Properties

Label 7.14.14.33
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)
Galois group 14T32

Related objects

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Defining polynomial

\( x^{14} - 14 x^{12} - 14 x^{11} - 7 x^{10} - 21 x^{9} - 21 x^{8} + 14 x^{7} - 14 x^{6} - 14 x^{4} + 21 x^{2} + 21 x + 14 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $14$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial:\( x^{14} - 14 x^{12} - 14 x^{11} - 7 x^{10} - 21 x^{9} - 21 x^{8} + 14 x^{7} - 14 x^{6} - 14 x^{4} + 21 x^{2} + 21 x + 14 \)

Invariants of the Galois closure

Galois group:14T32
Inertia group:14T23
Unramified degree:$2$
Tame degree:$12$
Wild slopes:[13/12, 13/12]
Galois mean slope:$635/588$
Galois splitting model:$x^{14} - 1918888626089065616581472305457 x^{12} - 114804285321544332637412828644599178544659100 x^{11} + 5226212759405122929210210000359177605155743964239008307033460 x^{10} + 9019583690448296745013921569025078107888123016992652852603378557155873157472 x^{9} + 8481211974320465020358147201937828169211434412547674341481814041943875169011897500868707408 x^{8} + 5365473592638556825808817348891552806865421770148804837013393449059377110677917647643712782085028391264832 x^{7} + 2443727794528207131664688316159070864003247655563692756905711208025996935797182050119698942518012049233147573706061489088 x^{6} + 819963712675500798259746610061529435296091876223117169554433710728214972980578211607100496861602761962320774778774873528662614141089792 x^{5} + 202361894503752186867974105566827013362878472305666097230197935274977601971822432063719889534499232685098263834170694320740302895172830997867240791040 x^{4} + 35882173111906048222442695272081265568541032161209354452267153294742477114323493829735839032671644011081592617117828684168312117582116176479150459437704588804046848 x^{3} + 4336633097738751296079079850992440659302246947851331316652219644115421676020382971965535793943181599452336814395823820143269762028504353222741607531612103404990077694725516787712 x^{2} + 320179406223300574648237148358046054467821971389171305330709570936467231007815854051885486302210675461277806436669617286150823768824188879995925601851877048281526381342811270639554408506720256 x + 10896397791580810223837163499069006646100612570332475289674936479804824857306995600759369487228767072286104737213002685112253280738932914979484123622174558919375139556817242375942422239787397879621789417472$