Properties

Label 7.14.14.9
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T24

Related objects

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Defining polynomial

\( x^{14} + 7 x^{13} + 35 x^{12} + 7 x^{11} + 42 x^{10} + 42 x^{9} + 42 x^{8} + 43 x^{7} + 14 x^{6} + 35 x^{5} + 14 x^{4} + 7 x + 3 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{7}(\sqrt{*})$
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} + \left(14 t + 35\right) x^{6} + 14 t x^{5} + 7 x^{4} + \left(21 t + 7\right) x^{3} + \left(14 t + 35\right) x^{2} + 35 x + 28 t + 28 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:14T24
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:$x^{14} - 42 x^{11} + 21 x^{10} + 42 x^{9} + 63 x^{8} + 390 x^{7} + 147 x^{6} + 294 x^{5} - 1764 x^{4} - 3045 x^{3} - 1043 x^{2} - 2919 x - 3141$