Defining polynomial
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\(x^{14} - 14 x^{9} - 84 x^{8} + 14 x^{7} - 1127 x^{4} - 4116 x^{3} - 3038 x^{2} - 588 x + 49\)
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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: |
\( x^{7} + \left(14 t + 35\right) x^{2} + \left(28 t + 42\right) x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 3t + 1$ |
| 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: |
$x^{14} - 564052017496651404892855580211604422 x^{12} - 118183932663131051367696896044123561251541254267233803 x^{11} + 80886822754670264644312342189865835983538633473760704825069167224045181 x^{10} + 34590775149055544748701097314647265772715289196994690588502935179758502847783028339101695 x^{9} + 1812241885343692464378810289771675952327024373345747158660717670495571391900520484945738127078767479735280 x^{8} - 1375648929263233334436841742610074869663255801443124973975359456379372626900430424666509657896238460033500206598336783360432 x^{7} - 332205289991390429626267070204231257783233127562695673325709480790252611833985431510537132039648739635751889693080007214592066709361871272327 x^{6} - 30264371695954826626150875075890805263542278881588840905760865156643302746944214658383898247712369062815012928976381060702991949237629218551112953833603362301 x^{5} - 713286361814286060742897609998185982587250187967825348470872965830156809033308575602153128113510816002794372729245264366342988025998893145529943528903782514596793497548111797 x^{4} + 69375830928800963638334976556038259001150774894674676100852640764820039711215300943608432711401395105014949129691343229342712731620383150492396523876118684157111233789821277743635432550068376 x^{3} + 4871468092968334070361330577632110007612081183498104447589335437088579286481787422322445597154897581751171015189808652829149387242625813578357998232692571284926111174203107386249805739017486524247332675205140 x^{2} + 75914642816991309385270385042745923143227121340720383742150123910083106869018687934079313964893051404462216351363956035734424283448958856896020986760967109170825020986870191222759881015719588376668638075464062316230472124129 x - 642134467099050807908330578073989806908064689373599695993709023386425906160725407108888077152390533386902540212659943335030520187301101130245852030099399068883258395335874877781920171493325493698599063461037806331382574477938251050545471311$
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