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:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 27 + \left(39 a + 10\right)\cdot 41 + \left(8 a + 20\right)\cdot 41^{2} + \left(19 a + 18\right)\cdot 41^{3} + \left(21 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 29 + \left(2 a + 8\right)\cdot 41 + \left(4 a + 29\right)\cdot 41^{2} + \left(3 a + 19\right)\cdot 41^{3} + \left(3 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 3 + \left(17 a + 16\right)\cdot 41 + \left(3 a + 37\right)\cdot 41^{2} + \left(9 a + 4\right)\cdot 41^{3} + \left(4 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 30 + \left(38 a + 1\right)\cdot 41 + \left(36 a + 39\right)\cdot 41^{2} + \left(37 a + 24\right)\cdot 41^{3} + \left(37 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 13 + \left(23 a + 10\right)\cdot 41 + \left(37 a + 30\right)\cdot 41^{2} + \left(31 a + 28\right)\cdot 41^{3} + \left(36 a + 39\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 22 + \left(a + 34\right)\cdot 41 + \left(32 a + 7\right)\cdot 41^{2} + \left(21 a + 26\right)\cdot 41^{3} + \left(19 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5,6,4,3)$ |
| $(1,6)(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,5,6,4,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,4,6,5,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
