Defining polynomial
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\(x^{14} + 28 x^{8} + 14 x^{7} - 98 x^{2} + 196 x + 49\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/6]$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{7} + \left(7 t + 35\right) x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 6t + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| Inertia group: | Intransitive group isomorphic to $C_7:F_7$ |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | $[7/6, 7/6]$ |
| Galois mean slope: | $341/294$ |
| Galois splitting model: |
$x^{14} - 4643636887097677602 x^{12} - 1443286778608578361627275785 x^{11} + 6718227867057528796987429463376792693 x^{10} + 3529053853519782968165435040187622080042494119 x^{9} - 2626691794418043870866606803193450499094651729778400594 x^{8} - 1414075916397590878498395147581368637689534854254083246677572468 x^{7} + 415123910212475925089730923014801557701610110581036459187400521656563331 x^{6} + 167024087357848463296702561369362099459818469496992887556269940465746181240958051 x^{5} - 26637862759471051975937392862907589616434828905665337471500468818395123531649302311637312 x^{4} - 2159633478622975679298387120540215721694236528482531790242413817546422066978111440214399356382558 x^{3} + 142799477781558446842542542628476481883297876818076019509301166395430043748711017205499734793224348479844 x^{2} + 8117664833999094473265640050208775005968307328115561535966219327047185991859485757524695309642294557627723598110 x + 92642747175887659868022379286736904675691422821608409863304969812995607176163358769523734616609312284579577622679758397$
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