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:
| $r_{ 1 }$ |
$=$ |
$ 39 + 22\cdot 41 + 39\cdot 41^{2} + 36\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 36 + \left(a + 15\right)\cdot 41 + \left(16 a + 18\right)\cdot 41^{2} + \left(19 a + 1\right)\cdot 41^{3} + \left(11 a + 21\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 8 + \left(39 a + 2\right)\cdot 41 + \left(24 a + 24\right)\cdot 41^{2} + \left(21 a + 2\right)\cdot 41^{3} + \left(29 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 28\cdot 41 + 12\cdot 41^{2} + 21\cdot 41^{3} + 31\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 4 + \left(29 a + 29\right)\cdot 41 + \left(40 a + 8\right)\cdot 41^{2} + \left(17 a + 3\right)\cdot 41^{3} + \left(7 a + 23\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 34 + \left(11 a + 24\right)\cdot 41 + 19\cdot 41^{2} + \left(23 a + 16\right)\cdot 41^{3} + \left(33 a + 27\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5,6)$ |
| $(1,4)(2,5)(3,6)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $-2$ |
| $9$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(2,3)(4,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
