Defining polynomial
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$( x^{2} + 6 x + 3 )^{7} + 35 x ( x^{2} + 6 x + 3 )^{3} + 7$
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\frac{3}{2}]$ |
| Visible Swan slopes: | $[\frac{1}{2}]$ |
| Means: | $\langle\frac{3}{7}\rangle$ |
| Rams: | $(\frac{1}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $48 = (7^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical 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: |
\( x^{7} + \left(14 t + 35\right) x^{3} + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + (3 t + 1)$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $588$ |
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| Inertia group: | Intransitive group isomorphic to $C_7:D_7$ |
| Wild inertia group: | $C_7^2$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{1}{2}]$ |
| Galois mean slope: | $1.4795918367346939$ |
| Galois splitting model: |
$x^{14} - 18768842783116526105215118850668580714 x^{12} - 4148643413574896039459848643688659305510095775622815632 x^{11} + 108985533556227493312302166771967620954011047073765936511562028755884889863 x^{10} + 43973835981881497092220139685491611469896925950994317175258458705794292505675549753744473968 x^{9} - 195932108308872841920313304354943252093593989218594711257346689274670683279036872681408637675091593187684213024 x^{8} - 87172680544348497885592764781896220418631777890998383329646597465851803681399241785851217651605539036972381822568979014001214496 x^{7} + 122656085499504030512334719534004721939899840846451464109582965115659000450306499097318115350896940229516726083760247795281269515214903443366081375 x^{6} + 61061294715797883309637701865594504504848365713349484466563677758026254552938244870748589318003303125716401006977899716762361253195735118901360440556766912129003360 x^{5} - 22188164909548147272406395354119824303689995646506179550434878973696553900259925395701343076313432424895424364500089958510588416443434283608208772028458774713862216354007820852371330 x^{4} - 13048626335274016767663131137327501447243875577528470920786036953603431294994578713405701541817380480292104271069360371237108783864418136868790538002677162886921253014660235394455645579001399804494416 x^{3} + 334743309078781818175994013656540488764121137812638577027218247500390022656534268821742828185882589725863895495186727285715044788244038969480627779560539680629569592627058958969087577711401433480186766667584368235193 x^{2} + 752365679145777718345584396602105860997404754354611636742326249640027122535991953167656035152371309408977211414452910688960164126168838779916487468482265556456162141557138757707420829792264843947953499494226331885073006935590994515760 x + 78450932606215081983035599120513157683783859087934733384383893045754606950674908965713064250478786977254386389764758904601398965371017841788047304850966547836371002792693552710922815383167607253026557517474972696962695358458636943970597882325367931628$
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