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