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 + 27 + \left(21 a + 35\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(9 a + 7\right)\cdot 47^{3} + \left(30 a + 17\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 43 + \left(21 a + 23\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(9 a + 34\right)\cdot 47^{3} + \left(30 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 34 + 25 a\cdot 47 + \left(17 a + 15\right)\cdot 47^{2} + \left(37 a + 24\right)\cdot 47^{3} + \left(16 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 29 + \left(25 a + 22\right)\cdot 47 + \left(17 a + 2\right)\cdot 47^{2} + \left(37 a + 10\right)\cdot 47^{3} + \left(16 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 18 + \left(25 a + 12\right)\cdot 47 + \left(17 a + 15\right)\cdot 47^{2} + \left(37 a + 44\right)\cdot 47^{3} + \left(16 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 38 + \left(21 a + 45\right)\cdot 47 + \left(29 a + 11\right)\cdot 47^{2} + \left(9 a + 20\right)\cdot 47^{3} + \left(30 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,2)(3,5,4)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
