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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 13 + \left(10 a + 17\right)\cdot 43 + 27\cdot 43^{2} + \left(26 a + 2\right)\cdot 43^{3} + \left(2 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 5 + \left(9 a + 10\right)\cdot 43 + \left(33 a + 12\right)\cdot 43^{2} + \left(22 a + 32\right)\cdot 43^{3} + \left(24 a + 11\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 4\cdot 43 + 36\cdot 43^{2} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 35 + \left(33 a + 32\right)\cdot 43 + \left(9 a + 35\right)\cdot 43^{2} + \left(20 a + 21\right)\cdot 43^{3} + \left(18 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 19 + \left(32 a + 21\right)\cdot 43 + \left(42 a + 17\right)\cdot 43^{2} + \left(16 a + 28\right)\cdot 43^{3} + \left(40 a + 25\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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
