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