Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 a + \left(13 a + 4\right)\cdot 41 + \left(24 a + 26\right)\cdot 41^{2} + \left(34 a + 2\right)\cdot 41^{3} + \left(28 a + 36\right)\cdot 41^{4} + \left(30 a + 15\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 31 + \left(27 a + 20\right)\cdot 41 + \left(16 a + 3\right)\cdot 41^{2} + 6 a\cdot 41^{3} + \left(12 a + 6\right)\cdot 41^{4} + \left(10 a + 38\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 31\cdot 41 + 28\cdot 41^{2} + 18\cdot 41^{3} + 34\cdot 41^{4} + 38\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 23 + \left(11 a + 22\right)\cdot 41 + \left(12 a + 35\right)\cdot 41^{2} + \left(28 a + 30\right)\cdot 41^{3} + \left(3 a + 13\right)\cdot 41^{4} + \left(4 a + 29\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 39\cdot 41 + 8\cdot 41^{2} + 8\cdot 41^{3} + 36\cdot 41^{4} + 3\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 15 + \left(29 a + 4\right)\cdot 41 + \left(28 a + 20\right)\cdot 41^{2} + \left(12 a + 21\right)\cdot 41^{3} + \left(37 a + 37\right)\cdot 41^{4} + \left(36 a + 37\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,3,4)(2,5,6)$ |
| $(4,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)$ |
$1$ |
| $3$ |
$2$ |
$(1,2)(3,5)$ |
$-1$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,4,3)(2,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,5,6,2,3,4)$ |
$0$ |
| $4$ |
$6$ |
$(1,4,3,2,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
