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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 9 + \left(42 a + 13\right)\cdot 43 + \left(27 a + 28\right)\cdot 43^{2} + \left(38 a + 2\right)\cdot 43^{3} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 40 + 4\cdot 43 + 15 a\cdot 43^{2} + \left(4 a + 24\right)\cdot 43^{3} + \left(42 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 7 + 12\cdot 43 + \left(15 a + 7\right)\cdot 43^{2} + 4 a\cdot 43^{3} + \left(42 a + 11\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 25 + \left(42 a + 37\right)\cdot 43 + \left(27 a + 21\right)\cdot 43^{2} + \left(38 a + 32\right)\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 34 + 30\cdot 43 + \left(15 a + 13\right)\cdot 43^{2} + \left(4 a + 13\right)\cdot 43^{3} + \left(42 a + 34\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 15 + \left(42 a + 30\right)\cdot 43 + \left(27 a + 14\right)\cdot 43^{2} + \left(38 a + 13\right)\cdot 43^{3} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,4,5,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,4,5,6,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,6,5,4,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
