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:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 15 + \left(19 a + 33\right)\cdot 41 + \left(3 a + 2\right)\cdot 41^{2} + \left(13 a + 17\right)\cdot 41^{3} + \left(a + 40\right)\cdot 41^{4} + \left(3 a + 6\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 16 + \left(16 a + 13\right)\cdot 41 + \left(22 a + 25\right)\cdot 41^{2} + \left(17 a + 30\right)\cdot 41^{3} + \left(28 a + 40\right)\cdot 41^{4} + \left(31 a + 25\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 28 + \left(21 a + 32\right)\cdot 41 + \left(37 a + 34\right)\cdot 41^{2} + \left(27 a + 11\right)\cdot 41^{3} + \left(39 a + 31\right)\cdot 41^{4} + \left(37 a + 14\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 5\cdot 41 + 7\cdot 41^{2} + 20\cdot 41^{3} + 13\cdot 41^{4} + 33\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + \left(31 a + 9\right)\cdot 41 + \left(40 a + 8\right)\cdot 41^{2} + \left(10 a + 21\right)\cdot 41^{3} + 24 a\cdot 41^{4} + \left(19 a + 34\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 12 + \left(9 a + 16\right)\cdot 41 + 17\cdot 41^{2} + \left(30 a + 13\right)\cdot 41^{3} + \left(16 a + 21\right)\cdot 41^{4} + \left(21 a + 27\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 + 10\cdot 41 + 33\cdot 41^{2} + 29\cdot 41^{3} + 30\cdot 41^{4} + 10\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 a + 35 + \left(24 a + 1\right)\cdot 41 + \left(18 a + 35\right)\cdot 41^{2} + \left(23 a + 19\right)\cdot 41^{3} + \left(12 a + 26\right)\cdot 41^{4} + \left(9 a + 10\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,8)(3,5)(4,7)$ |
| $(1,3,6,5)(2,7,8,4)$ |
| $(1,4,3)(5,6,7)$ |
| $(1,6)(3,7)(4,5)$ |
| $(1,2,6,8)(3,4,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,8)(3,5)(4,7)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,6)(3,7)(4,5)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(2,5,4)(3,7,8)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,3,6,5)(2,7,8,4)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,7,3,6,4,5)(2,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,7,8,5,6,4,2,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,4,8,3,6,7,2,5)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
