\(x^{16} + 2 x^{15} + 2 x^{13} + 4 x^{7} + 4 x^{3} + 2 x^{2} + 2\)
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| Base field: | $\Q_{2}$
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| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
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$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{5}{2}]$ |
| Visible Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{2}]$ |
| Means: | $\langle\frac{1}{14}, \frac{3}{28}, \frac{1}{8}, \frac{13}{16}\rangle$ |
| Rams: | $(\frac{1}{7}, \frac{1}{7}, \frac{1}{7}, 11)$ |
| Jump set: | $[1, 2, 4, 9, 25]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$21504$
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| Galois group: |
$C_2^7:F_8:C_3$ (as 16T1800)
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| Inertia group: |
$C_2^7:F_8$ (as 16T1694)
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| Wild inertia group: |
not computed
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| Galois unramified degree: |
$3$
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| Galois tame degree: |
$7$
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| Galois Artin slopes: |
$[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{16}{7}, \frac{16}{7}, \frac{16}{7}, \frac{17}{7}, \frac{17}{7}, \frac{17}{7}, \frac{5}{2}]$
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| Galois Swan slopes: |
$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{9}{7},\frac{9}{7},\frac{9}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7},\frac{3}{2}]$
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| Galois mean slope: |
$2.4461495535714284$
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| Galois splitting model: |
$x^{16} - 4 x^{15} + 22 x^{14} - 70 x^{13} + 168 x^{12} - 420 x^{11} + 56 x^{10} + 1580 x^{9} - 566 x^{8} - 3628 x^{7} + 1526 x^{6} + 5096 x^{5} - 3724 x^{4} - 2184 x^{3} + 3586 x^{2} - 1100 x + 254$
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