Defining polynomial
|
$( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$ $=$$\Gal(K/\Q_{5})$: | $C_4^2$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | $[1]$ |
| Roots of unity: | $3120 = (5^{ 4 } - 1) \cdot 5$ |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.4.1.0a1.1, 5.2.2.2a1.2, 5.2.2.2a1.1, 5.1.4.3a1.4, 5.1.4.3a1.1, 5.1.4.3a1.2, 5.1.4.3a1.3, 5.4.2.4a1.2, 5.2.4.6a1.2, 5.2.4.6a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.4.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{4} + 4 x^{2} + 4 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + 4 z^2 + z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $16$ |
| Galois group: | $C_4^2$ (as 16T4) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | not computed |