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 a + 28 + \left(39 a + 19\right)\cdot 41 + \left(14 a + 38\right)\cdot 41^{2} + \left(27 a + 27\right)\cdot 41^{3} + \left(29 a + 30\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 34 + \left(11 a + 18\right)\cdot 41 + \left(20 a + 16\right)\cdot 41^{2} + \left(33 a + 21\right)\cdot 41^{3} + \left(37 a + 21\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 10\cdot 41 + 25\cdot 41^{2} + 22\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 13 + \left(a + 21\right)\cdot 41 + \left(26 a + 2\right)\cdot 41^{2} + \left(13 a + 13\right)\cdot 41^{3} + \left(11 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 7 + \left(29 a + 22\right)\cdot 41 + \left(20 a + 24\right)\cdot 41^{2} + \left(7 a + 19\right)\cdot 41^{3} + \left(3 a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 30\cdot 41 + 15\cdot 41^{2} + 18\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(2,5)(3,6)$ |
| $(1,6,4,3)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
