Defining polynomial
| \( x^{12} + 39 x^{6} + 676 \) |
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{13}$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 13 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{13}$. | |
Intermediate fields
| $\Q_{13}(\sqrt{*})$, $\Q_{13}(\sqrt{13})$, $\Q_{13}(\sqrt{13*})$, 13.3.2.1, 13.4.2.1, 13.6.4.1, 13.6.5.3, 13.6.5.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 13 t^{2} \in\Q_{13}(t)[x]$ |