Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 1 + \left(18 a + 18\right)\cdot 41 + \left(15 a + 13\right)\cdot 41^{2} + \left(31 a + 20\right)\cdot 41^{3} + \left(6 a + 13\right)\cdot 41^{4} + \left(17 a + 21\right)\cdot 41^{5} + \left(12 a + 20\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 4 + \left(11 a + 40\right)\cdot 41 + \left(21 a + 35\right)\cdot 41^{2} + \left(40 a + 36\right)\cdot 41^{3} + \left(27 a + 18\right)\cdot 41^{4} + \left(8 a + 29\right)\cdot 41^{5} + \left(31 a + 36\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 15 + \left(2 a + 5\right)\cdot 41 + \left(8 a + 6\right)\cdot 41^{2} + \left(22 a + 40\right)\cdot 41^{3} + \left(11 a + 17\right)\cdot 41^{4} + \left(12 a + 31\right)\cdot 41^{5} + \left(33 a + 9\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 5 + \left(22 a + 17\right)\cdot 41 + 25 a\cdot 41^{2} + \left(9 a + 17\right)\cdot 41^{3} + \left(34 a + 2\right)\cdot 41^{4} + \left(23 a + 25\right)\cdot 41^{5} + \left(28 a + 40\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 19 + \left(38 a + 38\right)\cdot 41 + \left(32 a + 27\right)\cdot 41^{2} + \left(18 a + 16\right)\cdot 41^{3} + \left(29 a + 30\right)\cdot 41^{4} + \left(28 a + 15\right)\cdot 41^{5} + \left(7 a + 15\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 37 + \left(5 a + 24\right)\cdot 41 + \left(30 a + 35\right)\cdot 41^{2} + 15 a\cdot 41^{3} + \left(37 a + 2\right)\cdot 41^{4} + \left(18 a + 38\right)\cdot 41^{5} + \left(28 a + 28\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 a + 14 + \left(29 a + 16\right)\cdot 41 + \left(19 a + 6\right)\cdot 41^{2} + 14\cdot 41^{3} + \left(13 a + 21\right)\cdot 41^{4} + \left(32 a + 27\right)\cdot 41^{5} + \left(9 a + 39\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 2 a + 31 + \left(35 a + 3\right)\cdot 41 + \left(10 a + 38\right)\cdot 41^{2} + \left(25 a + 17\right)\cdot 41^{3} + \left(3 a + 16\right)\cdot 41^{4} + \left(22 a + 16\right)\cdot 41^{5} + \left(12 a + 13\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(4,7)$ |
| $(1,8)(2,5)(3,4)(6,7)$ |
| $(3,4,6,7)(5,8)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,7)(5,8)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(2,5)(3,4)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,8)(3,4)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,4)(3,8)(5,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,5,7)(2,6,8,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,5,3)(2,4,8,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,2,7)(3,8,6,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,2,8)(3,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,8,2,5)(3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
