Properties

Label 7.14.14.26
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 42 x^{13} + 28 x^{12} + 21 x^{11} + 14 x^{10} + 14 x^{8} + 29 x^{7} + 42 x^{6} + 28 x^{5} + 7 x^{4} + 35 x^{3} + 35 x^{2} + 35 x + 31 \)

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} + 28 x^{6} + \left(42 t + 7\right) x^{5} + \left(42 t + 7\right) x^{4} + \left(14 t + 7\right) x^{3} + 7 t x^{2} + \left(21 t + 21\right) x + 28 t + 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:$x^{14} - 8038823564374568007 x^{12} - 11774725679048237237576726986 x^{11} - 731191928246254490365530862004844836 x^{10} + 9892832027708628068999985633391322055990753808 x^{9} + 8235560509565882625005275711024211487559462101021362233 x^{8} + 2520548688490481291699335181763846697601975586081121954072534306 x^{7} + 65217457669075299405706581638387832043113986324244358643026210071282720 x^{6} - 135452801967195685184976240742008265035944258628311149392959345270720028184786088 x^{5} - 26285406081958776406257816513182493645696993640580058769278314615935676860949564122448503 x^{4} - 100305408760340645218538512486922019845876276287141540264789159194769850678392046530856985621282 x^{3} + 429425630783308916268926724184819613175361047610979294132241064707835483922441897546227531290796021823051 x^{2} + 43260130996049873726411770277451068878478186589939713591589596645542021665240667040988878175176218724064080844428 x + 1319569192071309670099844444864615157512505392643493301557702632154693290814991519445878645475148627465759580830153615884$