Defining polynomial
|
\(x^{14} + 84 x^{13} + 3038 x^{12} + 61488 x^{11} + 756084 x^{10} + 5714352 x^{9} + 25377688 x^{8} + 58198682 x^{7} + 50756468 x^{6} + 22895628 x^{5} + 6802152 x^{4} + 9898168 x^{3} + 63376432 x^{2} + 249556496 x + 421775625\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $14$ |
| This field is Galois over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{2})$, 13.7.6.1 x7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{2} + 12 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + 13 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{6} + 7z^{5} + 8z^{4} + 9z^{3} + 9z^{2} + 8z + 7$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_7$ (as 14T2) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | Not computed |