Properties

Label 2.14.21.34
Base \(\Q_{2}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(21\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\(x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $21$
Discriminant root field: $\Q_{2}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 2 })|$: $14$
This field is Galois and abelian over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{2})$, 2.7.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} + 4 t^{3} x + 2 t^{6} + 6 t^{5} + 6 t^{4} + 2 t^{3} + 2 t^{2} + 2 t \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$7$
Tame degree:$1$
Wild slopes:[3]
Galois mean slope:$3/2$
Galois splitting model:$x^{14} + 84 x^{12} + 2156 x^{10} + 19208 x^{8} + 60368 x^{6} + 47040 x^{4} + 12544 x^{2} + 896$