Defining polynomial
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\(x^{14} + 28 x^{13} + 147 x + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $26$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\frac{25}{12}]$ |
| Visible Swan slopes: | $[\frac{13}{12}]$ |
| Means: | $\langle\frac{13}{14}\rangle$ |
| Rams: | $(\frac{13}{6})$ |
| Jump set: | undefined |
| Roots of unity: | $6 = (7 - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 28 x^{13} + 147 x + 7 \)
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Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z + 4$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[13, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1176$ |
| Galois group: | $D_7^2:C_6$ (as 14T32) |
| Inertia group: | $C_7^2:C_{12}$ (as 14T23) |
| Wild inertia group: | $C_7^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $12$ |
| Galois Artin slopes: | $[\frac{25}{12}, \frac{25}{12}]$ |
| Galois Swan slopes: | $[\frac{13}{12},\frac{13}{12}]$ |
| Galois mean slope: | $2.0595238095238098$ |
| Galois splitting model: |
$x^{14} - 2426729695977866342210370953224475730403640353419038631872218863058698 x^{12} - 32256668007097722520539996542702325961741983621722703345684240643832187865977369972615613677019468032549 x^{11} + 1538309974049829201081936050219276099994094949606447409859462857618118081552055614281768692825108032690106650266350840226674733075039738207 x^{10} + 41721298429324715265187295037883784930809024668239431670742664121310139320280884746598268379422706506447000531597813755125050376147698870296874127190728731206288262175751480 x^{9} + 239712749155666429803760729281005470325443016198960376931115362265365606940947117745431182940642355913094559070723502885829673981967553929055454344648314231261315616558897905246736648897442486222850204057129 x^{8} + 971202388627884682824865900868240214463861500898817038116259865493938343716018735590034439084871139688867439459391480332032239579740335113999163146613705827912279709809834574679495113221640645310078541354318814687402033079586268589886072193 x^{7} + 98662037711668504174055682161399263952665371505252694048380692642445773330160406832584898627744188904658102577984126189258532594071864596830307491102511455923250980783606842096276236506041346900379027108559307500449775458181592795209655380801511452321103923653519829106344518 x^{6} - 1371324738089295288292745527749236609799930520921756946469816647200490035935180076884114901858928148809924563318543855886722652338022163648315051303329909726042949906869677512804822066880271425921011395668685290467154201290019262138032099787558152235808276546360680098300705789810423167004220478255803355573171 x^{5} - 120142479509778205225281961325193644550443785294454889398058924350698118041845629988924390881152701729963042920760960627807815369859305521944030852957034640107888813792535610948380079540021473935259439032503413608465764817769690139631236380293638910654376875381794673976810984870764656942317772244451882620920731298313203057698756851360881392927 x^{4} - 1088829998679568617663720701078238414486391490766460237400075435371451645398127190155805331470559340985748013908718035395526780802417553035684396962406225279933134835379880169568777732767592063108993130939980765868210950667228394761491286309685443841389015223828449504974840014139722306799098194382917789164210452760776288669554455394303655872044882231444818853004124174466125664 x^{3} + 25685886469300379963099866085053466958641896715267883863992008478208120278302927535135951875659490325291364734299087998569313912659516139514714656393600138795634590747270456523132797328002205899657117847443948325879757880980990830866538387034828399187571933356794280053742004443042444584520448151114866417244239499756853779717527341324025183557272646261753773514056105914955790603849659102996652102115809772905382 x^{2} + 558404219018151585295600744635429276763566881155883611917634649555981893073840733382787418476661144968162283312941863913874588855797116870416506822709073504521995463428971240626098838620873721930483827728063639233105739470585153849860571744803500839961658522892458340016655664670834755500555288756255587773900179817304593280910044213506467179375790190900535828503253911095498975996635876311506916816455344776825385269090981707046675732434235216951 x + 3161934067251385029512248098021583962879569404157158546887126772699513807144017732223358474391188118123720611417782799408277449969880336524560557379554994758744672698556507909328724619506827812613611519157550712495242500735453772762886276454393844727366823506372057311186427141549484565582919875317681433843193773094002966285523151293750811244034988186425855021045541047013939484525545049798829358432701178831444162412392587786657447124753374430232724021026173848236165552067782796$
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