Defining polynomial
| \( x^{12} + 5238 x^{6} + 147015625 \) |
Invariants
| Base field: | $\Q_{97}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{97}$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 97 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{97}$. | |
Intermediate fields
| $\Q_{97}(\sqrt{*})$, $\Q_{97}(\sqrt{97})$, $\Q_{97}(\sqrt{97*})$, 97.3.2.1, 97.4.2.1, 97.6.4.1, 97.6.5.1, 97.6.5.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{97}(\sqrt{*})$ $\cong \Q_{97}(t)$ where $t$ is a root of \( x^{2} - x + 5 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 97 t^{6} \in\Q_{97}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |