Defining polynomial
| \( x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $12$ |
| This field is Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{*})$, 2.3.2.1 x3, 2.4.0.1, 2.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{3} + \left(2 t^{3} + 2 t^{2} + 2 t + 2\right) x^{2} + 2 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{12} - 3 x^{11} - 888 x^{10} + 2660 x^{9} + 308151 x^{8} - 878220 x^{7} - 53209587 x^{6} + 132388596 x^{5} + 4800412971 x^{4} - 8975649010 x^{3} - 213443493450 x^{2} + 211625342025 x + 3644274033955$ |