Defining polynomial
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\(x^{14} - 84 x^{8} + 14 x^{7} + 588 x^{2} - 588 x + 49\)
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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: |
\( x^{7} + 14 t x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 5t$ |
| 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: |
$x^{14} - 510886853187457491230770760422357362 x^{12} - 15715591503366477449759939219514393453983237980161103 x^{11} + 68189240632410654332730030453113751342935303412252106549668034032771115 x^{10} + 3349282769840880609873479534950112604662440373623379575858124973379187002636030370908671 x^{9} - 2544591879307537173099355262911078358253355376936957467885155885099382873931438686547307602944789180232048 x^{8} - 56035502171456665760427000344241174422342851405522242377421821687294581768945168446732937234792766789407522893494484674914 x^{7} + 40793955177574246450626294719116280761086128507365716915658287017053622721725876667919140947351990187177575150813208443512311318492030114711 x^{6} - 498166483489620465813376177495886096393389568239244938978958702601312638129004794466277180586493489208962676960585122451867874649290552045229363198170929255 x^{5} - 256815422010060246215891301574988182953429024188301732137662345027314584468053068413033122366197599370502920155735797925297548547657403060424001932821103396351587331215259544 x^{4} + 10481312753427405308784348391616253700114704978694037085455463489174018339768615871136414179034341846622924119536961822769039560953769727969318925309485064344539961917379044255751050575618132 x^{3} + 187185167631216872317912798478272706573500264669987194059723070890890553110227014583478925056743945236305227806486749495258701865273116603987579205557257660071137351379514677516955550105917160899046038289960 x^{2} - 8769581328915772636920360550351747924084765347527259401713715443718284430683029191801012304239053341110901070354853013755183015049356861044895544735101164126404195106559857785104676293361147560738707414016843471833722255070 x - 87895526567760706670538912189253735701553742827432271499920840426329732130372450556570550929187292535073249740242687466606116348776380274746061475947342252820082961403144278325674595015990359752479451961745332159737310699049154298375317219$
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