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:
| $r_{ 1 }$ |
$=$ |
$ 19 a^{2} + 33 a + 38 + \left(12 a^{2} + 39 a + 24\right)\cdot 47 + \left(20 a^{2} + 22 a + 40\right)\cdot 47^{2} + \left(7 a^{2} + 23 a + 14\right)\cdot 47^{3} + \left(27 a^{2} + 34 a + 7\right)\cdot 47^{4} + \left(24 a^{2} + 19 a + 2\right)\cdot 47^{5} + \left(37 a^{2} + 9 a + 28\right)\cdot 47^{6} + \left(43 a^{2} + 17 a + 40\right)\cdot 47^{7} + \left(46 a^{2} + 11 a + 46\right)\cdot 47^{8} + \left(29 a^{2} + 38 a + 12\right)\cdot 47^{9} + \left(9 a^{2} + 9 a + 19\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a^{2} + 35 a + 14 + \left(37 a^{2} + 5 a + 27\right)\cdot 47 + \left(7 a^{2} + 34 a + 15\right)\cdot 47^{2} + \left(20 a^{2} + 39 a + 40\right)\cdot 47^{3} + \left(45 a^{2} + 17 a + 43\right)\cdot 47^{4} + \left(16 a^{2} + 14 a + 33\right)\cdot 47^{5} + \left(21 a^{2} + 8 a + 42\right)\cdot 47^{6} + \left(11 a^{2} + 24 a + 22\right)\cdot 47^{7} + \left(21 a^{2} + 46 a + 42\right)\cdot 47^{8} + \left(43 a^{2} + 38 a + 39\right)\cdot 47^{9} + \left(6 a^{2} + 25 a + 13\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 11\cdot 47 + 28\cdot 47^{2} + 11\cdot 47^{3} + 27\cdot 47^{4} + 14\cdot 47^{5} + 37\cdot 47^{6} + 41\cdot 47^{7} + 17\cdot 47^{8} + 40\cdot 47^{9} + 27\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + 46 a + 20 + \left(34 a^{2} + 44 a + 21\right)\cdot 47 + \left(2 a^{2} + 4 a + 5\right)\cdot 47^{2} + \left(45 a^{2} + 26 a + 43\right)\cdot 47^{3} + \left(28 a^{2} + 13 a + 10\right)\cdot 47^{4} + \left(3 a^{2} + 18 a + 7\right)\cdot 47^{5} + \left(27 a^{2} + 32 a + 7\right)\cdot 47^{6} + \left(17 a^{2} + 28 a + 35\right)\cdot 47^{7} + \left(19 a^{2} + 14 a + 38\right)\cdot 47^{8} + \left(14 a^{2} + 28\right)\cdot 47^{9} + \left(29 a^{2} + 18 a + 11\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 33\cdot 47 + 29\cdot 47^{2} + 26\cdot 47^{3} + 25\cdot 47^{4} + 19\cdot 47^{5} + 6\cdot 47^{6} + 38\cdot 47^{7} + 5\cdot 47^{8} + 19\cdot 47^{9} + 23\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 a^{2} + 18 a + 45 + \left(4 a^{2} + 32 a + 9\right)\cdot 47 + \left(28 a^{2} + 16 a + 9\right)\cdot 47^{2} + \left(28 a^{2} + 16 a + 10\right)\cdot 47^{3} + \left(33 a^{2} + 9 a + 20\right)\cdot 47^{4} + \left(3 a^{2} + 44 a + 7\right)\cdot 47^{5} + \left(42 a^{2} + 39 a + 37\right)\cdot 47^{6} + \left(32 a^{2} + 2 a + 18\right)\cdot 47^{7} + \left(33 a^{2} + 11 a + 20\right)\cdot 47^{8} + \left(36 a^{2} + 36 a + 26\right)\cdot 47^{9} + \left(46 a^{2} + 3 a + 46\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 4 + 2\cdot 47 + 36\cdot 47^{2} + 8\cdot 47^{3} + 41\cdot 47^{4} + 12\cdot 47^{5} + 3\cdot 47^{6} + 14\cdot 47^{7} + 23\cdot 47^{8} + 34\cdot 47^{9} + 42\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 a^{2} + 13 a + 13 + \left(22 a^{2} + 43 a + 45\right)\cdot 47 + \left(36 a^{2} + 7 a + 25\right)\cdot 47^{2} + \left(28 a^{2} + 28 a + 10\right)\cdot 47^{3} + \left(19 a^{2} + 15 a + 39\right)\cdot 47^{4} + \left(26 a^{2} + 14 a + 5\right)\cdot 47^{5} + \left(45 a^{2} + 6 a + 44\right)\cdot 47^{6} + \left(17 a^{2} + 41 a + 35\right)\cdot 47^{7} + \left(6 a^{2} + 32 a + 12\right)\cdot 47^{8} + \left(36 a^{2} + 7 a + 25\right)\cdot 47^{9} + \left(10 a^{2} + 3 a + 21\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 29 a^{2} + 43 a + 11 + \left(29 a^{2} + 21 a + 12\right)\cdot 47 + \left(45 a^{2} + 7 a + 44\right)\cdot 47^{2} + \left(10 a^{2} + 7 a + 21\right)\cdot 47^{3} + \left(33 a^{2} + 3 a + 19\right)\cdot 47^{4} + \left(18 a^{2} + 30 a + 37\right)\cdot 47^{5} + \left(14 a^{2} + 44 a + 28\right)\cdot 47^{6} + \left(17 a^{2} + 26 a + 34\right)\cdot 47^{7} + \left(13 a^{2} + 24 a + 26\right)\cdot 47^{8} + \left(27 a^{2} + 19 a + 7\right)\cdot 47^{9} + \left(37 a^{2} + 33 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 |
| $(2,5)(3,4)(6,9)(7,8)$ |
| $(1,6,9)(2,8,4)$ |
| $(2,4,8)(3,7,5)$ |
| $(1,3,8,9,7,4,6,5,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,8)(2,9)(3,7)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,9,6)(2,8,4)(3,7,5)$ |
$-3$ |
| $3$ |
$3$ |
$(1,6,9)(2,8,4)$ |
$0$ |
| $3$ |
$3$ |
$(1,9,6)(2,4,8)$ |
$0$ |
| $9$ |
$6$ |
$(1,4,9,8,6,2)(3,7)$ |
$0$ |
| $9$ |
$6$ |
$(1,2,6,8,9,4)(3,7)$ |
$0$ |
| $6$ |
$9$ |
$(1,3,8,9,7,4,6,5,2)$ |
$0$ |
| $6$ |
$9$ |
$(1,8,7,6,2,3,9,4,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,5,8,9,3,4,6,7,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
