Defining polynomial
|
$( x^{2} + 4 x + 2 )^{10} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{8} + 5 ( x^{2} + 4 x + 2 )^{7} + 5$
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $10$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $32$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| Visible Artin slopes: | $[\frac{15}{8}]$ |
| Visible Swan slopes: | $[\frac{7}{8}]$ |
| Means: | $\langle\frac{7}{10}\rangle$ |
| Rams: | $(\frac{7}{4})$ |
| Jump set: | undefined |
| Roots of unity: | $24 = (5^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.2.2.2a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{10} + \left(15 t + 15\right) x^{8} + 5 x^{7} + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z + (3 t + 1)$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[7, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| 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 |