Defining polynomial
|
\(x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{10} + 16 x^{9} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 18\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $66$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2^2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3, \frac{7}{2}, 4, 5]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},3,4]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{19}{8}, \frac{51}{16}\rangle$ |
| Rams: | $(2, 3, 5, 13)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.1.4.10a1.1, 2.1.4.11a1.13, 2.1.4.11a1.12, 2.1.8.31a1.184 x2, 2.1.8.26c1.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{10} + 16 x^{9} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 18 \)
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Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[51, 38, 28, 16, 0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $\OD_{16}:C_2$ (as 16T41) |
| Inertia group: | $\OD_{16}:C_2$ (as 16T41) |
| Wild inertia group: | $\OD_{16}:C_2$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, \frac{7}{2}, 4, 5]$ |
| Galois Swan slopes: | $[1,2,\frac{5}{2},3,4]$ |
| Galois mean slope: | $4.1875$ |
| Galois splitting model: | $x^{16} - 8 x^{14} + 44 x^{12} + 280 x^{10} + 994 x^{8} + 1016 x^{6} - 652 x^{4} - 1288 x^{2} + 961$ |