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:
| $r_{ 1 }$ |
$=$ |
$ 36 a^{2} + 30 a + 1 + \left(26 a^{2} + 31 a + 13\right)\cdot 41 + \left(39 a^{2} + 12 a + 35\right)\cdot 41^{2} + \left(27 a^{2} + 5 a + 5\right)\cdot 41^{3} + \left(38 a^{2} + 33 a + 20\right)\cdot 41^{4} + \left(29 a^{2} + a + 1\right)\cdot 41^{5} + \left(30 a^{2} + 9 a\right)\cdot 41^{6} + \left(14 a^{2} + 38 a + 20\right)\cdot 41^{7} + \left(9 a^{2} + 26 a + 10\right)\cdot 41^{8} + \left(32 a^{2} + 18 a + 21\right)\cdot 41^{9} + \left(8 a^{2} + 28 a + 31\right)\cdot 41^{10} + \left(31 a + 7\right)\cdot 41^{11} + \left(11 a^{2} + 37 a + 6\right)\cdot 41^{12} + \left(26 a^{2} + 26 a + 37\right)\cdot 41^{13} + \left(25 a^{2} + 34 a + 22\right)\cdot 41^{14} + \left(20 a^{2} + 33 a + 1\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 33\cdot 41 + 7\cdot 41^{2} + 40\cdot 41^{3} + 32\cdot 41^{4} + 32\cdot 41^{5} + 37\cdot 41^{6} + 38\cdot 41^{7} + 7\cdot 41^{8} + 9\cdot 41^{9} + 31\cdot 41^{10} + 14\cdot 41^{11} + 16\cdot 41^{12} + 40\cdot 41^{13} + 15\cdot 41^{14} + 27\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a^{2} + 33 a + 13 + \left(7 a^{2} + 27 a + 35\right)\cdot 41 + \left(37 a^{2} + 10 a + 24\right)\cdot 41^{2} + \left(9 a^{2} + 22 a + 27\right)\cdot 41^{3} + \left(23 a^{2} + 12 a + 13\right)\cdot 41^{4} + \left(36 a^{2} + 26 a + 31\right)\cdot 41^{5} + \left(2 a^{2} + 4 a + 35\right)\cdot 41^{6} + \left(30 a^{2} + 6 a + 24\right)\cdot 41^{7} + \left(10 a^{2} + a + 33\right)\cdot 41^{8} + \left(38 a^{2} + 18 a + 12\right)\cdot 41^{9} + \left(6 a^{2} + 15 a + 31\right)\cdot 41^{10} + \left(23 a^{2} + 13 a + 32\right)\cdot 41^{11} + \left(32 a^{2} + 17 a + 24\right)\cdot 41^{12} + \left(12 a^{2} + 32 a + 8\right)\cdot 41^{13} + \left(22 a^{2} + 2 a + 38\right)\cdot 41^{14} + \left(24 a + 7\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a^{2} + 13 a + 10 + \left(34 a^{2} + 12 a + 39\right)\cdot 41 + \left(15 a^{2} + 26 a + 37\right)\cdot 41^{2} + \left(37 a^{2} + 25 a + 4\right)\cdot 41^{3} + \left(12 a^{2} + 35 a + 34\right)\cdot 41^{4} + \left(11 a^{2} + 37 a\right)\cdot 41^{5} + \left(25 a^{2} + 18 a + 37\right)\cdot 41^{6} + \left(9 a^{2} + 6 a + 24\right)\cdot 41^{7} + \left(7 a^{2} + 3 a + 17\right)\cdot 41^{8} + \left(17 a^{2} + 32 a + 12\right)\cdot 41^{9} + \left(15 a^{2} + 2 a + 23\right)\cdot 41^{10} + \left(18 a + 17\right)\cdot 41^{11} + \left(30 a^{2} + 19 a + 9\right)\cdot 41^{12} + \left(40 a^{2} + 35 a + 27\right)\cdot 41^{13} + \left(32 a^{2} + 6 a + 31\right)\cdot 41^{14} + \left(26 a^{2} + 33 a + 11\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a^{2} + 34 a + 8 + \left(14 a^{2} + 8 a + 32\right)\cdot 41 + \left(36 a^{2} + 39 a + 5\right)\cdot 41^{2} + \left(12 a^{2} + 37 a + 23\right)\cdot 41^{3} + \left(34 a^{2} + 10 a + 3\right)\cdot 41^{4} + \left(33 a^{2} + 36 a + 4\right)\cdot 41^{5} + \left(29 a^{2} + 9 a + 13\right)\cdot 41^{6} + \left(17 a^{2} + 10 a + 8\right)\cdot 41^{7} + \left(14 a^{2} + 26 a\right)\cdot 41^{8} + \left(23 a^{2} + 11 a + 29\right)\cdot 41^{9} + \left(6 