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:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 31 + \left(33 a + 40\right)\cdot 41 + \left(14 a + 36\right)\cdot 41^{2} + \left(a + 2\right)\cdot 41^{3} + \left(a + 32\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 22\cdot 41 + 8\cdot 41^{2} + 3\cdot 41^{3} + 33\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 19 + \left(36 a + 28\right)\cdot 41 + \left(37 a + 7\right)\cdot 41^{2} + \left(40 a + 1\right)\cdot 41^{3} + \left(13 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 29 + \left(4 a + 38\right)\cdot 41 + \left(3 a + 2\right)\cdot 41^{2} + 4\cdot 41^{3} + \left(27 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 23\cdot 41 + 18\cdot 41^{2} + 37\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 14 + \left(7 a + 9\right)\cdot 41 + \left(26 a + 7\right)\cdot 41^{2} + \left(39 a + 33\right)\cdot 41^{3} + \left(39 a + 33\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,5,6,3,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $15$ | $2$ | $(1,5)(3,6)$ | $1$ |
| $20$ | $3$ | $(1,6,2)(3,4,5)$ | $-1$ |
| $30$ | $4$ | $(1,3,5,6)$ | $1$ |
| $24$ | $5$ | $(2,5,4,6,3)$ | $0$ |
| $20$ | $6$ | $(1,5,6,3,2,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
