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:
| $r_{ 1 }$ |
$=$ |
$ 36 + 33\cdot 41 + 17\cdot 41^{2} + 26\cdot 41^{3} + 3\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 38\cdot 41 + 10\cdot 41^{2} + 28\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 36 + \left(19 a + 14\right)\cdot 41 + \left(9 a + 10\right)\cdot 41^{2} + \left(12 a + 13\right)\cdot 41^{3} + \left(39 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 1 + \left(21 a + 29\right)\cdot 41 + \left(31 a + 19\right)\cdot 41^{2} + \left(28 a + 40\right)\cdot 41^{3} + \left(a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 17 + \left(34 a + 36\right)\cdot 41 + \left(30 a + 2\right)\cdot 41^{2} + \left(30 a + 38\right)\cdot 41^{3} + \left(7 a + 1\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 29 + \left(6 a + 11\right)\cdot 41 + \left(10 a + 20\right)\cdot 41^{2} + \left(10 a + 17\right)\cdot 41^{3} + \left(33 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 |
| $(1,2)(3,5)(4,6)$ |
| $(2,3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(1,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,5,6)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
