Defining polynomial
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\(x^{14} + 42 x^{13} + 147 x + 21\)
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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
| $\Q_{7}(\sqrt{7})$ |
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: |
\( x^{14} + 42 x^{13} + 147 x + 21 \)
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Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z + 2$ |
| 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} - 141640242483183957504928923038618027756659206299647712835681574434681299778 x^{12} - 1178834035689702747649161238642792000221315630256025946094862622730721527509055273219469146367743112342526175058 x^{11} - 488296593050515374976166066121880265482590376503362805747756547754357640107502536925172309560808606053148159512520136149800995040483499699912534305 x^{10} + 68436488125288572211557894325014207493145264133249650830469292714167381562447959541568139899665673211877775485215869265349408092565654176556891406747962493717038808120599979404822911520 x^{9} + 664755362067700359972222416191000955219754634661061243209698613083545103537526730107608121036196741134735577666398475974808502269426566511985411245778593835218342395259645583892335007806884318599849278052671372918514863599 x^{8} + 3392680159712291687475186010013927224064726519661212622421796868515830631905237196260217496393885062973823093905783440491435521974547461150535964211142435920246945195804790509145476839814101281355965091344982520481103440008483495720220447576367641225757911810 x^{7} + 10861621769711076909106610735553871396269929674256005765440321579729022157039940752230172445871145190919514543925292392347204559037300724662866324526948593009039760203957999111886139017773591433393372960402189183647132189714558675498109788386700451388164584370103039731741150720923648187456270743 x^{6} + 22680322768630983634826860497590287532713069975201370712042853216118993182254025542558700031045215744998786021192542632656412508894042081568835123782365520731360818986348115755835931694233295245517897911317553076588078446861901252693111715221908093293979825834910599307301028456544578368195144193219310581891032336027613144409629224 x^{5} + 30163404159442533137117449635471600409510174721390642477753120916627463483126261233355493712320736397524432502136249683529634101425156829339903515450458026664864724896228405353260928058806801958016466909652698224998875762738769373693337845536631801549132988788851023348356833735378420784253004913685214550446600400534004740798469651176808183700669536013119997161552871 x^{4} + 22530998871166724904564789788120267746413261085802554612910656249977135794669295529133896624523049822879470444030546729234286065865970390523481855991954282170263758165794247234441099456353232365175977017955512176118356469502416135898883959834785224637404025495846736354508872495721819294708143985312558471223622875656626622452843316027952627873648740290000044593081983418621343368655087280231490642351542 x^{3} + 4502038025822258810307949900272222228562711218036955050503057072971762995845736073662566509427831149070200115030011180713540781019421506180347314242768964232132776488366442236747499874748403517542445107198987461968948490691972605148571503058829258110397069636325360354631244595881325194752217960060809049511239499562738027803153239330166905973174652802679006262417641871607798779703758145274927946038984141363012204932935809638476774300822 x^{2} - 5603885499114079614163267214990725421948414081787783893833741061478712926039073625779139456592744338991655208480535644529081805139761661137205469820751561815606505854430242885734723100610104059366711510304730185517026275946594188846713320549080129066779889754781303883752304266023989608731814417927424000454690089628763842871956663957155014872720733426572153769830761229982354873027511139255812029141307801594599152514177538009796465383531041244668890088585776802285282458668 x - 3181206144737210735999106512750474095184279776892567616027431435371406320274925115058015498773516742031148644344759964140300054778181343683859060881055695646384947000683860347393509677084816599646812477017031767243036819176054110009721429166555129704542904377283520617318538571098433868382004117694127398129112996819804919466399188419164529952744557024768123845906313255008122076512766601165155115365275243165787136355373287340537135968265966863352393157680802854542604583421668089973755146053112260054549606351$
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