Defining polynomial
| \( x^{12} + 4 x^{10} + 4 x^{9} + 4 x^{7} - 2 x^{6} + 4 x^{2} + 4 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.3.2.1, 2.6.8.1, 2.6.10.3, 2.6.10.8 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{12} + 4 x^{10} + 4 x^{9} + 4 x^{7} - 2 x^{6} + 4 x^{2} + 4 x + 2 \) |