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:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 30 + \left(35 a + 19\right)\cdot 41 + \left(7 a + 35\right)\cdot 41^{2} + \left(18 a + 40\right)\cdot 41^{3} + \left(35 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 37 + \left(35 a + 21\right)\cdot 41 + \left(7 a + 20\right)\cdot 41^{2} + \left(18 a + 8\right)\cdot 41^{3} + \left(35 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 38 + \left(5 a + 31\right)\cdot 41 + \left(33 a + 8\right)\cdot 41^{2} + \left(22 a + 14\right)\cdot 41^{3} + \left(5 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 15 + \left(5 a + 35\right)\cdot 41 + \left(33 a + 31\right)\cdot 41^{2} + \left(22 a + 29\right)\cdot 41^{3} + \left(5 a + 32\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 31 + \left(5 a + 29\right)\cdot 41 + \left(33 a + 23\right)\cdot 41^{2} + \left(22 a + 5\right)\cdot 41^{3} + \left(5 a + 4\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 14 + \left(35 a + 25\right)\cdot 41 + \left(7 a + 2\right)\cdot 41^{2} + \left(18 a + 24\right)\cdot 41^{3} + \left(35 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,2)(3,5,4)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
