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:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 6 + \left(14 a + 29\right)\cdot 47 + \left(16 a + 34\right)\cdot 47^{2} + \left(2 a + 31\right)\cdot 47^{3} + \left(24 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 8 + \left(32 a + 7\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + 44 a\cdot 47^{3} + \left(22 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a + 17 + \left(14 a + 39\right)\cdot 47 + \left(16 a + 21\right)\cdot 47^{2} + \left(2 a + 44\right)\cdot 47^{3} + \left(24 a + 35\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 22 + \left(14 a + 17\right)\cdot 47 + \left(16 a + 34\right)\cdot 47^{2} + \left(2 a + 11\right)\cdot 47^{3} + \left(24 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 39 + \left(32 a + 18\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + \left(44 a + 20\right)\cdot 47^{3} + \left(22 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 3 + \left(32 a + 29\right)\cdot 47 + \left(30 a + 39\right)\cdot 47^{2} + \left(44 a + 32\right)\cdot 47^{3} + \left(22 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,4)(3,6)$ |
| $(1,2,3,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,4,3)(2,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,3,5,4,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,4,5,3,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
