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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 30\cdot 43 + 30\cdot 43^{2} + 3\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 21 + \left(26 a + 14\right)\cdot 43 + \left(16 a + 32\right)\cdot 43^{2} + \left(27 a + 12\right)\cdot 43^{3} + \left(37 a + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 28 + \left(17 a + 16\right)\cdot 43 + \left(33 a + 34\right)\cdot 43^{2} + 2\cdot 43^{3} + \left(15 a + 19\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 + 35\cdot 43 + 43^{2} + 30\cdot 43^{3} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 3 + \left(25 a + 16\right)\cdot 43 + \left(9 a + 7\right)\cdot 43^{2} + \left(42 a + 13\right)\cdot 43^{3} + \left(27 a + 33\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a + 3 + \left(16 a + 16\right)\cdot 43 + \left(26 a + 22\right)\cdot 43^{2} + \left(15 a + 23\right)\cdot 43^{3} + \left(5 a + 26\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4,3,2,5)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,4)(2,3)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $20$ | $3$ | $(1,4,2)(3,5,6)$ | $1$ |
| $30$ | $4$ | $(1,2,5,3)$ | $0$ |
| $24$ | $5$ | $(1,6,2,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,6,4,3,2,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
