Defining polynomial
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\(x^{14} + 28 x^{2} + 14 x + 21\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[13/12]$ |
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: |
\( x^{14} + 28 x^{2} + 14 x + 21 \)
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Ramification polygon
| Residual polynomials: | $2z + 4$,$z^{7} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_7^2:C_6$ (as 14T32) |
| Inertia group: | $C_7^2:C_{12}$ (as 14T23) |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | $[13/12, 13/12]$ |
| Galois mean slope: | $635/588$ |
| Galois splitting model: |
$x^{14} - 2902478607361245758445398681097284210955769932032030751078295 x^{12} - 558062567152808384781295233775481548483283482069615787559300988930517729722947249365740288 x^{11} + 835616890547751372793983055447156978864406122590596387338705892438204259096963524352705655799741902922895396332749004136 x^{10} + 558815514315443971718082899830510250191360430103130333171022378772702173544104273628583098023489876918889825854088021110003492737474943640410782253056 x^{9} - 91025817253146708308072963077065043829867333638601186104234050232436293882644564483327181867201997247045686613551021991622601477611556717656925549845867075804352438868244942642064 x^{8} - 106017369302834761005387750575149620267822298312420030572284840719833505260462714482098833109074328130567377421251361094355334131558483712747827057843990272050184419514856498327535895548949608992274148736229376 x^{7} - 11679811933729482020565353030207785676335876270466609196305652742264370965526109748227235237881273071651074195868818443627624674458754136061640970519636081533590075657599340342844484347550349696600460365094612260710462886505675692885356544 x^{6} + 9829629648411785134697200472335544201721206531525020776800585360884352338954678202140149245031280566738632155058927865263144111597588854610245975388211451789979369160609471224433361904767919722908774749793813663667747589836802938236078746764788519357068838731135057920 x^{5} + 1666342531478797543722972629923973870800324452390400155434819797728966260738906505452738475756069279983753018065576664349300253451482008957787719331873895911537645184251546858186362220324082379973530061031197725292674300763585870579609457259245083411895206224707416535793604504156961774306268006144 x^{4} - 267464242662648454183160218309975240841969141532865869857562486647446423963092904770020820012064416574503764326679524120788871112968728307684448942610039823466965900284754897142516060838418068010062262681286406696425567211077259731110604484266524751562371791860946344728878258982393909373751825127616282381376826626835087294464 x^{3} - 83619997980855189506949190624790374922889322229188279091747334321391897520891757583477686237446071856735853596140945601299851270870301245910574738714596178181017104273659603900103674755806350658782919596948215418694868242642690922756852529827975765791009384699015151791448050743125718305728943442594019725636421362175176022797818236537999748990493866719232 x^{2} + 9634420810452108630742227361352657366110353174974349543670891049227794232893119805066004734827244030886681095533553008323171544919366094853092293203661944444036158123354542198227185699251145499110406562518954228270769107374267859813746475957923586758120576505138337398913245309087466649613385668332120546470228132194200155216487063151526442446329086483233792502119733349000690094047232 x - 4813383512578696225954591092670772399049339686493350701855557721761496863912041457949214455352064275653321850803371360740279904641996870058693755935549871388694398664421554330465964932485229773195056482149276221511787291081951479870410808293185870327035564724984372407867067581816583114952953121363936154410128495095757936822542928103185795593656900215499225649422979267481630216125024378344870457260162145415168$
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