Defining polynomial
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\(x^{14} + 35 x^{2} + 14 x + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[13/12]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 35 x^{2} + 14 x + 7 \)
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Ramification polygon
| Residual polynomials: | $2z + 5$,$z^{7} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_7^2:C_6$ (as 14T32) |
| Inertia group: | $C_7^2:C_{12}$ (as 14T23) |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | $[13/12, 13/12]$ |
| Galois mean slope: | $635/588$ |
| Galois splitting model: |
$x^{14} + 8534045076809897554984974581269 x^{12} - 7505448240946387175352789752162565024711901122 x^{11} + 14151645481603066755599082561334449952057461772764698700980960 x^{10} - 10117878741554607696001349192297577904850074033731855714501264085683875320200 x^{9} + 9449968620389220850178138723991625358576916871196173641699668684729532655909934355643153376 x^{8} - 5050484463347093774654155548448445616233659651622933050355427311483336918026472175844999291300555018296864 x^{7} + 3009806018396367725063975836950605218735211513575158324101130603415707175840546152498249653296416598824829660601487992384 x^{6} - 1163855379594771800393736305645890581556742810562410939623396777675313875595548367623112885245040895982917561713158591511191522524605952 x^{5} + 467165670050763286711208555279328246481926387293213446634146138806386106560453817180202696315195207270462384871134415384066390357748105231930214976000 x^{4} - 121854427762676008507334557875111714533505287044802479974960164956159715609215071596893096620773290240940535520757646446704685185800159757188154949508315257592858624 x^{3} + 31321424157054585230741817866942164531630851005496701253057809086529729137427038706700142165348849699555795988447005531469633151400934755876462290814250159877446079791568894136320 x^{2} - 4439566593893841161815599126349340827771894496106342938308503907092011969371125885994719152783009685917710954351242999754499271558234748261825550743576876834979888846734811671100972472775344128 x + 557788892001850118882425984750085585722578561932799032414905377649632349391763786582105863136549299205747519205979805825118804985906069624157782691500953606308008933353886097688703991970060918225260443107328$
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