Properties

Label 7.14.14.8
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 7 x^{13} + 14 x^{12} + 14 x^{11} + 21 x^{10} + 7 x^{9} + 35 x^{8} + 8 x^{7} + 7 x^{5} + 21 x^{4} + 35 x^{2} + 28 x + 45 \)

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: $-1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{7} + \left(14 t + 14\right) x^{6} + \left(7 t + 42\right) x^{5} + \left(42 t + 21\right) x^{3} + \left(7 t + 42\right) x^{2} + 7 t x + 28 t + 14 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:14T23
Inertia group:Intransitive group isomorphic to $C_7^2:C_3:C_2$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:[7/6, 7/6]
Galois mean slope:$341/294$
Galois splitting model:$x^{14} - 564052017496651404892855580211604422 x^{12} - 118183932663131051367696896044123561251541254267233803 x^{11} + 80886822754670264644312342189865835983538633473760704825069167224045181 x^{10} + 34590775149055544748701097314647265772715289196994690588502935179758502847783028339101695 x^{9} + 1812241885343692464378810289771675952327024373345747158660717670495571391900520484945738127078767479735280 x^{8} - 1375648929263233334436841742610074869663255801443124973975359456379372626900430424666509657896238460033500206598336783360432 x^{7} - 332205289991390429626267070204231257783233127562695673325709480790252611833985431510537132039648739635751889693080007214592066709361871272327 x^{6} - 30264371695954826626150875075890805263542278881588840905760865156643302746944214658383898247712369062815012928976381060702991949237629218551112953833603362301 x^{5} - 713286361814286060742897609998185982587250187967825348470872965830156809033308575602153128113510816002794372729245264366342988025998893145529943528903782514596793497548111797 x^{4} + 69375830928800963638334976556038259001150774894674676100852640764820039711215300943608432711401395105014949129691343229342712731620383150492396523876118684157111233789821277743635432550068376 x^{3} + 4871468092968334070361330577632110007612081183498104447589335437088579286481787422322445597154897581751171015189808652829149387242625813578357998232692571284926111174203107386249805739017486524247332675205140 x^{2} + 75914642816991309385270385042745923143227121340720383742150123910083106869018687934079313964893051404462216351363956035734424283448958856896020986760967109170825020986870191222759881015719588376668638075464062316230472124129 x - 642134467099050807908330578073989806908064689373599695993709023386425906160725407108888077152390533386902540212659943335030520187301101130245852030099399068883258395335874877781920171493325493698599063461037806331382574477938251050545471311$