Defining polynomial
|
\(x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.5.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{5} + 4 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 20 x + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$ |