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:
| $r_{ 1 }$ |
$=$ |
$ 5 + 40\cdot 47 + 37\cdot 47^{2} + 14\cdot 47^{3} + 20\cdot 47^{4} + 34\cdot 47^{5} + 24\cdot 47^{6} + 45\cdot 47^{7} + 34\cdot 47^{8} + 16\cdot 47^{9} + 2\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a^{2} + 32 a + 8 + \left(13 a^{2} + 45 a\right)\cdot 47 + \left(42 a^{2} + a + 32\right)\cdot 47^{2} + \left(9 a^{2} + 9 a + 10\right)\cdot 47^{3} + \left(18 a^{2} + 25 a + 10\right)\cdot 47^{4} + \left(5 a^{2} + a + 7\right)\cdot 47^{5} + \left(25 a^{2} + 10 a + 38\right)\cdot 47^{6} + \left(12 a^{2} + 6 a + 5\right)\cdot 47^{7} + \left(4 a^{2} + 41 a + 20\right)\cdot 47^{8} + \left(25 a^{2} + 21 a + 40\right)\cdot 47^{9} + \left(39 a^{2} + 43 a + 19\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a^{2} + 34 a + 4 + \left(21 a^{2} + 12 a + 16\right)\cdot 47 + \left(46 a^{2} + 32 a + 40\right)\cdot 47^{2} + \left(21 a^{2} + 10 a + 34\right)\cdot 47^{3} + \left(30 a^{2} + 39 a + 34\right)\cdot 47^{4} + \left(34 a^{2} + 5 a + 18\right)\cdot 47^{5} + \left(a^{2} + 12 a + 38\right)\cdot 47^{6} + \left(46 a^{2} + 28 a + 25\right)\cdot 47^{7} + \left(18 a^{2} + 19 a + 2\right)\cdot 47^{8} + \left(26 a^{2} + 33 a + 43\right)\cdot 47^{9} + \left(12 a^{2} + 13 a + 12\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 36\cdot 47 + 46\cdot 47^{2} + 21\cdot 47^{3} + 45\cdot 47^{4} + 46\cdot 47^{5} + 30\cdot 47^{6} + 34\cdot 47^{7} + 11\cdot 47^{8} + 25\cdot 47^{9} + 18\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a^{2} + 28 a + 14 + \left(11 a^{2} + 35 a + 43\right)\cdot 47 + \left(5 a^{2} + 12 a + 4\right)\cdot 47^{2} + \left(15 a^{2} + 27 a + 21\right)\cdot 47^{3} + \left(45 a^{2} + 29 a + 17\right)\cdot 47^{4} + \left(6 a^{2} + 39 a + 10\right)\cdot 47^{5} + \left(20 a^{2} + 24 a + 28\right)\cdot 47^{6} + \left(35 a^{2} + 12 a + 4\right)\cdot 47^{7} + \left(23 a^{2} + 33 a + 12\right)\cdot 47^{8} + \left(42 a^{2} + 38 a + 28\right)\cdot 47^{9} + \left(41 a^{2} + 36 a + 24\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 a^{2} + 16 a + 32 + \left(a^{2} + 18 a + 13\right)\cdot 47 + \left(40 a^{2} + 24 a + 40\right)\cdot 47^{2} + \left(36 a^{2} + 18\right)\cdot 47^{3} + \left(30 a^{2} + 15 a + 42\right)\cdot 47^{4} + \left(30 a^{2} + 24 a + 28\right)\cdot 47^{5} + \left(23 a^{2} + 29 a + 19\right)\cdot 47^{6} + \left(9 a^{2} + 13 a + 11\right)\cdot 47^{7} + \left(4 a^{2} + 9 a + 13\right)\cdot 47^{8} + \left(16 a^{2} + 39 a + 17\right)\cdot 47^{9} + \left(17 a^{2} + 23 a + 46\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 10 a + 34 + \left(7 a^{2} + 26 a + 23\right)\cdot 47 + \left(2 a^{2} + 39 a + 11\right)\cdot 47^{2} + \left(13 a^{2} + 16 a + 18\right)\cdot 47^{3} + \left(39 a^{2} + 33 a + 12\right)\cdot 47^{4} + \left(28 a^{2} + 31 a + 25\right)\cdot 47^{5} + \left(32 a^{2} + 20 a + 37\right)\cdot 47^{6} + \left(28 a^{2} + 19 a + 2\right)\cdot 47^{7} + \left(18 a^{2} + 32 a + 42\right)\cdot 47^{8} + \left(18 a^{2} + 20 a + 21\right)\cdot 47^{9} + \left(37 a^{2} + 39\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 20 + 22\cdot 47 + 4\cdot 47^{2} + 14\cdot 47^{3} + 23\cdot 47^{4} + 26\cdot 47^{5} + 16\cdot 47^{6} + 45\cdot 47^{8} + 31\cdot 47^{9} + 27\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ a^{2} + 21 a + 36 + \left(38 a^{2} + 2 a + 38\right)\cdot 47 + \left(4 a^{2} + 30 a + 16\right)\cdot 47^{2} + \left(44 a^{2} + 29 a + 33\right)\cdot 47^{3} + \left(23 a^{2} + 45 a + 28\right)\cdot 47^{4} + \left(34 a^{2} + 37 a + 36\right)\cdot 47^{5} + \left(37 a^{2} + 43 a\right)\cdot 47^{6} + \left(8 a^{2} + 13 a + 10\right)\cdot 47^{7} + \left(24 a^{2} + 5 a + 6\right)\cdot 47^{8} + \left(12 a^{2} + 34 a + 10\right)\cdot 47^{9} + \left(39 a^{2} + 22 a + 43\right)\cdot 47^{10} +O\left(47^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,7,8,3,9,4,5,6)$ |
| $(1,8,4)(6,9,7)$ |
| $(1,8,4)(2,3,5)(6,7,9)$ |
| $(1,6)(2,5)(4,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,6)(2,5)(4,7)(8,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,8,4)(2,3,5)(6,7,9)$ |
$-3$ |
| $3$ |
$3$ |
$(2,3,5)(6,9,7)$ |
$0$ |
| $3$ |
$3$ |
$(2,5,3)(6,7,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,8)(2,9,3,7,5,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,8)(2,6,5,7,3,9)$ |
$0$ |
| $6$ |
$9$ |
$(1,2,7,8,3,9,4,5,6)$ |
$0$ |
| $6$ |
$9$ |
$(1,2,6,8,3,7,4,5,9)$ |
$0$ |
| $6$ |
$9$ |
$(1,6,3,4,9,2,8,7,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
