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(35 a + 15\right)\cdot 41 + \left(21 a + 32\right)\cdot 41^{2} + \left(35 a + 25\right)\cdot 41^{3} + \left(11 a + 27\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 3 + \left(5 a + 37\right)\cdot 41 + \left(19 a + 21\right)\cdot 41^{2} + \left(5 a + 28\right)\cdot 41^{3} + \left(29 a + 27\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 25 + \left(22 a + 14\right)\cdot 41 + \left(37 a + 23\right)\cdot 41^{2} + \left(23 a + 18\right)\cdot 41^{3} + \left(a + 1\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 29\cdot 41 + 27\cdot 41^{2} + 27\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 34\cdot 41 + 26\cdot 41^{2} + 10\cdot 41^{3} + 16\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 5 + \left(18 a + 33\right)\cdot 41 + \left(3 a + 31\right)\cdot 41^{2} + \left(17 a + 11\right)\cdot 41^{3} + \left(39 a + 23\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,5)(4,6)$ |
| $(3,5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,4)$ | $-2$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $0$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
