Properties

Label 7.14.14.32
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)
Galois group 14T32

Related objects

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Defining polynomial

\( x^{14} + 21 x^{13} - 14 x^{12} + 21 x^{11} + 7 x^{10} + 14 x^{8} - 21 x^{7} - 14 x^{6} + 14 x^{5} - 14 x^{4} + 21 x^{3} + 7 x^{2} + 7 x - 14 \)

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:\( x^{14} + 21 x^{13} - 14 x^{12} + 21 x^{11} + 7 x^{10} + 14 x^{8} - 21 x^{7} - 14 x^{6} + 14 x^{5} - 14 x^{4} + 21 x^{3} + 7 x^{2} + 7 x - 14 \)

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} - 13875874175697957193073362681855666064768624869903701930291667 x^{12} - 1973304589297447679433544391356676980058521651385793338348590277223755693810656762414014632 x^{11} + 26982565776009561791816252167888561051117251065669237475512952801188389137455816875763131883272197415756642356533347741004 x^{10} + 9974120702736716373106614538050349637845829632230436271018556267308074064841978646078374594729277329512502474779453706149497291469400234645126593608688 x^{9} - 20055901405577851859027945128132586834274625014077497154960703314366821977096936069722913512824429806693114152597325460747941255291767980431680659389679561650939182028486570702805024 x^{8} - 10358186548770978044630518254124036813067477526848084166685208703415384899269664644062570220937921408613748995626570105714141924610908705525865651760549296997214141873706765342911231209476781119445680186608783360 x^{7} + 6907328027837720222541678885643078716545750317280354960182790594762572312682151172100338362437426507437455497743626101303719025340222344705579273571314870769745907826896710757444336312861305120679138650166780853794814534558187822412280904384 x^{6} + 4480573334259057541778651541171740221967924171169593835347657236768272305368910469668743512244873055546719995816128642943210387256951740730397240079872128749347646341509583980968922251973637019040949907654091747913906020087160078850410834649067392597503830387491683863808 x^{5} - 1079031162856723316723106463166585187838196452828569544050860654722104127385352693590256652956379533409499552763356285253167256695662191838781848288583975464676761654771461875330505755098174863932320453364767653258914603987019543383668996012019936736767095220503908726255310870186971149485154019286528 x^{4} - 894328031867662564218919014677344039315286284468785295364263118387500506645259371137039505912001661404266830715112839353611554054396079535398786759729852819867202614660414810949131609168346275446338474598156338063220573145297040296004911137336174624245525603501839250699733619978715668434807553465721530435507620257782264230461440 x^{3} + 56757515858218923506184905308209748528041200043333074890745273196684750683647507056038847439458701198957618355254999372171860997813367923935922727449290599668347934930911833757032752691496871576615241422550842599713309553636577478823415618173409018610069526249240982671983370613986710221237001648840102848305053351194716020577673577636617404784793548431002624 x^{2} + 68949601140060183142932703687549209664161263965169351812088437617799817407629523146655647194594745172372269938074543205767130923142231626095457240750047060130040599044683853957452586180686192422155159945770676952473107458903862228162595184995548314022669045568436383989636655430333104127698953040524465015236799281871305930485725565397861850026554182248713213434342244752245778688818917376 x + 266365518318610754150668283465164412723260343187510699344858197819347521676496747955753691489901088735307619436746310299107331605188796818763691551226330859747872835751487274121416428812561624580868894877602555143208292371509023184743816961628586021175074393129307494173929206886711804295605026420694550289264506394546881054869456356486523374246645176094694134938831438566690218681845237740793222566095558562698088448$