Defining polynomial
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\(x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $50$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{20}{7}, \frac{20}{7}, \frac{20}{7}, \frac{15}{4}]$ |
| Visible Swan slopes: | $[\frac{13}{7},\frac{13}{7},\frac{13}{7},\frac{11}{4}]$ |
| Means: | $\langle\frac{13}{14}, \frac{39}{28}, \frac{13}{8}, \frac{35}{16}\rangle$ |
| Rams: | $(\frac{13}{7}, \frac{13}{7}, \frac{13}{7}, 9)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.8.20a1.2 |
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: |
\( x^{16} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 2 \)
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Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[35, 26, 26, 16, 0]$ |
Invariants of the Galois closure
| Galois degree: | $21504$ |
| Galois group: | $C_2^7:F_8:C_3$ (as 16T1800) |
| Inertia group: | $C_2^7:F_8$ (as 16T1694) |
| Wild inertia group: | not computed |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: |
$x^{16} - 16 x^{14} + 168 x^{12} - 1456 x^{10} + 9156 x^{8} - 22176 x^{6} + 54320 x^{4} - 60544 x^{2} + 58564$
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