Defining polynomial
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\(x^{16} + 16 x^{15} + 12 x^{12} + 26 x^{8} + 48 x^{6} + 32 x^{4} + 32 x^{3} + 16 x^{2} + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $71$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 7/2, 9/2, 11/2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.4.9.8, 2.8.27.117 |
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: |
\( x^{16} + 16 x^{15} + 12 x^{12} + 26 x^{8} + 48 x^{6} + 32 x^{4} + 32 x^{3} + 16 x^{2} + 6 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$,$z^{8} + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[56, 40, 24, 8, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^6.\SD_{16}$ (as 16T1285) |
| Inertia group: | $C_2^5.\SD_{16}$ (as 16T1001) |
| Wild inertia group: | data not computed |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 3, 7/2, 7/2, 15/4, 9/2, 9/2, 11/2]$ |
| Galois mean slope: | $1247/256$ |
| Galois splitting model: | $x^{16} - 12 x^{12} - 96 x^{10} - 438 x^{8} + 1248 x^{6} - 1008 x^{4} - 4080 x^{2} + 726$ |