Defining polynomial
| \( x^{12} + 4 x^{11} + 8 x^{10} - 4 x^{7} + 4 x^{6} + 8 x^{5} - 2 x^{4} - 4 x^{2} + 8 x + 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $30$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.11.4 |
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: | \( x^{12} + 4 x^{11} - 440 x^{10} - 4592 x^{9} - 32624 x^{8} - 1102148 x^{7} - 4397036 x^{6} + 107696312 x^{5} - 162236050 x^{4} - 6096737856 x^{3} + 69022998236 x^{2} - 152538984136 x + 25946042822 \) |
Invariants of the Galois closure
| Galois group: | 12T223 |
| Inertia group: | 12T187 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 8/3, 8/3, 3, 3, 19/6, 19/6] |
| Galois mean slope: | $1183/384$ |
| Galois splitting model: | $x^{12} - 12 x^{10} - 4 x^{9} + 48 x^{8} + 36 x^{7} - 64 x^{6} - 80 x^{5} - 8 x^{4} + 40 x^{3} + 56 x^{2} + 32 x + 2$ |