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(5 a + 23\right)\cdot 43 + \left(9 a + 19\right)\cdot 43^{2} + \left(10 a + 42\right)\cdot 43^{3} + \left(2 a + 3\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 23\cdot 43 + 40\cdot 43^{2} + 35\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 8 + \left(25 a + 22\right)\cdot 43 + \left(30 a + 40\right)\cdot 43^{2} + \left(10 a + 9\right)\cdot 43^{3} + \left(8 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 26 + \left(37 a + 19\right)\cdot 43 + \left(33 a + 23\right)\cdot 43^{2} + 32 a\cdot 43^{3} + \left(40 a + 39\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 19\cdot 43 + 2\cdot 43^{2} + 7\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 35 + \left(17 a + 20\right)\cdot 43 + \left(12 a + 2\right)\cdot 43^{2} + \left(32 a + 33\right)\cdot 43^{3} + \left(34 a + 41\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(2,5)(3,6)$ |
| $(1,6,4,3)$ |
| $(1,3,5)(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,5)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
