Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 14\cdot 41 + 13\cdot 41^{2} + 30\cdot 41^{3} + 13\cdot 41^{4} + 30\cdot 41^{5} + 32\cdot 41^{6} + 14\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a + 14 + \left(5 a + 40\right)\cdot 41 + \left(24 a + 7\right)\cdot 41^{2} + \left(26 a + 2\right)\cdot 41^{3} + \left(28 a + 13\right)\cdot 41^{4} + \left(26 a + 14\right)\cdot 41^{5} + \left(a + 22\right)\cdot 41^{6} + \left(27 a + 14\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 34 + \left(35 a + 23\right)\cdot 41 + \left(16 a + 33\right)\cdot 41^{2} + \left(14 a + 16\right)\cdot 41^{3} + \left(12 a + 31\right)\cdot 41^{4} + \left(14 a + 24\right)\cdot 41^{5} + 39 a\cdot 41^{6} + \left(13 a + 12\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 28 + 35 a\cdot 41 + \left(16 a + 33\right)\cdot 41^{2} + \left(14 a + 38\right)\cdot 41^{3} + \left(12 a + 27\right)\cdot 41^{4} + \left(14 a + 26\right)\cdot 41^{5} + \left(39 a + 18\right)\cdot 41^{6} + \left(13 a + 26\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 24 + \left(26 a + 40\right)\cdot 41 + \left(9 a + 39\right)\cdot 41^{2} + \left(12 a + 6\right)\cdot 41^{3} + \left(25 a + 9\right)\cdot 41^{4} + \left(23 a + 18\right)\cdot 41^{5} + \left(22 a + 39\right)\cdot 41^{6} + \left(36 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 26\cdot 41 + 27\cdot 41^{2} + 10\cdot 41^{3} + 27\cdot 41^{4} + 10\cdot 41^{5} + 8\cdot 41^{6} + 26\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 a + 8 + \left(5 a + 17\right)\cdot 41 + \left(24 a + 7\right)\cdot 41^{2} + \left(26 a + 24\right)\cdot 41^{3} + \left(28 a + 9\right)\cdot 41^{4} + \left(26 a + 16\right)\cdot 41^{5} + \left(a + 40\right)\cdot 41^{6} + \left(27 a + 28\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 2 a + 18 + 14 a\cdot 41 + \left(31 a + 1\right)\cdot 41^{2} + \left(28 a + 34\right)\cdot 41^{3} + \left(15 a + 31\right)\cdot 41^{4} + \left(17 a + 22\right)\cdot 41^{5} + \left(18 a + 1\right)\cdot 41^{6} + \left(4 a + 23\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,6,2)(3,5,7,8)$ |
| $(1,8,6,5)(2,3,4,7)$ |
| $(1,6)(2,4)(3,7)(5,8)$ |
| $(2,8,7)(3,4,5)$ |
| $(2,5)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $-2$ |
| $12$ | $2$ | $(2,5)(3,7)(4,8)$ | $0$ |
| $8$ | $3$ | $(1,4,7)(2,3,6)$ | $-1$ |
| $6$ | $4$ | $(1,4,6,2)(3,5,7,8)$ | $0$ |
| $8$ | $6$ | $(1,3,4,6,7,2)(5,8)$ | $1$ |
| $6$ | $8$ | $(1,4,3,8,6,2,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,3,5,6,4,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
