\right)\cdot 43^{2} + \left(38 a + 38\right)\cdot 43^{3} + \left(21 a + 39\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 33 + \left(36 a + 32\right)\cdot 43 + \left(12 a + 4\right)\cdot 43^{2} + \left(38 a + 25\right)\cdot 43^{3} + \left(21 a + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 23 + \left(36 a + 25\right)\cdot 43 + \left(12 a + 40\right)\cdot 43^{2} + \left(38 a + 5\right)\cdot 43^{3} + \left(21 a + 11\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 32 + \left(6 a + 9\right)\cdot 43 + \left(30 a + 17\right)\cdot 43^{2} + \left(4 a + 31\right)\cdot 43^{3} + \left(21 a + 37\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 26 + \left(6 a + 35\right)\cdot 43 + \left(30 a + 30\right)\cdot 43^{2} + \left(4 a + 20\right)\cdot 43^{3} + \left(21 a + 23\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 42 + \left(6 a + 16\right)\cdot 43 + \left(30 a + 24\right)\cdot 43^{2} + \left(4 a + 7\right)\cdot 43^{3} + 21 a\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,3,5,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,2,5,3,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
