Defining polynomial
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\(x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $54$ |
| 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: | $[2, 3, \frac{7}{2}, 4]$ |
| Visible Swan slopes: | $[1,2,\frac{5}{2},3]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{15}{8}, \frac{39}{16}\rangle$ |
| Rams: | $(1, 3, 5, 9)$ |
| Jump set: | $[1, 2, 15, 31, 47]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.1.4.8b1.3, 2.1.4.9a1.1, 2.1.4.9a1.4, 2.1.8.22d1.1, 2.1.8.24c1.21, 2.1.8.24c1.27 |
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: |
\( x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 2 \)
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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: | $[39, 30, 20, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $64$ |
| Galois group: | $D_4:D_4$ (as 16T126) |
| Inertia group: | $C_4:D_4$ (as 16T43) |
| Wild inertia group: | $C_4:D_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, \frac{7}{2}, 4]$ |
| Galois Swan slopes: | $[1,1,2,\frac{5}{2},3]$ |
| Galois mean slope: | $3.4375$ |
| Galois splitting model: |
$x^{16} - 8 x^{15} + 28 x^{14} - 48 x^{13} + 20 x^{12} + 136 x^{11} - 400 x^{10} + 624 x^{9} - 474 x^{8} - 184 x^{7} + 1240 x^{6} - 2112 x^{5} + 2396 x^{4} - 1936 x^{3} + 1176 x^{2} - 480 x + 130$
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