Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 + 46\cdot 47 + 29\cdot 47^{2} + 4\cdot 47^{3} + 9\cdot 47^{4} + 15\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 13 + \left(20 a + 45\right)\cdot 47 + \left(2 a + 20\right)\cdot 47^{2} + \left(27 a + 30\right)\cdot 47^{3} + \left(4 a + 21\right)\cdot 47^{4} + \left(45 a + 7\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 21 + \left(26 a + 34\right)\cdot 47 + \left(44 a + 5\right)\cdot 47^{2} + \left(19 a + 35\right)\cdot 47^{3} + \left(42 a + 3\right)\cdot 47^{4} + \left(a + 46\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 40\cdot 47 + 7\cdot 47^{2} + 36\cdot 47^{3} + 47^{4} + 32\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a + 25 + \left(26 a + 28\right)\cdot 47 + \left(44 a + 30\right)\cdot 47^{2} + \left(19 a + 19\right)\cdot 47^{3} + \left(42 a + 43\right)\cdot 47^{4} + \left(a + 15\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 17 + \left(20 a + 39\right)\cdot 47 + \left(2 a + 45\right)\cdot 47^{2} + \left(27 a + 14\right)\cdot 47^{3} + \left(4 a + 14\right)\cdot 47^{4} + \left(45 a + 24\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,6)$ |
| $(2,3)(5,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,2,4,3,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
