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