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:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 2 + \left(a + 39\right)\cdot 43 + \left(27 a + 24\right)\cdot 43^{2} + 15 a\cdot 43^{3} + \left(7 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 6 + \left(a + 18\right)\cdot 43 + \left(27 a + 37\right)\cdot 43^{2} + \left(15 a + 30\right)\cdot 43^{3} + \left(7 a + 3\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 22 + \left(41 a + 41\right)\cdot 43 + \left(15 a + 10\right)\cdot 43^{2} + \left(27 a + 17\right)\cdot 43^{3} + \left(35 a + 12\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 38 + \left(a + 23\right)\cdot 43 + \left(27 a + 28\right)\cdot 43^{2} + \left(15 a + 28\right)\cdot 43^{3} + \left(7 a + 20\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 33 + \left(41 a + 35\right)\cdot 43 + \left(15 a + 19\right)\cdot 43^{2} + \left(27 a + 19\right)\cdot 43^{3} + \left(35 a + 38\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 29 + \left(41 a + 13\right)\cdot 43 + \left(15 a + 7\right)\cdot 43^{2} + \left(27 a + 32\right)\cdot 43^{3} + \left(35 a + 22\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,4,6,2,3)$ |
| $(1,6)(2,5)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,4,6,2,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,2,6,4,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
