Properties

Label 7.14.14.31
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)
Galois group 14T32

Related objects

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Defining polynomial

\( x^{14} - 14 x^{13} + 14 x^{12} + 21 x^{10} - 21 x^{9} + 14 x^{8} - 7 x^{6} - 14 x^{5} - 14 x^{4} - 14 x^{3} - 14 x^{2} + 7 x + 14 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $14$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{7*})$

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:\( x^{14} - 14 x^{13} + 14 x^{12} + 21 x^{10} - 21 x^{9} + 14 x^{8} - 7 x^{6} - 14 x^{5} - 14 x^{4} - 14 x^{3} - 14 x^{2} + 7 x + 14 \)

Invariants of the Galois closure

Galois group:14T32
Inertia group:14T23
Unramified degree:$2$
Tame degree:$12$
Wild slopes:[13/12, 13/12]
Galois mean slope:$635/588$
Galois splitting model:$x^{14} + 8534045076809897554984974581269 x^{12} - 7505448240946387175352789752162565024711901122 x^{11} + 14151645481603066755599082561334449952057461772764698700980960 x^{10} - 10117878741554607696001349192297577904850074033731855714501264085683875320200 x^{9} + 9449968620389220850178138723991625358576916871196173641699668684729532655909934355643153376 x^{8} - 5050484463347093774654155548448445616233659651622933050355427311483336918026472175844999291300555018296864 x^{7} + 3009806018396367725063975836950605218735211513575158324101130603415707175840546152498249653296416598824829660601487992384 x^{6} - 1163855379594771800393736305645890581556742810562410939623396777675313875595548367623112885245040895982917561713158591511191522524605952 x^{5} + 467165670050763286711208555279328246481926387293213446634146138806386106560453817180202696315195207270462384871134415384066390357748105231930214976000 x^{4} - 121854427762676008507334557875111714533505287044802479974960164956159715609215071596893096620773290240940535520757646446704685185800159757188154949508315257592858624 x^{3} + 31321424157054585230741817866942164531630851005496701253057809086529729137427038706700142165348849699555795988447005531469633151400934755876462290814250159877446079791568894136320 x^{2} - 4439566593893841161815599126349340827771894496106342938308503907092011969371125885994719152783009685917710954351242999754499271558234748261825550743576876834979888846734811671100972472775344128 x + 557788892001850118882425984750085585722578561932799032414905377649632349391763786582105863136549299205747519205979805825118804985906069624157782691500953606308008933353886097688703991970060918225260443107328$