Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{3} + 3 x + 42 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a^{2} + 5 a + 3 + \left(34 a^{2} + 46 a + 20\right)\cdot 47 + \left(5 a^{2} + 44 a + 24\right)\cdot 47^{2} + \left(25 a^{2} + 17 a + 36\right)\cdot 47^{3} + \left(43 a^{2} + 23 a + 28\right)\cdot 47^{4} + \left(17 a^{2} + 26 a + 43\right)\cdot 47^{5} + \left(44 a^{2} + 23 a + 10\right)\cdot 47^{6} + \left(10 a^{2} + 31\right)\cdot 47^{7} + \left(14 a^{2} + 17\right)\cdot 47^{8} + \left(2 a^{2} + 27 a + 46\right)\cdot 47^{9} + \left(36 a^{2} + 17 a + 8\right)\cdot 47^{10} + \left(30 a^{2} + 3 a + 12\right)\cdot 47^{11} + \left(21 a^{2} + 30 a + 29\right)\cdot 47^{12} + \left(46 a^{2} + 4 a + 44\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a^{2} + 17 a + 39 + \left(3 a^{2} + 39 a + 5\right)\cdot 47 + \left(39 a^{2} + 27 a + 44\right)\cdot 47^{2} + \left(8 a^{2} + 3 a + 3\right)\cdot 47^{3} + \left(25 a^{2} + 35 a + 39\right)\cdot 47^{4} + \left(29 a^{2} + 29 a + 19\right)\cdot 47^{5} + \left(16 a^{2} + 26 a + 2\right)\cdot 47^{6} + \left(14 a^{2} + 23 a + 38\right)\cdot 47^{7} + \left(28 a^{2} + 40 a + 45\right)\cdot 47^{8} + \left(28 a^{2} + 5 a + 4\right)\cdot 47^{9} + \left(5 a^{2} + 15 a + 42\right)\cdot 47^{10} + \left(20 a^{2} + 23 a + 37\right)\cdot 47^{11} + \left(3 a^{2} + 28 a + 39\right)\cdot 47^{12} + \left(31 a^{2} + 10 a + 13\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a^{2} + 2 a + 34 + \left(6 a^{2} + 8 a + 20\right)\cdot 47 + \left(15 a^{2} + 5 a + 27\right)\cdot 47^{2} + \left(9 a^{2} + 45 a + 29\right)\cdot 47^{3} + \left(21 a^{2} + 17 a + 6\right)\cdot 47^{4} + \left(38 a^{2} + 9 a + 39\right)\cdot 47^{5} + \left(29 a^{2} + 20 a + 24\right)\cdot 47^{6} + \left(10 a^{2} + 9 a + 5\right)\cdot 47^{7} + \left(6 a^{2} + 4 a + 9\right)\cdot 47^{8} + \left(39 a^{2} + 40 a + 35\right)\cdot 47^{9} + \left(17 a^{2} + 31 a + 3\right)\cdot 47^{10} + \left(9 a^{2} + 46 a + 15\right)\cdot 47^{11} + \left(17 a^{2} + 4 a + 19\right)\cdot 47^{12} + \left(24 a^{2} + 10 a + 3\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a^{2} + 41 a + 6 + \left(40 a^{2} + 13 a + 41\right)\cdot 47 + \left(a^{2} + 34 a\right)\cdot 47^{2} + \left(22 a^{2} + 13 a + 8\right)\cdot 47^{3} + \left(13 a^{2} + 42 a + 38\right)\cdot 47^{4} + \left(36 a^{2} + 6 a + 34\right)\cdot 47^{5} + \left(26 a^{2} + 5 a + 18\right)\cdot 47^{6} + \left(11 a^{2} + 2 a + 7\right)\cdot 47^{7} + \left(6 a^{2} + 44 a + 9\right)\cdot 47^{8} + \left(25 a + 4\right)\cdot 47^{9} + \left(14 a^{2} + 29 a + 43\right)\cdot 47^{10} + \left(33 a^{2} + 8 a + 15\right)\cdot 47^{11} + \left(10 a^{2} + 5 a + 6\right)\cdot 47^{12} + \left(36 a^{2} + 18 a + 27\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a^{2} + 