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