Defining polynomial
| \( x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $|\Gal(K/\Q_{ 3 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.7.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{7} + x^{2} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 3 t^{2} \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $7$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |