Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 5 + \left(3 a + 23\right)\cdot 41 + \left(39 a + 5\right)\cdot 41^{2} + \left(24 a + 24\right)\cdot 41^{3} + \left(9 a + 40\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 a + 11 + \left(37 a + 30\right)\cdot 41 + \left(a + 37\right)\cdot 41^{2} + \left(16 a + 18\right)\cdot 41^{3} + \left(31 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 18 + \left(17 a + 28\right)\cdot 41 + \left(28 a + 1\right)\cdot 41^{2} + \left(35 a + 40\right)\cdot 41^{3} + \left(2 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 20\cdot 41 + 8\cdot 41^{2} + 3\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 34 + \left(23 a + 20\right)\cdot 41 + \left(12 a + 28\right)\cdot 41^{2} + \left(5 a + 36\right)\cdot 41^{3} + \left(38 a + 31\right)\cdot 41^{4} +O\left(41^{ 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 |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
