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:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 22 + \left(21 a + 14\right)\cdot 41 + \left(3 a + 5\right)\cdot 41^{2} + \left(2 a + 21\right)\cdot 41^{3} + \left(40 a + 31\right)\cdot 41^{4} + \left(7 a + 32\right)\cdot 41^{5} + \left(a + 5\right)\cdot 41^{6} + \left(28 a + 33\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 20 + \left(19 a + 26\right)\cdot 41 + \left(37 a + 35\right)\cdot 41^{2} + \left(38 a + 19\right)\cdot 41^{3} + 9\cdot 41^{4} + \left(33 a + 8\right)\cdot 41^{5} + \left(39 a + 35\right)\cdot 41^{6} + \left(12 a + 7\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 37 + \left(21 a + 11\right)\cdot 41 + \left(3 a + 5\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(40 a + 14\right)\cdot 41^{4} + \left(7 a + 24\right)\cdot 41^{5} + \left(a + 39\right)\cdot 41^{6} + \left(28 a + 6\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 19 + \left(26 a + 29\right)\cdot 41 + \left(23 a + 18\right)\cdot 41^{2} + \left(29 a + 8\right)\cdot 41^{3} + \left(12 a + 16\right)\cdot 41^{4} + \left(6 a + 17\right)\cdot 41^{5} + \left(25 a + 6\right)\cdot 41^{6} + \left(10 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 32\cdot 41 + 18\cdot 41^{2} + 17\cdot 41^{3} + 8\cdot 41^{4} + 2\cdot 41^{5} + 28\cdot 41^{6} + 33\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 5 + \left(19 a + 29\right)\cdot 41 + \left(37 a + 35\right)\cdot 41^{2} + \left(38 a + 23\right)\cdot 41^{3} + 26\cdot 41^{4} + \left(33 a + 16\right)\cdot 41^{5} + \left(39 a + 1\right)\cdot 41^{6} + \left(12 a + 34\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 8\cdot 41 + 22\cdot 41^{2} + 23\cdot 41^{3} + 32\cdot 41^{4} + 38\cdot 41^{5} + 12\cdot 41^{6} + 7\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 26 a + 23 + \left(14 a + 11\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(11 a + 32\right)\cdot 41^{3} + \left(28 a + 24\right)\cdot 41^{4} + \left(34 a + 23\right)\cdot 41^{5} + \left(15 a + 34\right)\cdot 41^{6} + \left(30 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 |
| $(1,4,5)(2,8,7)$ |
| $(1,7,2,5)(3,4,6,8)$ |
| $(1,6,2,3)(4,5,8,7)$ |
| $(1,2)(3,6)(4,8)(5,7)$ |
| $(1,2)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,8)(5,7)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,2)(4,7)(5,8)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,6,7)(2,3,5)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,6,2,3)(4,5,8,7)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,5,6,2,7,3)(4,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,5,3,4,2,7,6,8)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,7,3,8,2,5,6,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
