Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 14 + \left(23 a + 3\right)\cdot 41 + 14 a\cdot 41^{2} + \left(40 a + 35\right)\cdot 41^{3} + \left(13 a + 33\right)\cdot 41^{4} + \left(7 a + 24\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 8\cdot 41 + 36\cdot 41^{2} + 21\cdot 41^{3} + 5\cdot 41^{4} + 35\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 4 + \left(15 a + 15\right)\cdot 41 + \left(39 a + 7\right)\cdot 41^{2} + \left(24 a + 33\right)\cdot 41^{3} + \left(36 a + 27\right)\cdot 41^{4} + \left(13 a + 24\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 35 + \left(9 a + 31\right)\cdot 41 + \left(34 a + 38\right)\cdot 41^{2} + \left(20 a + 19\right)\cdot 41^{3} + \left(6 a + 20\right)\cdot 41^{4} + \left(33 a + 10\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 37 + \left(25 a + 8\right)\cdot 41 + \left(a + 28\right)\cdot 41^{2} + \left(16 a + 27\right)\cdot 41^{3} + \left(4 a + 30\right)\cdot 41^{4} + \left(27 a + 29\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 6 + \left(31 a + 14\right)\cdot 41 + \left(6 a + 9\right)\cdot 41^{2} + \left(20 a + 7\right)\cdot 41^{3} + \left(34 a + 19\right)\cdot 41^{4} + \left(7 a + 21\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 + 3\cdot 41 + 24\cdot 41^{2} + 32\cdot 41^{4} + 25\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 a + 5 + \left(17 a + 37\right)\cdot 41 + \left(26 a + 19\right)\cdot 41^{2} + 18\cdot 41^{3} + \left(27 a + 35\right)\cdot 41^{4} + \left(33 a + 32\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(3,6)(4,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,3)(5,6,8)$ |
| $(1,6,8,3)(2,5,7,4)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $12$ | $2$ | $(1,5)(3,6)(4,8)$ | $0$ |
| $8$ | $3$ | $(1,4,3)(5,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $8$ | $6$ | $(1,2,5,8,7,4)(3,6)$ | $1$ |
| $6$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,6,5,2,8,3,4,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
