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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 39 + 8\cdot 41 + \left(37 a + 8\right)\cdot 41^{2} + \left(31 a + 31\right)\cdot 41^{3} + \left(21 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 30 + \left(40 a + 14\right)\cdot 41 + \left(3 a + 36\right)\cdot 41^{2} + \left(9 a + 7\right)\cdot 41^{3} + \left(19 a + 9\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 23 + \left(33 a + 12\right)\cdot 41 + \left(40 a + 14\right)\cdot 41^{2} + \left(21 a + 10\right)\cdot 41^{3} + \left(35 a + 18\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a + 9 + \left(7 a + 21\right)\cdot 41 + 21\cdot 41^{2} + \left(19 a + 35\right)\cdot 41^{3} + \left(5 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 20 + \left(6 a + 19\right)\cdot 41 + \left(4 a + 18\right)\cdot 41^{2} + \left(28 a + 40\right)\cdot 41^{3} + \left(24 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 2 + \left(34 a + 5\right)\cdot 41 + \left(36 a + 24\right)\cdot 41^{2} + \left(12 a + 38\right)\cdot 41^{3} + \left(16 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 |
| $(1,2)(3,4)(5,6)$ |
| $(1,5,3)(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,3,2,5,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,5,2,3,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
