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