Defining polynomial
| \( x^{12} + 28561 x^{4} - 742586 x^{2} + 9653618 \) |
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $6$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{13}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 13 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{13}$. | |
Intermediate fields
| $\Q_{13}(\sqrt{*})$, 13.3.0.1, 13.4.2.2, 13.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.6.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{6} + x^{2} - 2 x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 13 t \in\Q_{13}(t)[x]$ |