Defining polynomial
| \( x^{12} - 8 x^{10} - 28 x^{8} + 40 x^{6} - 44 x^{4} + 48 x^{2} + 40 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $33$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2*})$, 2.3.0.1, 2.4.11.7, 2.6.9.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{4} - 4 t^{2} x^{2} - 6 t^{2} + 6 t \in\Q_{2}(t)[x]$ |