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