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:
| $r_{ 1 }$ |
$=$ |
$ 28 + 12\cdot 43^{2} + 16\cdot 43^{3} + 33\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 1 + \left(10 a + 6\right)\cdot 43 + 24 a\cdot 43^{2} + \left(37 a + 13\right)\cdot 43^{3} + \left(10 a + 37\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 a + 6 + \left(12 a + 26\right)\cdot 43 + \left(11 a + 30\right)\cdot 43^{2} + \left(10 a + 15\right)\cdot 43^{3} + \left(23 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 10 + \left(32 a + 7\right)\cdot 43 + \left(18 a + 14\right)\cdot 43^{2} + \left(5 a + 26\right)\cdot 43^{3} + \left(32 a + 10\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 42 + \left(30 a + 2\right)\cdot 43 + \left(31 a + 29\right)\cdot 43^{2} + \left(32 a + 14\right)\cdot 43^{3} + \left(19 a + 30\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
