Defining polynomial
| \( x^{12} + 26 x^{9} + 845 x^{6} + 6591 x^{3} + 114244 \) |
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $8$ |
| 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
| $\Q_{13}(\sqrt{*})$, 13.3.2.3, 13.4.0.1, 13.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{4} + x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{3} - 13 t^{2} \in\Q_{13}(t)[x]$ |