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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 16 + \left(33 a + 15\right)\cdot 41 + \left(14 a + 29\right)\cdot 41^{2} + \left(21 a + 23\right)\cdot 41^{3} + \left(11 a + 40\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 24 + \left(33 a + 37\right)\cdot 41 + \left(14 a + 27\right)\cdot 41^{2} + \left(21 a + 12\right)\cdot 41^{3} + \left(11 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 33 + \left(7 a + 29\right)\cdot 41 + \left(26 a + 39\right)\cdot 41^{2} + \left(19 a + 20\right)\cdot 41^{3} + \left(29 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 35 + \left(7 a + 13\right)\cdot 41 + \left(26 a + 40\right)\cdot 41^{2} + \left(19 a + 31\right)\cdot 41^{3} + \left(29 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 2 + \left(7 a + 36\right)\cdot 41 + \left(26 a + 38\right)\cdot 41^{2} + \left(19 a + 20\right)\cdot 41^{3} + \left(29 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 14 + \left(33 a + 31\right)\cdot 41 + \left(14 a + 28\right)\cdot 41^{2} + \left(21 a + 12\right)\cdot 41^{3} + \left(11 a + 18\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,6,2)(3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
