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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 30\cdot 41 + 38\cdot 41^{2} + 37\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 24 + \left(31 a + 25\right)\cdot 41 + \left(38 a + 9\right)\cdot 41^{2} + \left(28 a + 3\right)\cdot 41^{3} + \left(37 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 28 + \left(38 a + 7\right)\cdot 41 + \left(28 a + 7\right)\cdot 41^{2} + \left(11 a + 37\right)\cdot 41^{3} + \left(26 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 a + 40 + \left(2 a + 35\right)\cdot 41 + \left(12 a + 14\right)\cdot 41^{2} + \left(29 a + 2\right)\cdot 41^{3} + \left(14 a + 32\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 38 + \left(9 a + 6\right)\cdot 41 + \left(2 a + 12\right)\cdot 41^{2} + \left(12 a + 10\right)\cdot 41^{3} + \left(3 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 + 16\cdot 41 + 40\cdot 41^{2} + 31\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,5)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $-1$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
