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 + 14 + \left(6 a + 36\right)\cdot 43 + \left(23 a + 4\right)\cdot 43^{2} + \left(34 a + 40\right)\cdot 43^{3} + \left(6 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 25\cdot 43 + 37\cdot 43^{2} + 12\cdot 43^{3} + 19\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 7 + \left(36 a + 7\right)\cdot 43 + \left(19 a + 21\right)\cdot 43^{2} + \left(8 a + 8\right)\cdot 43^{3} + \left(36 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 3 + \left(6 a + 25\right)\cdot 43 + \left(8 a + 34\right)\cdot 43^{2} + \left(41 a + 9\right)\cdot 43^{3} + \left(22 a + 14\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 9\cdot 43 + 37\cdot 43^{2} + 14\cdot 43^{3} + 9\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 9 + \left(36 a + 25\right)\cdot 43 + \left(34 a + 36\right)\cdot 43^{2} + \left(a + 42\right)\cdot 43^{3} + \left(20 a + 38\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
