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 + 9 + \left(a + 2\right)\cdot 41 + \left(29 a + 36\right)\cdot 41^{2} + \left(34 a + 9\right)\cdot 41^{3} + \left(15 a + 23\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 32 + \left(a + 39\right)\cdot 41 + \left(29 a + 12\right)\cdot 41^{2} + \left(34 a + 35\right)\cdot 41^{3} + \left(15 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 25 + \left(a + 37\right)\cdot 41 + \left(29 a + 27\right)\cdot 41^{2} + \left(34 a + 26\right)\cdot 41^{3} + \left(15 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 2 + \left(39 a + 14\right)\cdot 41 + \left(11 a + 16\right)\cdot 41^{2} + \left(6 a + 28\right)\cdot 41^{3} + \left(25 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 36 + \left(39 a + 11\right)\cdot 41 + \left(11 a + 31\right)\cdot 41^{2} + \left(6 a + 19\right)\cdot 41^{3} + \left(25 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 20 + \left(39 a + 17\right)\cdot 41 + \left(11 a + 39\right)\cdot 41^{2} + \left(6 a + 2\right)\cdot 41^{3} + \left(25 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,4)(3,5)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,2,6,3,4)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,4,3,6,2,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
