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^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 8 + \left(24 a + 38\right)\cdot 47 + 47^{2} + \left(11 a + 45\right)\cdot 47^{3} + \left(37 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 17 + \left(42 a + 37\right)\cdot 47 + \left(43 a + 37\right)\cdot 47^{2} + \left(10 a + 9\right)\cdot 47^{3} + \left(43 a + 23\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 2 + \left(23 a + 21\right)\cdot 47 + \left(37 a + 17\right)\cdot 47^{2} + \left(32 a + 25\right)\cdot 47^{3} + \left(12 a + 9\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 1 + \left(22 a + 20\right)\cdot 47 + \left(46 a + 25\right)\cdot 47^{2} + \left(35 a + 19\right)\cdot 47^{3} + \left(9 a + 35\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 8 + \left(4 a + 9\right)\cdot 47 + \left(3 a + 36\right)\cdot 47^{2} + \left(36 a + 34\right)\cdot 47^{3} + \left(3 a + 4\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 12 + \left(23 a + 15\right)\cdot 47 + \left(9 a + 22\right)\cdot 47^{2} + \left(14 a + 6\right)\cdot 47^{3} + \left(34 a + 2\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,3,4,2,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,6,2,4,3,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
