\(x^{16} + 8 x^{15} + 16 x^{12} + 16 x^{9} + 8 x^{8} + 16 x^{3} + 8 x^{2} + 10\)
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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 $c$: | $62$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
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$C_2^2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, \frac{7}{2}, \frac{7}{2}, \frac{19}{4}]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},\frac{5}{2},\frac{15}{4}]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{17}{8}, \frac{47}{16}\rangle$ |
| Rams: | $(2, 3, 3, 13)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$512$
|
| Galois group: |
$C_2^6:D_4$ (as 16T969)
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| Inertia group: |
$C_2^6:C_4$ (as 16T648)
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| Wild inertia group: |
$C_2^6:C_4$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
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| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]$
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| Galois mean slope: |
$4.3359375$
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| Galois splitting model: |
$x^{16} - 8 x^{15} + 8 x^{14} + 64 x^{13} + 136 x^{12} - 1216 x^{11} - 2320 x^{10} + 13376 x^{9} + 15172 x^{8} - 48240 x^{7} - 205512 x^{6} + 364656 x^{5} + 862032 x^{4} - 992880 x^{3} - 974088 x^{2} - 4099680 x - 1922850$
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