$( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 8 x + 2$
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| Base field: | $\Q_{2}$
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| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $44$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
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$C_1$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, \frac{19}{6}, \frac{19}{6}]$ |
| Visible Swan slopes: | $[2,\frac{13}{6},\frac{13}{6}]$ |
| Means: | $\langle1, \frac{19}{12}, \frac{15}{8}\rangle$ |
| Rams: | $(2, \frac{7}{3}, \frac{7}{3})$ |
| 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: |
\( x^{8} + 4 x^{7} + 4 t x^{6} + 4 x^{4} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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| Galois degree: |
$1536$
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| Galois group: |
$C_2^6:S_4$ (as 16T1315)
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| Inertia group: |
Intransitive group isomorphic to $C_2^6:A_4$
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| Wild inertia group: |
$C_2^6:C_2^2$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$3$
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| Galois Artin slopes: |
$[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3, \frac{19}{6}, \frac{19}{6}]$
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| Galois Swan slopes: |
$[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2,\frac{13}{6},\frac{13}{6}]$
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| Galois mean slope: |
$3.0494791666666665$
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| Galois splitting model: |
$x^{16} - 8 x^{15} - 12 x^{14} + 180 x^{13} + 54 x^{12} - 1452 x^{11} - 880 x^{10} + 3264 x^{9} + 10055 x^{8} + 18596 x^{7} - 40112 x^{6} - 104488 x^{5} + 81002 x^{4} + 177892 x^{3} - 97028 x^{2} - 102028 x + 56663$
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