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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 21 + \left(21 a + 6\right)\cdot 47 + \left(29 a + 5\right)\cdot 47^{2} + \left(9 a + 41\right)\cdot 47^{3} + \left(30 a + 27\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 29 + \left(25 a + 4\right)\cdot 47 + \left(17 a + 14\right)\cdot 47^{2} + \left(37 a + 30\right)\cdot 47^{3} + \left(16 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 2 + \left(21 a + 24\right)\cdot 47 + \left(29 a + 32\right)\cdot 47^{2} + \left(9 a + 27\right)\cdot 47^{3} + \left(30 a + 15\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 12 + \left(25 a + 30\right)\cdot 47 + \left(17 a + 42\right)\cdot 47^{2} + \left(37 a + 30\right)\cdot 47^{3} + \left(16 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 40 + 25 a\cdot 47 + \left(17 a + 23\right)\cdot 47^{2} + \left(37 a + 17\right)\cdot 47^{3} + \left(16 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 38 + \left(21 a + 27\right)\cdot 47 + \left(29 a + 23\right)\cdot 47^{2} + \left(9 a + 40\right)\cdot 47^{3} + \left(30 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6,4,3,2)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,6,3)(2,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,6)(2,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,3,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,4,6,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
