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:
| $r_{ 1 }$ |
$=$ |
$ 36 a + 20 + \left(12 a + 23\right)\cdot 43 + \left(32 a + 21\right)\cdot 43^{2} + \left(33 a + 40\right)\cdot 43^{3} + \left(38 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 1 + \left(30 a + 9\right)\cdot 43 + \left(10 a + 19\right)\cdot 43^{2} + \left(9 a + 2\right)\cdot 43^{3} + \left(4 a + 37\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 13 + 30 a\cdot 43 + \left(10 a + 41\right)\cdot 43^{2} + \left(9 a + 41\right)\cdot 43^{3} + \left(4 a + 36\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 19 + \left(30 a + 20\right)\cdot 43 + \left(10 a + 33\right)\cdot 43^{2} + 9 a\cdot 43^{3} + \left(4 a + 41\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 26 + 12 a\cdot 43 + \left(32 a + 14\right)\cdot 43^{2} + \left(33 a + 42\right)\cdot 43^{3} + \left(38 a + 35\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 8 + \left(12 a + 32\right)\cdot 43 + \left(32 a + 42\right)\cdot 43^{2} + 33 a\cdot 43^{3} + \left(38 a + 32\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5,3,6,4)$ |
| $(1,3)(2,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,6)(4,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,5,6)(2,3,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,6,5)(2,4,3)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,5,3,6,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,6,3,5,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
