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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 31 + \left(25 a + 6\right)\cdot 41 + \left(15 a + 25\right)\cdot 41^{2} + \left(28 a + 38\right)\cdot 41^{3} + \left(7 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 33 + \left(11 a + 10\right)\cdot 41 + \left(27 a + 29\right)\cdot 41^{2} + \left(23 a + 9\right)\cdot 41^{3} + \left(35 a + 23\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 22 + \left(17 a + 15\right)\cdot 41 + \left(8 a + 39\right)\cdot 41^{2} + \left(34 a + 11\right)\cdot 41^{3} + \left(23 a + 28\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 30 + \left(23 a + 38\right)\cdot 41 + \left(32 a + 5\right)\cdot 41^{2} + \left(6 a + 24\right)\cdot 41^{3} + \left(17 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 1 + \left(15 a + 12\right)\cdot 41 + \left(25 a + 5\right)\cdot 41^{2} + \left(12 a + 26\right)\cdot 41^{3} + \left(33 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 7 + \left(29 a + 39\right)\cdot 41 + \left(13 a + 17\right)\cdot 41^{2} + \left(17 a + 12\right)\cdot 41^{3} + \left(5 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(3,6,5)$ |
| $(1,4,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
| $1$ | $3$ | $(1,4,2)(3,6,5)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,2,4)(3,5,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(3,6,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(3,5,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ |
| $3$ | $6$ | $(1,5,4,3,2,6)$ | $0$ |
| $3$ | $6$ | $(1,6,2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
