Defining polynomial
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\(x^{16} + 2 x^{9} + 2 x^{4} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $24$ |
| 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: | $[\frac{4}{3}, \frac{4}{3}, \frac{5}{3}, \frac{5}{3}]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}]$ |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}, \frac{11}{24}, \frac{9}{16}\rangle$ |
| Rams: | $(\frac{1}{3}, \frac{1}{3}, \frac{5}{3}, \frac{5}{3})$ |
| Jump set: | $[1, 2, 5, 10, 25]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.4.4a1.1 |
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: |
\( x^{16} + 2 x^{9} + 2 x^{4} + 2 \)
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Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[9, 9, 4, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $96$ |
| Galois group: | $C_4^2:S_3$ (as 16T195) |
| Inertia group: | $C_4^2:C_3$ (as 16T63) |
| Wild inertia group: | $C_4^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, \frac{5}{3}, \frac{5}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}]$ |
| Galois mean slope: | $1.5416666666666667$ |
| Galois splitting model: |
$x^{16} - 2 x^{15} - 176 x^{14} - 76 x^{13} + 10584 x^{12} + 24580 x^{11} - 236488 x^{10} - 925212 x^{9} + 1185218 x^{8} + 9368638 x^{7} + 6137672 x^{6} - 27314420 x^{5} - 43923786 x^{4} - 1203456 x^{3} + 35879804 x^{2} + 23272968 x + 4217441$
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