Defining polynomial
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\(x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $56$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, 4, 4]$ |
| Visible Swan slopes: | $[1,2,3,3]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}, \frac{41}{16}\rangle$ |
| Rams: | $(1, 3, 7, 7)$ |
| Jump set: | $[1, 7, 15, 31, 47]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.1.4.8b1.3 |
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: |
\( x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 2 \)
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Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^3 + z + 1$ |
| Associated inertia: | $1$,$1$,$3$ |
| Indices of inseparability: | $[41, 34, 20, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $6144$ |
| Galois group: | $C_2^6.(D_4\times A_4)$ (as 16T1655) |
| Inertia group: | $C_2^2\wr C_2^2$ (as 16T1152) |
| Wild inertia group: | not computed |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: |
$x^{16} - 24 x^{14} - 16 x^{13} + 344 x^{12} - 224 x^{11} - 1704 x^{10} + 2264 x^{9} + 454 x^{8} + 4960 x^{7} - 20056 x^{6} - 31360 x^{5} + 24636 x^{4} + 9376 x^{3} + 200 x^{2} - 10256 x + 3914$
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