Properties

Label 7.14.14.3
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 28 x^{13} + 14 x^{12} + 21 x^{11} + 21 x^{8} + 15 x^{7} + 35 x^{6} + 21 x^{5} + 28 x^{4} + 42 x^{3} + 28 x^{2} + 3 \)

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} + 7 x^{6} + 35 t x^{5} + 21 t x^{4} + \left(28 t + 7\right) x^{3} + 14 x^{2} + 42 t x + 35 \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:$x^{14} - 564052017496651404892855580211604422 x^{12} - 154865641108811054995009263860974583302189621494546283 x^{11} + 68274493059286391931952662785126184806268034083698188288919870656005193 x^{10} + 43847589619610288791764730575958309535305546399471039007303416720226413425171311857176085 x^{9} + 9148355250370158987218946199764221034455885689967512406635041602757026317291968557918421422254976762784826 x^{8} + 794479778500099572292327210076367104352751202560205733855902933278345854118642269122134869278822866545664936046356404027544 x^{7} + 5761513882929886990680537639806027314200648939646226236793632641509838968648205018822920628021170653206906306159493412875103066639203431855 x^{6} - 3125472782046491957350201539596069103278782427172319082751881579749405623040308642803107121879514458224004927346965069834514879902106136396185725087120039987 x^{5} - 91521547870604903085709374725135913076381858272931381577430075109889993712449215537354862199844234498087304983925508643867754681059381869850717251560441887482945072937990417 x^{4} + 5603036721451203330015559608281941817568015607448589218114237392626863694398643920288288291092033389294137829402301553859306081762435954744883476148325237523665981920807477413019439577537072 x^{3} + 109020836519736346319459629667823398432516572802485985957811105117205961861811304660384352734054094141502906071762773241696462635710179761538792913021946099931584029571440777583042899407321831880756693967562 x^{2} - 5967498622224037431865658871468901980953759601132816707360001312227773233473002490513606845581729208599942882071000618393043614548112322226970103348453898377258843415415887263148085464980913119958848084287985013478495823847 x + 51250465851001016825268129989367400133728170308779441838490740889876330928544118818264922776427236230887724359549174121904872469357732745273692270474242452280316080126646141179785965621564275949718114744620273626266780717395534203262681967$