Properties

Label 7.14.14.12
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 21 x^{13} + 35 x^{11} + 42 x^{10} + 14 x^{9} + 14 x^{8} + 22 x^{7} + 42 x^{6} + 14 x^{5} + 42 x^{4} + 7 x^{3} + 35 x^{2} + 14 x + 17 \)

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(7 t + 35\right) x^{6} + \left(28 t + 7\right) x^{5} + \left(7 t + 35\right) x^{4} + \left(7 t + 35\right) x^{3} + \left(42 t + 14\right) x + 35 t + 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:$x^{14} - 510886853187457491230770760422357362 x^{12} - 15715591503366477449759939219514393453983237980161103 x^{11} + 68189240632410654332730030453113751342935303412252106549668034032771115 x^{10} + 3349282769840880609873479534950112604662440373623379575858124973379187002636030370908671 x^{9} - 2544591879307537173099355262911078358253355376936957467885155885099382873931438686547307602944789180232048 x^{8} - 56035502171456665760427000344241174422342851405522242377421821687294581768945168446732937234792766789407522893494484674914 x^{7} + 40793955177574246450626294719116280761086128507365716915658287017053622721725876667919140947351990187177575150813208443512311318492030114711 x^{6} - 498166483489620465813376177495886096393389568239244938978958702601312638129004794466277180586493489208962676960585122451867874649290552045229363198170929255 x^{5} - 256815422010060246215891301574988182953429024188301732137662345027314584468053068413033122366197599370502920155735797925297548547657403060424001932821103396351587331215259544 x^{4} + 10481312753427405308784348391616253700114704978694037085455463489174018339768615871136414179034341846622924119536961822769039560953769727969318925309485064344539961917379044255751050575618132 x^{3} + 187185167631216872317912798478272706573500264669987194059723070890890553110227014583478925056743945236305227806486749495258701865273116603987579205557257660071137351379514677516955550105917160899046038289960 x^{2} - 8769581328915772636920360550351747924084765347527259401713715443718284430683029191801012304239053341110901070354853013755183015049356861044895544735101164126404195106559857785104676293361147560738707414016843471833722255070 x - 87895526567760706670538912189253735701553742827432271499920840426329732130372450556570550929187292535073249740242687466606116348776380274746061475947342252820082961403144278325674595015990359752479451961745332159737310699049154298375317219$