Defining polynomial
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\(x^{14} + 28 x^{10} + 21 x^{9} + 7\)
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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: |
\( x^{14} + 28 x^{10} + 21 x^{9} + 7 \)
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Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z^3 + 1$ |
| 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: |
$x^{14} + 524007546021825533868711126 x^{12} - 246357903033113610102539260995767942287 x^{11} + 144534820006540453758861608410771732108310602138905069 x^{10} - 44753780782181081016903242563734376240485165758355250227877786770 x^{9} + 30852715922809665114121047831323952086424476016127052051810100498236905788619258 x^{8} + 18687600300236774174156149759438347284913491803045267236388077812705722689166058551825176015 x^{7} + 4282137993286657409146439644362139768725980811804778164150514293227414601148931495596736296877573530878396 x^{6} + 4690142979919130568998020347364599408262480144280263363681827472334686651705098647328736318750102516236812722000156431 x^{5} + 295731019836322551136929707546635164824219990246994331361554992191071515534835267949251720200238945975836042429373711687169156313612 x^{4} + 369852472383119928543073192367916018193829905376405525447983099531104948618009212015109770561929053266386295688150628975599603971612408787416919 x^{3} + 7806936724049307306590229350698076792439035022475128494574486418642882631261797215703532081533167672575973278853916648286898870098373128226268564502877428399 x^{2} + 3002161375607999399184285028312680551547500397891150385501683853093621403391673680992298324088464276574315204530244085823587989656736893917841115779750045430660716500928 x + 94559480214754183437397500009314678477448798697139796003494936638158749924396376893712449077278486501665445676064811557340099139057867482473426186027334180578727343789540076823455004$
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