Properties

Label 7.14.14.5
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group $F_7 \times C_2$ (as 14T7)

Related objects

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Defining polynomial

\( x^{14} + 35 x^{12} + 28 x^{11} + 14 x^{10} + 42 x^{9} + 21 x^{8} + x^{7} + 21 x^{6} + 42 x^{5} + 7 x^{3} + 21 x^{2} + 28 x + 10 \)

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: $1$
$|\Aut(K/\Q_{ 7 })|$: $2$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.7.7.6

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 + 7\right) x^{6} + \left(14 t + 42\right) x^{5} + \left(21 t + 42\right) x^{4} + \left(28 t + 21\right) x^{3} + \left(21 t + 28\right) x^{2} + 21 x + 42 t + 28 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times F_7$ (as 14T7)
Inertia group:Intransitive group isomorphic to $F_7$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:[7/6]
Galois mean slope:$47/42$
Galois splitting model:$x^{14} + 42 x^{12} + 693 x^{10} + 9856 x^{8} - 26460 x^{6} + 41328 x^{4} - 44352 x^{2} + 20736$