Defining polynomial
| \( x^{12} - 3 x^{10} + 4 x^{8} - 3 x^{6} + 4 x^{4} + x^{2} + 3 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $6$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{*})$, 2.4.4.4, 2.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{6} + 2 t x^{5} + 2 t x^{3} + 2 t x^{2} + 2 t \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times C_3:D_4$ (as 12T42) |
| Inertia group: | Intransitive group isomorphic to $C_2\times C_6$ |
| Unramified degree: | $6$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 2] |
| Galois mean slope: | $5/3$ |
| Galois splitting model: | $x^{12} - 10 x^{9} + 30 x^{6} - 16 x^{3} + 4$ |