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