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