Defining polynomial
| \( x^{12} - 104 x^{8} - 45968 x^{4} - 2847312 \) |
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $9$ |
| Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 13 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{13}$. | |
Intermediate fields
| $\Q_{13}(\sqrt{13})$, 13.3.0.1, 13.4.3.1, 13.6.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.3.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{3} - 2 x + 6 \) |
| Relative Eisenstein polynomial: | $ x^{4} - 13 t^{4} \in\Q_{13}(t)[x]$ |