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:
| $r_{ 1 }$ |
$=$ |
$ a + \left(2 a + 29\right)\cdot 41 + \left(a + 30\right)\cdot 41^{2} + \left(38 a + 2\right)\cdot 41^{3} + \left(37 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 3 + \left(38 a + 34\right)\cdot 41 + \left(39 a + 31\right)\cdot 41^{2} + \left(2 a + 33\right)\cdot 41^{3} + \left(3 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 4 + \left(13 a + 2\right)\cdot 41 + \left(11 a + 5\right)\cdot 41^{2} + \left(19 a + 33\right)\cdot 41^{3} + \left(16 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 22 + \left(27 a + 35\right)\cdot 41 + \left(29 a + 25\right)\cdot 41^{2} + \left(21 a + 38\right)\cdot 41^{3} + \left(24 a + 33\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 24 + \left(3 a + 40\right)\cdot 41 + 24 a\cdot 41^{2} + \left(15 a + 37\right)\cdot 41^{3} + \left(2 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 29 + \left(37 a + 22\right)\cdot 41 + \left(16 a + 28\right)\cdot 41^{2} + \left(25 a + 18\right)\cdot 41^{3} + \left(38 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
