Defining polynomial
| \( x^{12} - 54 x^{10} - 509 x^{8} - 964 x^{6} - 777 x^{4} - 934 x^{2} + 357 \) |
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: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| 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} + 4 x^{5} + 4 x^{4} - 2 x^{3} - 2 x^{2} + 4 t x + 4 t + 2 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $A_4:C_4$ (as 12T30) |
| Inertia group: | Intransitive group isomorphic to $C_2\times A_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 2] |
| Galois mean slope: | $19/12$ |
| Galois splitting model: | $x^{12} - 4 x^{11} + 9 x^{10} + 2 x^{9} - 42 x^{8} + 112 x^{7} - 99 x^{6} - 84 x^{5} + 366 x^{4} - 314 x^{3} + 137 x^{2} - 16 x - 1$ |