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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 40 + \left(15 a + 21\right)\cdot 41 + \left(31 a + 15\right)\cdot 41^{2} + \left(20 a + 1\right)\cdot 41^{3} + \left(8 a + 2\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 1 + \left(15 a + 6\right)\cdot 41 + \left(31 a + 16\right)\cdot 41^{2} + \left(20 a + 12\right)\cdot 41^{3} + \left(8 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 9 + \left(15 a + 28\right)\cdot 41 + \left(31 a + 14\right)\cdot 41^{2} + \left(20 a + 1\right)\cdot 41^{3} + \left(8 a + 28\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 17 + \left(25 a + 4\right)\cdot 41 + \left(9 a + 11\right)\cdot 41^{2} + \left(20 a + 32\right)\cdot 41^{3} + \left(32 a + 32\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 7 + \left(25 a + 39\right)\cdot 41 + \left(9 a + 11\right)\cdot 41^{2} + \left(20 a + 32\right)\cdot 41^{3} + \left(32 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 9 + \left(25 a + 23\right)\cdot 41 + \left(9 a + 12\right)\cdot 41^{2} + \left(20 a + 2\right)\cdot 41^{3} + \left(32 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,3,5,2,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,5,3,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
