Defining polynomial
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\(x^{16} + 2 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{7} + 24 x^{4} + 24 x^{2} + 22\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $54$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 2, 7/2, 17/4]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.4.9.6, 2.4.9.8, 2.4.6.9, 2.8.26.86, 2.8.20.86, 2.8.26.87 |
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: |
\( x^{16} + 2 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{7} + 24 x^{4} + 24 x^{2} + 22 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{12} + 1$ |
| Associated inertia: | $1$,$1$,$2$ |
| Indices of inseparability: | $[39, 26, 12, 12, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4:D_4$ (as 16T392) |
| Inertia group: | $C_2^4:C_4$ (as 16T76) |
| Wild inertia group: | $C_2^4:C_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 3, 7/2, 4, 17/4]$ |
| Galois mean slope: | $123/32$ |
| Galois splitting model: | $x^{16} - 12 x^{14} + 56 x^{12} - 144 x^{10} + 204 x^{8} - 120 x^{6} + 8 x^{4} + 4$ |