Defining polynomial
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\(x^{14} + 7 x^{13} + 49 x + 21\)
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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: |
\( x^{14} + 7 x^{13} + 49 x + 21 \)
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Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z + 5$ |
| 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: |
$x^{14} - 40965545615349084598638 x^{12} - 6688002228360516181838922809746216 x^{11} - 451040661003022103953656229743184836499596993 x^{10} - 7445161854528935202220287981797794743706060948934444424 x^{9} + 817939998153164679058611901768644349283179455534212695229958609968 x^{8} + 50616993565499021906119834424696751194878310379050509870210199475955682396016 x^{7} + 575554511102105867450664329112060921105763808180786473425468030227768034548074990907535 x^{6} - 33826853530664438741305677598696196493422556457557431591382504203533624945484406178755360413165584 x^{5} - 976278346474254869464042331876399146037424170521760078781896177014517410960367400438153701140564727825388918 x^{4} + 2343121740817329582308642948968477251499223809616118748170549227392940698189705254268849496277759207247548857888518968 x^{3} + 245785007264690294323125474732576127395467340411166496963902465426258191231463442965335116472831492918672056412126822538633940305 x^{2} + 1272909351096610316142923561818406556561678740672922620869181236636502788138504416725905606274627188976341365599341191345948751734861514008 x + 1481890681832802928474015141096152757270454048979751400739225812061212909438656759849039069806705830802234784691415030905104997246426702896509813204$
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