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:
| $r_{ 1 }$ |
$=$ |
$ 29 + 11\cdot 41 + 18\cdot 41^{2} + 29\cdot 41^{3} + 28\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 27\cdot 41 + 2\cdot 41^{2} + 29\cdot 41^{3} + 35\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 3 + \left(13 a + 35\right)\cdot 41 + \left(26 a + 8\right)\cdot 41^{2} + \left(36 a + 10\right)\cdot 41^{3} + \left(4 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 16 + \left(27 a + 16\right)\cdot 41 + \left(14 a + 33\right)\cdot 41^{2} + \left(4 a + 11\right)\cdot 41^{3} + \left(36 a + 9\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 17 + \left(34 a + 28\right)\cdot 41 + \left(13 a + 5\right)\cdot 41^{2} + \left(9 a + 14\right)\cdot 41^{3} + \left(12 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 29 + \left(6 a + 3\right)\cdot 41 + \left(27 a + 13\right)\cdot 41^{2} + \left(31 a + 28\right)\cdot 41^{3} + \left(28 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$-2$ |
| $6$ |
$2$ |
$(3,4)$ |
$0$ |
| $9$ |
$2$ |
$(3,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$-2$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,2)(3,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,3,6,4,2)$ |
$1$ |
| $12$ |
$6$ |
$(2,5,6)(3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
