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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 24 + \left(11 a + 37\right)\cdot 41 + \left(11 a + 34\right)\cdot 41^{2} + \left(11 a + 22\right)\cdot 41^{3} + \left(30 a + 4\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 28 + \left(24 a + 35\right)\cdot 41 + \left(27 a + 29\right)\cdot 41^{2} + \left(22 a + 1\right)\cdot 41^{3} + \left(8 a + 18\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 21\cdot 41 + 4\cdot 41^{2} + 38\cdot 41^{3} + 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 31 + \left(16 a + 24\right)\cdot 41 + \left(13 a + 6\right)\cdot 41^{2} + \left(18 a + 1\right)\cdot 41^{3} + \left(32 a + 21\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 33\cdot 41 + 30\cdot 41^{2} + 13\cdot 41^{3} + 34\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 2 + \left(29 a + 11\right)\cdot 41 + \left(29 a + 16\right)\cdot 41^{2} + \left(29 a + 4\right)\cdot 41^{3} + \left(10 a + 2\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(1,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,5,6)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
