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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,6,2)(3,5,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,4,2,5,6,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,6,5,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
