Basic invariants
Defining polynomial
| $f(x)$ | $=$ | $ x^{6} - 86 x^{4} + 1376 x^{2} - 5504 $. |
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^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 15 + \left(a + 14\right)\cdot 47 + \left(22 a + 2\right)\cdot 47^{2} + \left(28 a + 6\right)\cdot 47^{3} + \left(29 a + 8\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 28 + \left(24 a + 8\right)\cdot 47 + \left(28 a + 7\right)\cdot 47^{2} + \left(18 a + 19\right)\cdot 47^{3} + \left(40 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 36 + \left(46 a + 29\right)\cdot 47 + \left(29 a + 16\right)\cdot 47^{2} + \left(5 a + 9\right)\cdot 47^{3} + \left(a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 32 + \left(45 a + 32\right)\cdot 47 + \left(24 a + 44\right)\cdot 47^{2} + \left(18 a + 40\right)\cdot 47^{3} + \left(17 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 19 + \left(22 a + 38\right)\cdot 47 + \left(18 a + 39\right)\cdot 47^{2} + \left(28 a + 27\right)\cdot 47^{3} + \left(6 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 11 + 17\cdot 47 + \left(17 a + 30\right)\cdot 47^{2} + \left(41 a + 37\right)\cdot 47^{3} + \left(45 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,6,2)(3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
