Defining polynomial
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\(x^{14} + 14 x^{9} + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| 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: | $[\frac{7}{4}]$ |
| Visible Swan slopes: | $[\frac{3}{4}]$ |
| Means: | $\langle\frac{9}{14}\rangle$ |
| Rams: | $(\frac{3}{2})$ |
| 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: |
\( x^{14} + 14 x^{9} + 7 \)
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Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z^3 + 3$ |
| Associated inertia: | $1$,$3$ |
| Indices of inseparability: | $[9, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1176$ |
| Galois group: | $D_7^2:C_6$ (as 14T32) |
| Inertia group: | $C_7^2:C_4$ (as 14T12) |
| Wild inertia group: | $C_7^2$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{7}{4}, \frac{7}{4}]$ |
| Galois Swan slopes: | $[\frac{3}{4},\frac{3}{4}]$ |
| Galois mean slope: | $1.7295918367346939$ |
| Galois splitting model: |
$x^{14} - 2031866514994496235590452844381076043920547970592875592460437114 x^{12} - 48772766996873795996285580683153277746844286653652583124892726313979288457814454062589966403260 x^{11} + 494792758564205952761253066868483779133808120714479365172879357203760819339703788145767465109777827362450520341117775955279895 x^{10} + 46346774388559513795590049448158221115747947535787025838203678688729738206026903880213433821702081709205941682930983751776904224022096317929817989432046456644 x^{9} + 1131741183153761925554800393473169801344701614936189477518206762559781092668666720629091515629511884278827325116824557359460856801708075765698551588195073319845698397261660298152079934066264 x^{8} + 16411379399849355848769725048483658240868596996439242784772386090869309329965366701655376455677843768299218686085070130863300780483445861419370329006048523893577608426584563119345869309184266930420399975744377600848104008 x^{7} + 162144284989757545933035487427415567245362030965732716838830425201032412797934266831035243691022418583170747898921728107094745786398630862281741015854695662401518624629500735909889149064123981520001080631400821332409283287372390627835192392311128724255 x^{6} + 1142259008160137414270031487951491537874986497398042219173785572196152917492248501836900364540389889104304054415852177541577242022064783509827345103098783006749982995756627219847985317325757607351783187857518258296240714207980330621229027456511459106877516416784564572325067907004136 x^{5} + 5793614404630648614866497473452745304551566873284571590574180379862046006941034046286006482878072027852717544676354838165798336988315960569267736545869810929335398920667722059377108141633984684191128935117740773197179832430285728327887887173368669099549177931248777683685685180256837080097136398923533365974987710 x^{4} + 20828391263430610758649084923685946408947758577013947582078314410743764750419646917628748514482296665273586913787049700085358906882790434019151265320294288492785169982820635379429513127862035507288145923775265803936151607798795043007466079026039433593882371699047611578833632723085314035694758095007237158210199649469123017234028937867188040692 x^{3} + 50676300486823790349397643066091878010397386537408074555505642534347759054666265054422781866613849130042316713105558715327531500922869606661066944593295460616301032207138965781400600104773464511290336622337814639987964908177709807371845725273394733568963914149765484445663501790138533177151670027225049049726198337912221683795790065514825852049457246777238053971234124155945 x^{2} + 75231772967656117966965127947033651276114741470768046896691815446592712240302086555659477889392139213874602081853687193200084637437798969443375685964117301626470259380498565672895987995205941949473488846822808143940258533278054339955990337632090358629509930867991966365430147528398034560096092464227431614885328344886291536361084124581158657053619957318019353903301470822531417134825948871169275348814420 x + 51700331099018625458682415494565276092796607288121233675589447216268134154703883920709512906765833841719319670311841371323969703908155234499766476405168899056803425743049785004157403391372298456703697331900928723980770996703352064109317957083285897521916491968269296400197976046220967122984844504408570312930213110695156336027462465620172393880012752816493370366827490163749064818884254803223105627324965908199348565963719195035282692$
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