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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 23 + \left(17 a + 39\right)\cdot 41 + \left(5 a + 6\right)\cdot 41^{2} + \left(36 a + 10\right)\cdot 41^{3} + \left(17 a + 37\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 6 + \left(13 a + 10\right)\cdot 41 + \left(12 a + 4\right)\cdot 41^{2} + \left(37 a + 21\right)\cdot 41^{3} + \left(5 a + 9\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 7 + \left(23 a + 29\right)\cdot 41 + \left(35 a + 5\right)\cdot 41^{2} + \left(4 a + 31\right)\cdot 41^{3} + \left(23 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 20\cdot 41 + 2\cdot 41^{2} + 30\cdot 41^{3} + 10\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 20\cdot 41 + 34\cdot 41^{2} + 32\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 24 + \left(27 a + 2\right)\cdot 41 + \left(28 a + 28\right)\cdot 41^{2} + \left(3 a + 38\right)\cdot 41^{3} + \left(35 a + 30\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)$ |
$3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
