Defining polynomial
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\(x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 18\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $52$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2\times C_4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, 3, 4]$ |
| Visible Swan slopes: | $[1,2,2,3]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{13}{8}, \frac{37}{16}\rangle$ |
| Rams: | $(1, 3, 3, 11)$ |
| Jump set: | $[1, 5, 25, 41, 57]$ |
| Roots of unity: | $8 = 2^{ 3 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2})$, 2.1.4.8b1.1, 2.1.4.11a1.1 x2, 2.1.4.11a1.14 x2, 2.1.8.20d1.5, 2.1.8.24c1.9, 2.1.8.24c1.6 |
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: |
\( x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 18 \)
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Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^6 + 1$,$z + 1$ |
| Associated inertia: | $1$,$2$,$1$ |
| Indices of inseparability: | $[37, 26, 24, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $C_2^2.D_4$ (as 16T54) |
| Inertia group: | $C_4:C_4$ (as 16T8) |
| Wild inertia group: | $C_4:C_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, 3, 4]$ |
| Galois Swan slopes: | $[1,2,2,3]$ |
| Galois mean slope: | $3.25$ |
| Galois splitting model: | $x^{16} - 8 x^{14} + 20 x^{12} - 8 x^{10} - 24 x^{8} + 40 x^{4} - 16 x^{2} + 4$ |