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