Defining polynomial
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\(x^{16} + 16 x^{15} + 48 x^{14} + 32 x^{13} + 56 x^{12} + 32 x^{11} + 48 x^{10} + 40 x^{8} + 48 x^{6} + 40 x^{4} + 34\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $78$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 4, 5, 47/8]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.4.11.18, 2.8.31.71 |
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: |
\( x^{16} + 16 x^{15} + 48 x^{14} + 32 x^{13} + 56 x^{12} + 32 x^{11} + 48 x^{10} + 40 x^{8} + 48 x^{6} + 40 x^{4} + 34 \)
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Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_2^6.C_2\wr D_4$ (as 16T1737) |
| Inertia group: | $C_2^6.C_2\wr C_4$ (as 16T1585) |
| Wild inertia group: | Not computed |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 45/8, 47/8]$ |
| Galois mean slope: | $11523/2048$ |
| Galois splitting model: | $x^{16} - 8 x^{14} + 60 x^{12} - 360 x^{10} + 1986 x^{8} - 744 x^{6} + 1124 x^{4} - 952 x^{2} + 243$ |