4 a + 16 + \left(46 a^{2} + 25 a + 6\right)\cdot 47 + \left(29 a^{2} + 7 a + 10\right)\cdot 47^{2} + \left(15 a^{2} + 35 a + 42\right)\cdot 47^{3} + \left(12 a^{2} + 33 a + 35\right)\cdot 47^{4} + \left(19 a^{2} + 30 a\right)\cdot 47^{5} + \left(37 a^{2} + 21 a + 40\right)\cdot 47^{6} + \left(24 a^{2} + 35 a + 33\right)\cdot 47^{7} + \left(34 a^{2} + 45 a + 18\right)\cdot 47^{8} + \left(7 a^{2} + 27 a + 19\right)\cdot 47^{9} + \left(15 a^{2} + 32 a + 45\right)\cdot 47^{10} + \left(4 a^{2} + 38 a + 4\right)\cdot 47^{11} + \left(19 a^{2} + 36 a + 23\right)\cdot 47^{12} + \left(33 a^{2} + 18 a + 21\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 3\cdot 47 + 38\cdot 47^{2} + 24\cdot 47^{3} + 39\cdot 47^{4} + 32\cdot 47^{5} + 29\cdot 47^{6} + 6\cdot 47^{7} + 13\cdot 47^{8} + 27\cdot 47^{9} + 14\cdot 47^{10} + 31\cdot 47^{11} + 47^{12} + 23\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 a^{2} + 25 a + 37 + \left(9 a^{2} + 8 a + 17\right)\cdot 47 + \left(2 a^{2} + 21 a + 17\right)\cdot 47^{2} + \left(13 a^{2} + 25 a + 12\right)\cdot 47^{3} + \left(25 a^{2} + 35 a + 39\right)\cdot 47^{4} + \left(46 a^{2} + 37 a + 6\right)\cdot 47^{5} + \left(32 a^{2} + 43 a + 35\right)\cdot 47^{6} + \left(21 a^{2} + 22 a + 5\right)\cdot 47^{7} + \left(4 a^{2} + 6 a + 45\right)\cdot 47^{8} + \left(16 a^{2} + 14 a + 26\right)\cdot 47^{9} + \left(5 a^{2} + 14 a + 41\right)\cdot 47^{10} + \left(43 a^{2} + 20 a + 36\right)\cdot 47^{11} + \left(21 a^{2} + 35 a + 29\right)\cdot 47^{12} + \left(16 a^{2} + 31 a + 31\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 4 + 7\cdot 47 + 30\cdot 47^{2} + 43\cdot 47^{3} + 19\cdot 47^{4} + 28\cdot 47^{5} + 26\cdot 47^{6} + 14\cdot 47^{7} + 2\cdot 47^{8} + 4\cdot 47^{9} + 7\cdot 47^{10} + 3\cdot 47^{12} + 46\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 9 + 18\cdot 47 + 42\cdot 47^{2} + 33\cdot 47^{3} + 34\cdot 47^{4} + 28\cdot 47^{5} + 46\cdot 47^{6} + 44\cdot 47^{7} + 26\cdot 47^{8} + 19\cdot 47^{9} + 28\cdot 47^{10} + 33\cdot 47^{11} + 35\cdot 47^{12} + 23\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(3,8)(4,9)(5,6)$ |
| $(2,7)(3,4)(8,9)$ |
| $(3,5,4)(6,8,9)$ |
| $(1,3,8)(2,4,9)(5,6,7)$ |
| $(1,2,7)(3,4,5)(6,8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,7)(3,5)(6,9)$ | $0$ |
| $9$ | $2$ | $(3,8)(4,9)(5,6)$ | $2$ |
| $9$ | $2$ | $(1,5)(2,4)(3,7)(8,9)$ | $0$ |
| $2$ | $3$ | $(1,2,7)(3,4,5)(6,8,9)$ | $-3$ |
| $6$ | $3$ | $(1,8,4)(2,9,5)(3,7,6)$ | $0$ |
| $6$ | $3$ | $(1,7,2)(3,4,5)$ | $0$ |
| $12$ | $3$ | $(1,3,8)(2,4,9)(5,6,7)$ | $0$ |
| $18$ | $6$ | $(1,4,8)(2,3,9,7,5,6)$ | $0$ |
| $18$ | $6$ | $(1,2,7)(3,9,5,8,4,6)$ | $-1$ |
| $18$ | $6$ | $(1,4,7,5,2,3)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
