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 + 20 + \left(a + 29\right)\cdot 41 + \left(34 a + 37\right)\cdot 41^{2} + \left(24 a + 5\right)\cdot 41^{3} + \left(31 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 5 + \left(8 a + 31\right)\cdot 41 + \left(33 a + 40\right)\cdot 41^{2} + \left(15 a + 12\right)\cdot 41^{3} + \left(5 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 11 + \left(39 a + 38\right)\cdot 41 + \left(6 a + 14\right)\cdot 41^{2} + \left(16 a + 5\right)\cdot 41^{3} + \left(9 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 5 + \left(11 a + 6\right)\cdot 41 + \left(21 a + 25\right)\cdot 41^{2} + \left(23 a + 31\right)\cdot 41^{3} + \left(25 a + 40\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 30 + \left(29 a + 18\right)\cdot 41 + \left(19 a + 36\right)\cdot 41^{2} + \left(17 a + 39\right)\cdot 41^{3} + \left(15 a + 11\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 12 + \left(32 a + 40\right)\cdot 41 + \left(7 a + 8\right)\cdot 41^{2} + \left(25 a + 27\right)\cdot 41^{3} + \left(35 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4,3,2,5)$ |
| $(1,2,4)(3,6,5)$ |
| $(3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(3,5,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(3,6,5)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-1$ |
| $3$ | $6$ | $(1,6,4,3,2,5)$ | $0$ |
| $3$ | $6$ | $(1,5,2,3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
