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:
| $r_{ 1 }$ |
$=$ |
$ 25 + 24\cdot 41 + 19\cdot 41^{2} + 4\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 11 + \left(13 a + 38\right)\cdot 41 + \left(33 a + 16\right)\cdot 41^{2} + \left(9 a + 1\right)\cdot 41^{3} + \left(6 a + 9\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 28\cdot 41 + 2\cdot 41^{2} + 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 39 + \left(27 a + 14\right)\cdot 41 + \left(7 a + 21\right)\cdot 41^{2} + \left(31 a + 38\right)\cdot 41^{3} + \left(34 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 8 + \left(21 a + 24\right)\cdot 41 + \left(22 a + 28\right)\cdot 41^{2} + \left(8 a + 16\right)\cdot 41^{3} + \left(29 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 9 + \left(19 a + 33\right)\cdot 41 + \left(18 a + 33\right)\cdot 41^{2} + \left(32 a + 19\right)\cdot 41^{3} + \left(11 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,5)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$-2$ |
| $6$ |
$2$ |
$(3,4)$ |
$0$ |
| $9$ |
$2$ |
$(3,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,5,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,5,6)(2,3,4)$ |
$1$ |
| $18$ |
$4$ |
$(1,2)(3,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,5,4,6,2)$ |
$1$ |
| $12$ |
$6$ |
$(1,5,6)(3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
