Defining polynomial
|
$( x^{5} + 4 x + 11 )^{2} + \left(650 x^{4} + 169065 x^{3} + 22003800 x^{2} + 1434642690 x + 37701179620\right) ( x^{5} + 4 x + 11 ) + 18650354190 x^{4} + 3718368225 x^{3} + 335340785 x^{2} - 2390665550 x + 6946094727$
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{13})$, 13.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.5.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{5} + 4 x + 11 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 130 x + 13 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
Not computedInvariants 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: | $x^{10} - 3 x^{9} - 21 x^{8} + 56 x^{7} + 130 x^{6} - 304 x^{5} - 204 x^{4} + 471 x^{3} - 61 x^{2} - 89 x + 23$ |