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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 36 + \left(6 a + 6\right)\cdot 41 + \left(7 a + 12\right)\cdot 41^{2} + \left(15 a + 26\right)\cdot 41^{3} + \left(37 a + 26\right)\cdot 41^{4} + \left(40 a + 12\right)\cdot 41^{5} + \left(10 a + 1\right)\cdot 41^{6} + \left(39 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 1 + \left(6 a + 30\right)\cdot 41 + \left(7 a + 13\right)\cdot 41^{2} + \left(15 a + 17\right)\cdot 41^{3} + \left(37 a + 40\right)\cdot 41^{4} + \left(40 a + 24\right)\cdot 41^{5} + \left(10 a + 6\right)\cdot 41^{6} + \left(39 a + 40\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 5 + \left(14 a + 31\right)\cdot 41 + \left(36 a + 34\right)\cdot 41^{2} + \left(35 a + 25\right)\cdot 41^{3} + \left(7 a + 26\right)\cdot 41^{4} + \left(28 a + 2\right)\cdot 41^{5} + \left(25 a + 37\right)\cdot 41^{6} + \left(2 a + 8\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 7\cdot 41 + 31\cdot 41^{2} + 19\cdot 41^{3} + 25\cdot 41^{4} + 18\cdot 41^{5} + 7\cdot 41^{6} + 12\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 5 + \left(34 a + 34\right)\cdot 41 + \left(33 a + 28\right)\cdot 41^{2} + \left(25 a + 14\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} + 28\cdot 41^{5} + \left(30 a + 39\right)\cdot 41^{6} + \left(a + 23\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 40 + \left(34 a + 10\right)\cdot 41 + \left(33 a + 27\right)\cdot 41^{2} + \left(25 a + 23\right)\cdot 41^{3} + 3 a\cdot 41^{4} + 16\cdot 41^{5} + \left(30 a + 34\right)\cdot 41^{6} + a\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 a + 36 + \left(26 a + 9\right)\cdot 41 + \left(4 a + 6\right)\cdot 41^{2} + \left(5 a + 15\right)\cdot 41^{3} + \left(33 a + 14\right)\cdot 41^{4} + \left(12 a + 38\right)\cdot 41^{5} + \left(15 a + 3\right)\cdot 41^{6} + \left(38 a + 32\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 23 + 33\cdot 41 + 9\cdot 41^{2} + 21\cdot 41^{3} + 15\cdot 41^{4} + 22\cdot 41^{5} + 33\cdot 41^{6} + 28\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,6)(2,5,4)$ |
| $(1,3,5,7)(2,8,6,4)$ |
| $(1,8,5,4)(2,7,6,3)$ |
| $(1,5)(2,8)(4,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-4$ |
| $12$ |
$2$ |
$(1,5)(2,8)(4,6)$ |
$0$ |
| $8$ |
$3$ |
$(1,4,7)(3,5,8)$ |
$1$ |
| $6$ |
$4$ |
$(1,3,5,7)(2,8,6,4)$ |
$0$ |
| $8$ |
$6$ |
$(1,3,4,5,7,8)(2,6)$ |
$-1$ |
| $6$ |
$8$ |
$(1,4,3,2,5,8,7,6)$ |
$0$ |
| $6$ |
$8$ |
$(1,8,3,6,5,4,7,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
