Properties

Label 7.14.14.17
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 14 x^{13} + 42 x^{12} + 14 x^{11} + 28 x^{10} + 35 x^{8} + 8 x^{7} + 14 x^{3} + 35 x + 10 \)

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} + 28 x^{6} + 28 t x^{5} + \left(35 t + 21\right) x^{4} + \left(14 t + 28\right) x^{3} + 42 t x^{2} + \left(14 t + 35\right) x + 28 t + 7 \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:$x^{14} - 510886853187457491230770760422357362 x^{12} - 20061456448006274049998995568204029921135570879718589 x^{11} + 53492031089442271628613100215958342495392429349087260387564024829536531 x^{10} + 4483142054817153088234773587613548759503393034062885384387934200534974255848614595518545 x^{9} - 1756524418418610446727040153816646325499951794166628746527327958272415815456762567334901141884706113170680 x^{8} - 213909917612297031602737345189960953418818094430902409419785572294545437538979600660033524291214398641344990479230661434312 x^{7} + 7775311272049941408059066571812339831236662357223933565661575777018677352697936936865066497907958227725257598249720386101436610465735657761 x^{6} + 1354753175482626595647665095780133206220812959191048592531288114791520116828777914687151523576324511977152077797632888311500719940943652521573038864544092201 x^{5} + 8547204621831558633853275156301431199676169195002310616206154362932133121903032994818810344887109402736988674844282001280784065051348619740971531422997147423758907584952340 x^{4} - 2260046472039162804514292770946600765174138254475135226516540847464245019611453448406837212001553650840115751223474393751691559438558456770832743717045089215385415436907811440577148703127804 x^{3} - 55904516053455225142706375306455827165631560933487880639791921410184221937632907863753989079465692539235666883379704997186675411016423297779286012807946594656822814052246399406456036264119009657143874560948 x^{2} - 50528433271445914590662546617693785312327089658768457675719167854213130099248230705082049358049954370167865505375643331332128319431991198381459647900812547347169800711501572171660019680933609829907731938786757755751303130 x + 5093834982769670333817804748426756783999688938726684816903619390099265156961094977158340735839973171400222625935195217384806478226323094896386309238165921785473685043643458465297012708729498982377826712372508962956672790844003027549940421$