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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 9 + \left(21 a + 18\right)\cdot 43 + \left(14 a + 7\right)\cdot 43^{2} + \left(2 a + 20\right)\cdot 43^{3} + \left(18 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 24 + \left(5 a + 35\right)\cdot 43 + \left(7 a + 11\right)\cdot 43^{2} + \left(21 a + 27\right)\cdot 43^{3} + \left(9 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 29 + \left(37 a + 13\right)\cdot 43 + \left(25 a + 18\right)\cdot 43^{2} + \left(18 a + 38\right)\cdot 43^{3} + \left(35 a + 39\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 38 + \left(37 a + 26\right)\cdot 43 + \left(35 a + 13\right)\cdot 43^{2} + \left(21 a + 41\right)\cdot 43^{3} + \left(33 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 4\cdot 43 + 28\cdot 43^{2} + 5\cdot 43^{3} + 28\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 17 + \left(5 a + 20\right)\cdot 43 + \left(17 a + 6\right)\cdot 43^{2} + \left(24 a + 31\right)\cdot 43^{3} + \left(7 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 39 + \left(21 a + 9\right)\cdot 43 + 28 a\cdot 43^{2} + \left(40 a + 8\right)\cdot 43^{3} + \left(24 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
