$( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $40$ |
| 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: | $[\frac{8}{3}, \frac{8}{3}, 3]$ |
| Visible Swan slopes: | $[\frac{5}{3},\frac{5}{3},2]$ |
| Means: | $\langle\frac{5}{6}, \frac{5}{4}, \frac{13}{8}\rangle$ |
| Rams: | $(\frac{5}{3}, \frac{5}{3}, 3)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{7} + 4 t x^{5} + 4 x^{4} + 4 x^{2} + 10 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$384$
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| Galois group: |
$C_2^4.S_4$ (as 16T750)
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| Inertia group: |
Intransitive group isomorphic to $C_2^4:A_4$
|
| Wild inertia group: |
$D_4:C_2^3$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$3$
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| Galois Artin slopes: |
$[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]$
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| Galois Swan slopes: |
$[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]$
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| Galois mean slope: |
$2.8229166666666665$
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| Galois splitting model: |
$x^{16} - 8 x^{15} - 16 x^{14} + 204 x^{13} - 1504 x^{12} - 304 x^{11} + 21400 x^{10} - 66840 x^{9} - 8978 x^{8} + 1031640 x^{7} - 161360 x^{6} - 4471448 x^{5} + 19415936 x^{4} + 4235472 x^{3} - 146951704 x^{2} + 156967424 x - 23726804$
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