\(x^{16} + 4 x^{15} + 2 x^{12} + 4 x^{11} + 4 x^{8} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 6\)
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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 $c$: | $42$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_4$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, 2, \frac{5}{2}, \frac{13}{4}]$ |
| Visible Swan slopes: | $[1,1,\frac{3}{2},\frac{9}{4}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{9}{8}, \frac{27}{16}\rangle$ |
| Rams: | $(1, 1, 3, 9)$ |
| Jump set: | $[1, 3, 6, 12, 32]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$256$
|
| Galois group: |
$C_8^2:C_2^2$ (as 16T568)
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| Inertia group: |
$C_2^3.Q_8$ (as 16T103)
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| Wild inertia group: |
$C_2^3.Q_8$
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| Galois unramified degree: |
$4$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 2, \frac{5}{2}, 3, 3, \frac{13}{4}]$
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| Galois Swan slopes: |
$[1,1,\frac{3}{2},2,2,\frac{9}{4}]$
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| Galois mean slope: |
$3.0$
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| Galois splitting model: |
$x^{16} + 40 x^{12} - 120 x^{11} + 20 x^{10} + 40 x^{9} + 560 x^{8} - 960 x^{7} + 120 x^{6} + 400 x^{5} + 480 x^{4} - 880 x^{3} + 200 x^{2} + 80 x + 20$
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