Defining polynomial
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\(x^{14} - 28 x^{9} - 42 x^{8} + 14 x^{7} - 98 x^{4} - 49 x^{2} - 294 x + 49\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/6]$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{7} + \left(7 t + 7\right) x^{2} + 7 t x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 6t$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| Inertia group: | Intransitive group isomorphic to $C_7:F_7$ |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | $[7/6, 7/6]$ |
| Galois mean slope: | $341/294$ |
| Galois splitting model: |
$x^{14} - 564052017496651404892855580211604422 x^{12} - 135825411020953218867930241787417423659980874582494881 x^{11} + 75811883519589942674299379767637656214353258344276668422835062549820961 x^{10} + 39144153636003887008208063010032349479586405132063739601420378359787890257290341025023547 x^{9} + 5047663587170598765567501868196069691755092163157839431659063575235758532787752902274895484503979045094246 x^{8} - 470907362822897887577718440842322058985865332058029697064688554277938164615929247973284124563301367884717369888270005722050 x^{7} - 199320172392387345671748993421356310771073770793192606669349776203335442094606783472097645954616094621839779854919580372154921931261455931607 x^{6} - 21025251627770012417755818336994308045897443210652231728012319935854436543066762583426310469788030616254801940453453245528294159939110522319032355324005060839 x^{5} - 751826141595142513581310138947645247667131938049639934301126443034366267939685155016801629771751120321293462947839690240463801603027079503122776635615558006387469125894664587 x^{4} + 17257831124493053974186183593098924557874277864946824673755099745768201179596505061618653597843545143086468641365005622855605481990872712845243621315832251440455190007489380267259097587717248 x^{3} + 1664547093729054922876561083905083230663102036744307636433972327382900097602038742676077885534851177109892079982942254013930449549661711997313996852317524536857675776431894720058536088301304052205251855271786 x^{2} + 14499476047975901154178026690776462368807232737114586705225398160195364038861937990525542097189912540577754192830728491091652184028817626427960501691243317101590048194127606470422729925675814264816004094765600039079677544625 x - 490048607778372600714507891532907490580339413112615848851678891443769113698093596786701938869534995472555918865290505706943145573737672024363721024537849564053640214381300880276735261555501662660044239249930738720475994612327310853051786681$
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