Properties

Label 7.14.14.18
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 42 x^{13} + 21 x^{12} + 28 x^{11} + 35 x^{10} + 14 x^{8} + 22 x^{7} + 21 x^{6} + 21 x^{5} + 21 x^{4} + 21 x^{3} + 21 x^{2} + 14 x + 31 \)

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: $-1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{7} + \left(14 t + 14\right) x^{6} + \left(35 t + 42\right) x^{5} + 14 t x^{4} + \left(35 t + 7\right) x^{3} + \left(7 t + 35\right) x + 21 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:14T23
Inertia group:Intransitive group isomorphic to $C_7^2:C_3:C_2$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:[7/6, 7/6]
Galois mean slope:$341/294$
Galois splitting model:$x^{14} - 4643636887097677602 x^{12} - 1443286778608578361627275785 x^{11} + 6718227867057528796987429463376792693 x^{10} + 3529053853519782968165435040187622080042494119 x^{9} - 2626691794418043870866606803193450499094651729778400594 x^{8} - 1414075916397590878498395147581368637689534854254083246677572468 x^{7} + 415123910212475925089730923014801557701610110581036459187400521656563331 x^{6} + 167024087357848463296702561369362099459818469496992887556269940465746181240958051 x^{5} - 26637862759471051975937392862907589616434828905665337471500468818395123531649302311637312 x^{4} - 2159633478622975679298387120540215721694236528482531790242413817546422066978111440214399356382558 x^{3} + 142799477781558446842542542628476481883297876818076019509301166395430043748711017205499734793224348479844 x^{2} + 8117664833999094473265640050208775005968307328115561535966219327047185991859485757524695309642294557627723598110 x + 92642747175887659868022379286736904675691422821608409863304969812995607176163358769523734616609312284579577622679758397$