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:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 26 + \left(5 a + 4\right)\cdot 41 + \left(21 a + 16\right)\cdot 41^{2} + \left(26 a + 16\right)\cdot 41^{3} + \left(30 a + 18\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 20\cdot 41 + 9\cdot 41^{2} + 15\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 2\cdot 41 + 33\cdot 41^{2} + 31\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 31 + \left(35 a + 33\right)\cdot 41 + \left(19 a + 32\right)\cdot 41^{2} + \left(14 a + 33\right)\cdot 41^{3} + \left(10 a + 1\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 12 + \left(30 a + 24\right)\cdot 41 + \left(11 a + 13\right)\cdot 41^{2} + \left(10 a + 3\right)\cdot 41^{3} + \left(24 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 6 + \left(10 a + 37\right)\cdot 41 + \left(29 a + 17\right)\cdot 41^{2} + \left(30 a + 22\right)\cdot 41^{3} + \left(16 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(2,5,6)$ | $1$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,2,3,5,4,6)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
