Defining polynomial
|
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 15 ( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5$
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$: | $C_4$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\frac{7}{4}]$ |
| Visible Swan slopes: | $[\frac{3}{4}]$ |
| Means: | $\langle\frac{3}{5}\rangle$ |
| Rams: | $(\frac{3}{4})$ |
| Jump set: | undefined |
| Roots of unity: | $624 = (5^{ 4 } - 1)$ |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, 5.4.1.0a1.1, 5.1.5.7a1.3, 5.2.5.14a1.1 |
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: |
\( x^{5} + 15 x^{3} + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + (2 t^3 + t^2)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $80$ |
| Galois group: | $C_4\times F_5$ (as 20T20) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |