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 + 11 + \left(39 a + 3\right)\cdot 47 + \left(16 a + 3\right)\cdot 47^{2} + \left(16 a + 46\right)\cdot 47^{3} + \left(13 a + 43\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 27 + \left(7 a + 42\right)\cdot 47 + \left(30 a + 32\right)\cdot 47^{2} + \left(30 a + 9\right)\cdot 47^{3} + \left(33 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 37 + \left(39 a + 18\right)\cdot 47 + \left(16 a + 37\right)\cdot 47^{2} + \left(16 a + 6\right)\cdot 47^{3} + \left(13 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 34 + \left(39 a + 30\right)\cdot 47 + \left(16 a + 38\right)\cdot 47^{2} + \left(16 a + 40\right)\cdot 47^{3} + \left(13 a + 17\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 30 + \left(7 a + 30\right)\cdot 47 + \left(30 a + 31\right)\cdot 47^{2} + \left(30 a + 22\right)\cdot 47^{3} + \left(33 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 4 + \left(7 a + 15\right)\cdot 47 + \left(30 a + 44\right)\cdot 47^{2} + \left(30 a + 14\right)\cdot 47^{3} + \left(33 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3,6,4,5)$ |
| $(1,6)(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,3,6,4,5)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,5,4,6,3,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
