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