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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 1 + \left(26 a + 38\right)\cdot 41 + \left(9 a + 20\right)\cdot 41^{2} + \left(14 a + 31\right)\cdot 41^{3} + \left(40 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 15 + \left(6 a + 29\right)\cdot 41 + \left(23 a + 35\right)\cdot 41^{2} + 33 a\cdot 41^{3} + \left(11 a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 26 + \left(14 a + 13\right)\cdot 41 + \left(31 a + 23\right)\cdot 41^{2} + \left(26 a + 23\right)\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 37 + \left(34 a + 27\right)\cdot 41 + \left(17 a + 16\right)\cdot 41^{2} + \left(7 a + 37\right)\cdot 41^{3} + \left(29 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + \left(2 a + 11\right)\cdot 41 + \left(25 a + 38\right)\cdot 41^{2} + \left(30 a + 1\right)\cdot 41^{3} + \left(20 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 4 + \left(38 a + 3\right)\cdot 41 + \left(15 a + 29\right)\cdot 41^{2} + \left(10 a + 27\right)\cdot 41^{3} + \left(20 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,5,3,2,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,6,2,3,5,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
