Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 5 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 5 }$: $ x^{2} + 4 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2\cdot 5 + 4\cdot 5^{2} + 5^{3} + 4\cdot 5^{4} +O\left(5^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 3 + \left(4 a + 2\right)\cdot 5 + \left(3 a + 1\right)\cdot 5^{2} + \left(3 a + 3\right)\cdot 5^{3} + a\cdot 5^{4} +O\left(5^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 4 + \left(a + 1\right)\cdot 5^{2} + \left(a + 3\right)\cdot 5^{3} + \left(3 a + 3\right)\cdot 5^{4} +O\left(5^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 4\cdot 5 + 2\cdot 5^{2} + 5^{3} + 5^{4} +O\left(5^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
