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:
| $r_{ 1 }$ |
$=$ |
$ a + 3 + \left(2 a + 1\right)\cdot 5 + \left(a + 2\right)\cdot 5^{2} + \left(3 a + 2\right)\cdot 5^{3} + 3\cdot 5^{4} +O\left(5^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 1 + 4\cdot 5 + \left(2 a + 4\right)\cdot 5^{2} + \left(a + 1\right)\cdot 5^{3} + 3\cdot 5^{4} +O\left(5^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 4 + \left(2 a + 2\right)\cdot 5 + \left(3 a + 1\right)\cdot 5^{2} + \left(a + 4\right)\cdot 5^{3} + 4 a\cdot 5^{4} +O\left(5^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 4 + \left(4 a + 1\right)\cdot 5 + \left(2 a + 1\right)\cdot 5^{2} + \left(3 a + 1\right)\cdot 5^{3} + \left(4 a + 2\right)\cdot 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 value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
