Properties

Label 7.14.14.6
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 35 x^{13} + 14 x^{11} + 35 x^{10} + 14 x^{9} + 7 x^{8} + 43 x^{7} + 42 x^{6} + 42 x^{4} + 7 x^{3} + 42 x^{2} + 28 x + 24 \)

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} + 21 x^{6} + \left(42 t + 7\right) x^{5} + \left(42 t + 14\right) x^{4} + \left(21 t + 21\right) x^{3} + \left(35 t + 28\right) x^{2} + 14 t x + 28 t + 28 \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:$x^{14} - 510886853187457491230770760422357362 x^{12} - 52404445504601955047231318259610227615118278480867515 x^{11} + 82188729523025073909321016579762174917848959758468438180359926835872251 x^{10} + 14256958963404874578179647576818484389847394736998019056078861183147523289228330316809205 x^{9} - 3888373459775684891990577144728715836618817088080808973528220112708806473611688472025444229178639507443286 x^{8} - 689494517246822492855692785696221930466871329857152644705233788366942341418030224728499040450667870481410027864077561074542 x^{7} + 95849026482668142529126745688140208772902419435864196943871467410563685728581365259390562905001319720465983393430023896029037558488991294261 x^{6} + 12821642800622279198557943559210371485782857430804958247512223570238421418663086389601692247610268823307775725963259696858334444556423716364273359407623897527 x^{5} - 1390880173218588255205153573811253642410860758341568615752077072886885844039547154507815761679968274195810226734560790667541739209415689500980634875067712688262266775468919498 x^{4} - 87802895461081599412621905002774295327626213679467231333320440903488988848656654954141425838990782252660254435870741313229620854909830411751252025361818567128550406622934390764561043789695404 x^{3} + 9784074535969771818763757206093275406670787493463358668766015775854484032192224828493475077682741575748784839264194181737472574796628761514591510343291454834370140499963100644341687616930298247374892280142432 x^{2} + 43716042863067447959926868684006968395860639140853050161739425401235992385499606373404961253903758958522008766450141728964372894169718279599289068333186606152447035476016130128087453617632671831397104114054965881072005221240 x - 13128179668162432392688983837494875753734411342912216813112671278441308025881560228978639206672914599052284795870985276095543978605510974134360945873798710726647833124507470223369340989657361528979704941666400220185312175970804693188526615097$