Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 a + \left(8 a + 12\right)\cdot 41 + \left(a + 7\right)\cdot 41^{2} + \left(5 a + 31\right)\cdot 41^{3} + \left(25 a + 11\right)\cdot 41^{4} + \left(22 a + 37\right)\cdot 41^{5} + \left(27 a + 23\right)\cdot 41^{6} + \left(23 a + 38\right)\cdot 41^{7} + \left(30 a + 25\right)\cdot 41^{8} + \left(40 a + 20\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 5 + \left(32 a + 9\right)\cdot 41 + \left(39 a + 2\right)\cdot 41^{2} + \left(35 a + 4\right)\cdot 41^{3} + 15 a\cdot 41^{4} + \left(18 a + 39\right)\cdot 41^{5} + \left(13 a + 1\right)\cdot 41^{6} + 17 a\cdot 41^{7} + \left(10 a + 12\right)\cdot 41^{8} + 30\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 9 + \left(6 a + 27\right)\cdot 41 + \left(30 a + 17\right)\cdot 41^{2} + \left(27 a + 14\right)\cdot 41^{3} + \left(34 a + 18\right)\cdot 41^{4} + \left(40 a + 4\right)\cdot 41^{5} + \left(11 a + 30\right)\cdot 41^{6} + \left(28 a + 1\right)\cdot 41^{7} + \left(34 a + 10\right)\cdot 41^{8} + \left(33 a + 28\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 14\cdot 41 + 12\cdot 41^{2} + 6\cdot 41^{3} + 29\cdot 41^{4} + 26\cdot 41^{5} + 28\cdot 41^{6} + 34\cdot 41^{7} + 24\cdot 41^{8} + 34\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 40\cdot 41 + 10\cdot 41^{2} + 33\cdot 41^{3} + 34\cdot 41^{4} + 28\cdot 41^{5} + 21\cdot 41^{6} + 38\cdot 41^{7} + 20\cdot 41^{8} + 10\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a + 5 + \left(34 a + 21\right)\cdot 41 + \left(10 a + 19\right)\cdot 41^{2} + \left(13 a + 26\right)\cdot 41^{3} + \left(6 a + 12\right)\cdot 41^{4} + 10\cdot 41^{5} + \left(29 a + 25\right)\cdot 41^{6} + \left(12 a + 33\right)\cdot 41^{7} + \left(6 a + 3\right)\cdot 41^{8} + \left(7 a + 13\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 a + \left(22 a + 4\right)\cdot 41 + 37\cdot 41^{2} + \left(9 a + 10\right)\cdot 41^{3} + \left(20 a + 23\right)\cdot 41^{4} + \left(35 a + 6\right)\cdot 41^{5} + \left(38 a + 37\right)\cdot 41^{6} + \left(30 a + 1\right)\cdot 41^{7} + \left(12 a + 9\right)\cdot 41^{8} + \left(26 a + 21\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 4 a + 29 + \left(18 a + 35\right)\cdot 41 + \left(40 a + 15\right)\cdot 41^{2} + \left(31 a + 37\right)\cdot 41^{3} + \left(20 a + 33\right)\cdot 41^{4} + \left(5 a + 10\right)\cdot 41^{5} + \left(2 a + 36\right)\cdot 41^{6} + \left(10 a + 14\right)\cdot 41^{7} + \left(28 a + 16\right)\cdot 41^{8} + \left(14 a + 5\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(2,5)(3,6)(4,7)$ |
| $(2,6,4)(3,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $12$ | $2$ | $(2,5)(3,6)(4,7)$ | $0$ |
| $8$ | $3$ | $(1,4,3)(5,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $8$ | $6$ | $(1,6,4,8,3,5)(2,7)$ | $1$ |
| $6$ | $8$ | $(1,4,6,7,8,5,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
