Properties

Label 7.14.14.29
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)
Galois group 14T32

Related objects

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Defining polynomial

\( x^{14} + 7 x^{13} - 21 x^{11} - 14 x^{10} - 21 x^{9} + 7 x^{7} + 21 x^{6} + 14 x^{5} + 7 x^{4} - 14 x^{3} - 14 x^{2} - 14 x - 14 \)

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:\( x^{14} + 7 x^{13} - 21 x^{11} - 14 x^{10} - 21 x^{9} + 7 x^{7} + 21 x^{6} + 14 x^{5} + 7 x^{4} - 14 x^{3} - 14 x^{2} - 14 x - 14 \)

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:$x^{14} - 38944360535138045563543900828 x^{12} - 56826440797329383292333028928464129402862 x^{11} + 495906523712616886419788746586968445353150753823450864361 x^{10} - 3292556781880287643831038914102186697994130655611048139791156459420650 x^{9} - 2498723633424174068013491074778398845895453376498488456942507215266018418563483957252 x^{8} + 74212733381082990920591506963267062684841506634530950084762768832674997862620430042190992456081674 x^{7} + 8100595463068283576557653229310405618516909858411157974197296977341392302650999727426607270193635887281451355917 x^{6} - 93685635092982229221125776509168499797296196043723767588885133992939985916263206579628733206608206415309896200980582464495508 x^{5} - 9046115220138924994161256264783756111224459180987572412792934882734941540590805178485885510725975032145068893807261563396898508291061913352 x^{4} + 47876265316603927131733783747214830443518246338587039912329336379946801734897890821647152758977215983821433274428294834541568872961055111755769577315484 x^{3} + 4163499806601583760068574770601365421994585437654351140182345871295318911255939670137504778221897432079322242728461181328838501678205100071154271852999324340437691760 x^{2} - 431594231556406304319999803920907693609898526292659791400366328815449900505761012106422736283606513670112605367331702294646960709428731645386843573648568840825661660209896215888 x - 423403613449259706803579028650154818956522873055573981070168881489879444837763341558081557774660287514500278529794977794506411129801608269416622660965482482983808205168318279445674986725780984$