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