a^{2} + 8 a + 2\right)\cdot 41^{10} + \left(14 a^{2} + 30 a + 17\right)\cdot 41^{11} + \left(12 a^{2} + 32 a + 34\right)\cdot 41^{12} + \left(11 a^{2} + 37 a + 40\right)\cdot 41^{13} + \left(35 a^{2} + 8 a + 1\right)\cdot 41^{14} + \left(28 a^{2} + 29 a + 7\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a^{2} + 18 a + 4 + \left(40 a^{2} + 22\right)\cdot 41 + \left(5 a^{2} + 30 a + 26\right)\cdot 41^{2} + \left(38 a + 14\right)\cdot 41^{3} + \left(9 a^{2} + 37 a\right)\cdot 41^{4} + \left(18 a^{2} + 2 a + 21\right)\cdot 41^{5} + \left(21 a^{2} + 22 a + 7\right)\cdot 41^{6} + \left(8 a^{2} + 33 a + 2\right)\cdot 41^{7} + \left(17 a^{2} + 28 a + 2\right)\cdot 41^{8} + \left(26 a^{2} + 10 a + 31\right)\cdot 41^{9} + \left(25 a^{2} + 4 a + 1\right)\cdot 41^{10} + \left(26 a^{2} + 20 a + 39\right)\cdot 41^{11} + \left(17 a^{2} + 11 a + 37\right)\cdot 41^{12} + \left(3 a^{2} + 17 a + 21\right)\cdot 41^{13} + \left(21 a^{2} + 38 a + 33\right)\cdot 41^{14} + \left(32 a^{2} + 18 a + 36\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 15\cdot 41 + 36\cdot 41^{2} + 13\cdot 41^{3} + 11\cdot 41^{4} + 12\cdot 41^{5} + 35\cdot 41^{6} + 18\cdot 41^{7} + 4\cdot 41^{8} + 17\cdot 41^{9} + 4\cdot 41^{10} + 38\cdot 41^{11} + 6\cdot 41^{12} + 25\cdot 41^{13} + 36\cdot 41^{14} + 6\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 + 39\cdot 41 + 10\cdot 41^{2} + 3\cdot 41^{3} + 19\cdot 41^{4} + 30\cdot 41^{5} + 9\cdot 41^{6} + 20\cdot 41^{7} + 18\cdot 41^{8} + 12\cdot 41^{9} + 12\cdot 41^{10} + 36\cdot 41^{11} + 11\cdot 41^{12} + 39\cdot 41^{13} + 23\cdot 41^{14} + 20\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 36 a + 32 + \left(40 a^{2} + 15\right)\cdot 41 + \left(28 a^{2} + 4 a + 19\right)\cdot 41^{2} + \left(34 a^{2} + 34 a + 30\right)\cdot 41^{3} + \left(4 a^{2} + 33 a + 28\right)\cdot 41^{4} + \left(34 a^{2} + 17 a + 29\right)\cdot 41^{5} + \left(12 a^{2} + 17 a + 28\right)\cdot 41^{6} + \left(a^{2} + 28 a + 5\right)\cdot 41^{7} + \left(23 a^{2} + 36 a + 28\right)\cdot 41^{8} + \left(26 a^{2} + 31 a + 18\right)\cdot 41^{9} + \left(18 a^{2} + 22 a + 25\right)\cdot 41^{10} + \left(17 a^{2} + 9 a + 1\right)\cdot 41^{11} + \left(19 a^{2} + 4 a + 16\right)\cdot 41^{12} + \left(28 a^{2} + 14 a + 5\right)\cdot 41^{13} + \left(26 a^{2} + 31 a\right)\cdot 41^{14} + \left(13 a^{2} + 24 a + 3\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3)(2,7)(4,6)(5,9)$ |
| $(2,7,8)(3,9,4)$ |
| $(1,5,6)(3,9,4)$ |
| $(1,2,9)(3,5,7)(4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,3)(2,7)(4,6)(5,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,6)(2,7,8)(3,4,9)$ |
$-3$ |
| $3$ |
$3$ |
$(2,7,8)(3,9,4)$ |
$0$ |
| $3$ |
$3$ |
$(2,8,7)(3,4,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,2,9)(3,5,7)(4,6,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,7,9)(2,4,6)(3,5,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,9,7)(2,6,4)(3,8,5)$ |
$0$ |
| $9$ |
$6$ |
$(1,9,5,4,6,3)(2,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,3,6,4,5,9)(2,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
