Defining polynomial
| \( x^{12} - 7 x^{8} + 15 x^{4} + 3 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{*})$, 2.3.2.1 x3, 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: | $\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{6} + 2 x^{5} + \left(2 t + 2\right) x^{4} + 4 t x^{2} + 4 t + 2 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $\GL(2,Z/4)$ (as 12T50) |
| Inertia group: | Intransitive group isomorphic to $C_2^2\times A_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 2, 8/3, 8/3] |
| Galois mean slope: | $29/12$ |
| Galois splitting model: | $x^{12} - 9 x^{10} + 30 x^{8} - 45 x^{6} + 30 x^{4} - 9 x^{2} + 3$ |