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