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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 16 + \left(13 a + 13\right)\cdot 41 + \left(10 a + 14\right)\cdot 41^{2} + \left(12 a + 5\right)\cdot 41^{3} + \left(a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 34 + \left(38 a + 36\right)\cdot 41 + \left(17 a + 27\right)\cdot 41^{2} + \left(39 a + 28\right)\cdot 41^{3} + \left(5 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 34\cdot 41 + 35\cdot 41^{2} + 3\cdot 41^{3} + 11\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 40 + \left(2 a + 25\right)\cdot 41 + \left(23 a + 2\right)\cdot 41^{2} + \left(a + 6\right)\cdot 41^{3} + \left(35 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 + 19\cdot 41 + 10\cdot 41^{2} + 6\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 35 + \left(27 a + 33\right)\cdot 41 + \left(30 a + 31\right)\cdot 41^{2} + \left(28 a + 31\right)\cdot 41^{3} + \left(39 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $-2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $0$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
