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:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 25 + \left(27 a + 2\right)\cdot 41 + \left(a + 36\right)\cdot 41^{2} + \left(15 a + 32\right)\cdot 41^{3} + \left(5 a + 32\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 36 + \left(22 a + 26\right)\cdot 41 + \left(34 a + 1\right)\cdot 41^{2} + \left(39 a + 6\right)\cdot 41^{3} + \left(30 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 35 + \left(13 a + 26\right)\cdot 41 + \left(39 a + 13\right)\cdot 41^{2} + \left(25 a + 35\right)\cdot 41^{3} + \left(35 a + 33\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 28 + 18 a\cdot 41 + \left(6 a + 1\right)\cdot 41^{2} + \left(a + 9\right)\cdot 41^{3} + \left(10 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 25 + \left(24 a + 28\right)\cdot 41 + \left(38 a + 30\right)\cdot 41^{2} + \left(19 a + 29\right)\cdot 41^{3} + \left(34 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 15 + \left(16 a + 37\right)\cdot 41 + \left(2 a + 39\right)\cdot 41^{2} + \left(21 a + 9\right)\cdot 41^{3} + \left(6 a + 38\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$16$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $15$ |
$2$ |
$(1,2)$ |
$0$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
