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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 5 + \left(30 a + 21\right)\cdot 43 + \left(10 a + 29\right)\cdot 43^{2} + \left(9 a + 23\right)\cdot 43^{3} + \left(4 a + 40\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 23 + \left(12 a + 38\right)\cdot 43 + \left(32 a + 18\right)\cdot 43^{2} + \left(33 a + 24\right)\cdot 43^{3} + \left(38 a + 18\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 a + 19 + \left(12 a + 16\right)\cdot 43 + \left(32 a + 6\right)\cdot 43^{2} + \left(33 a + 37\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 12 + \left(12 a + 1\right)\cdot 43 + \left(32 a + 10\right)\cdot 43^{2} + \left(33 a + 22\right)\cdot 43^{3} + \left(38 a + 35\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 16 + \left(30 a + 15\right)\cdot 43 + \left(10 a + 38\right)\cdot 43^{2} + \left(9 a + 25\right)\cdot 43^{3} + \left(4 a + 23\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 12 + \left(30 a + 36\right)\cdot 43 + \left(10 a + 25\right)\cdot 43^{2} + \left(9 a + 38\right)\cdot 43^{3} + \left(4 a + 7\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,3,5,4,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
