Defining polynomial
|
$( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 ) + 5$
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $10$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $20$ |
|
| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_2$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{9}{8}]$ | |
| Visible Swan slopes: | $[\frac{1}{8}]$ | |
| Means: | $\langle\frac{1}{10}\rangle$ | |
| Rams: | $(\frac{1}{4})$ | |
| 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, 5.1.10.10a1.1 |
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: |
\( x^{10} + 5 x + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $400$ |
| Galois group: | $C_5^2:\OD_{16}$ (as 20T115) |
| Inertia group: | Intransitive group isomorphic to $C_5^2:C_8$ |
| Wild inertia group: | $C_5^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $8$ |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | $x^{20} - 10 x^{18} + 65 x^{16} - 280 x^{14} + 910 x^{12} - 2174 x^{10} + 3920 x^{8} - 4940 x^{6} + 3865 x^{4} - 1380 x^{2} + 144$ |