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:
| $r_{ 1 }$ |
$=$ |
$ a + 34 + \left(42 a + 3\right)\cdot 47 + \left(45 a + 2\right)\cdot 47^{2} + \left(27 a + 36\right)\cdot 47^{3} + \left(32 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 30 + \left(14 a + 21\right)\cdot 47 + \left(23 a + 31\right)\cdot 47^{2} + \left(14 a + 38\right)\cdot 47^{3} + \left(8 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 6 + \left(32 a + 16\right)\cdot 47 + \left(23 a + 16\right)\cdot 47^{2} + \left(32 a + 44\right)\cdot 47^{3} + \left(38 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 a + 36 + \left(4 a + 39\right)\cdot 47 + \left(a + 4\right)\cdot 47^{2} + \left(19 a + 46\right)\cdot 47^{3} + \left(14 a + 24\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 7 + \left(19 a + 27\right)\cdot 47 + \left(24 a + 28\right)\cdot 47^{2} + \left(33 a + 13\right)\cdot 47^{3} + \left(22 a + 14\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 28 + \left(27 a + 32\right)\cdot 47 + \left(22 a + 10\right)\cdot 47^{2} + \left(13 a + 9\right)\cdot 47^{3} + \left(24 a + 26\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,3)(5,6)$ |
| $(1,2,5,4,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,5,4,3,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,3,4,5,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
