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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 18\cdot 41 + 23\cdot 41^{2} + 19\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 19 + \left(17 a + 36\right)\cdot 41 + \left(3 a + 23\right)\cdot 41^{2} + \left(17 a + 37\right)\cdot 41^{3} + \left(36 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 2 + \left(40 a + 34\right)\cdot 41 + \left(25 a + 1\right)\cdot 41^{2} + \left(30 a + 8\right)\cdot 41^{3} + \left(9 a + 21\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 22\cdot 41 + 17\cdot 41^{2} + 21\cdot 41^{3} + 32\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 22 + \left(23 a + 4\right)\cdot 41 + \left(37 a + 17\right)\cdot 41^{2} + \left(23 a + 3\right)\cdot 41^{3} + \left(4 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a + 39 + 6\cdot 41 + \left(15 a + 39\right)\cdot 41^{2} + \left(10 a + 32\right)\cdot 41^{3} + \left(31 a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,6,5)(2,4,3)$ |
| $(1,3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,6,5)(2,4,3)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
