Defining polynomial
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\(x^{18} + 2 x^{15} + 2 x^{9} + 6\)
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $18$ |
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| Ramification index $e$: | $18$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $26$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[2]$ | |
| Visible Swan slopes: | $[1]$ | |
| Means: | $\langle\frac{1}{2}\rangle$ | |
| Rams: | $(9)$ | |
| Jump set: | $[9, 33]$ | |
| Roots of unity: | $2$ |
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Intermediate fields
| 2.1.3.2a1.1, 2.1.6.8a1.4, 2.1.9.8a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
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| Relative Eisenstein polynomial: |
\( x^{18} + 2 x^{15} + 2 x^{9} + 6 \)
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Ramification polygon
| Residual polynomials: | $z^{16} + z^{14} + 1$,$z + 1$ |
| Associated inertia: | $6$,$1$ |
| Indices of inseparability: | $[9, 0]$ |
Invariants of the Galois closure
| Galois degree: | $432$ |
| Galois group: | $C_6^2.D_6$ (as 18T147) |
| Inertia group: | $C_2^2:C_{18}$ (as 18T26) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $9$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1]$ |
| Galois mean slope: | $1.6111111111111112$ |
| Galois splitting model: |
$x^{18} - 6 x^{17} - 8551 x^{16} + 20206 x^{15} + 28305398 x^{14} - 140318122 x^{13} - 81705550668 x^{12} + 469529571214 x^{11} + 148224534194692 x^{10} - 864489697676922 x^{9} - 170118196455494530 x^{8} + 728605897907014926 x^{7} + 174777137314285606506 x^{6} - 584135111007902734098 x^{5} - 120626583223836586046496 x^{4} + 574797656184149939009718 x^{3} + 25366861392342429657839167 x^{2} - 57512541483738805328473852 x - 5300972427648457742422245739$
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