Defining polynomial
|
$( x^{4} + 3 x^{2} + 12 x + 2 )^{2} + \left(-6 x^{2} - 24 x + 512\right) ( x^{4} + 3 x^{2} + 12 x + 2 ) + 72 x^{3} - 27756 x^{2} - 6144 x + 56094$
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{2})$, 13.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{4} + 3 x^{2} + 12 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 13 t \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$ |