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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 38\cdot 41 + 38\cdot 41^{2} + 31\cdot 41^{3} + 4\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 40 + \left(26 a + 14\right)\cdot 41 + \left(34 a + 2\right)\cdot 41^{2} + \left(31 a + 31\right)\cdot 41^{3} + \left(37 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 16 + \left(16 a + 18\right)\cdot 41 + \left(40 a + 29\right)\cdot 41^{2} + \left(31 a + 33\right)\cdot 41^{3} + \left(12 a + 37\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 2\cdot 41 + 2\cdot 41^{2} + 9\cdot 41^{3} + 36\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 1 + \left(14 a + 26\right)\cdot 41 + \left(6 a + 38\right)\cdot 41^{2} + \left(9 a + 9\right)\cdot 41^{3} + \left(3 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a + 25 + \left(24 a + 22\right)\cdot 41 + 11\cdot 41^{2} + \left(9 a + 7\right)\cdot 41^{3} + \left(28 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,3,5)(2,4,6)$ |
| $(1,4)(2,3)(5,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(2,5)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(2,6,5,3)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
