Defining polynomial
| \( x^{12} - 122 x^{8} - 1484679 x^{4} - 2269810000 \) |
Invariants
| Base field: | $\Q_{61}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $9$ |
| Discriminant root field: | $\Q_{61}(\sqrt{61})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 61 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{61}$. | |
Intermediate fields
| $\Q_{61}(\sqrt{61})$, 61.3.0.1, 61.4.3.1, 61.6.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 61.3.0.1 $\cong \Q_{61}(t)$ where $t$ is a root of \( x^{3} - x + 10 \) |
| Relative Eisenstein polynomial: | $ x^{4} - 61 t^{4} \in\Q_{61}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | Not computed |