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:
| $r_{ 1 }$ |
$=$ |
$ 34 a + 14 + \left(2 a + 41\right)\cdot 47 + \left(43 a + 21\right)\cdot 47^{2} + \left(a + 31\right)\cdot 47^{3} + 11 a\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a + 2 + \left(2 a + 25\right)\cdot 47 + \left(43 a + 36\right)\cdot 47^{2} + \left(a + 43\right)\cdot 47^{3} + \left(11 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 22 + \left(44 a + 18\right)\cdot 47 + \left(3 a + 17\right)\cdot 47^{2} + \left(45 a + 38\right)\cdot 47^{3} + \left(35 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 1 + 2 a\cdot 47 + \left(43 a + 28\right)\cdot 47^{2} + \left(a + 30\right)\cdot 47^{3} + \left(11 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 23 + \left(44 a + 43\right)\cdot 47 + \left(3 a + 25\right)\cdot 47^{2} + \left(45 a + 4\right)\cdot 47^{3} + \left(35 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 35 + \left(44 a + 12\right)\cdot 47 + \left(3 a + 11\right)\cdot 47^{2} + \left(45 a + 39\right)\cdot 47^{3} + \left(35 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,4,6,2,3)$ |
| $(1,6)(2,5)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,4,6,2,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,2,6,4,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
