Properties

Label 2.14.14.2
Base \(\Q_{2}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(14\)
Galois group $C_2\times F_8$ (as 14T9)

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Defining polynomial

\(x^{14} + 2 x^{13} - x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{7} + 2 x + 1\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields

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:\( x^{2} + \left(2 t^{5} + 2 t^{3} + 2 t^{2} + 2\right) x + 2 t^{4} + 2 t^{3} + 2 t \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times F_8$ (as 14T9)
Inertia group:Intransitive group isomorphic to $C_2^3$
Unramified degree:$14$
Tame degree:$1$
Wild slopes:[2, 2, 2]
Galois mean slope:$7/4$
Galois splitting model:$x^{14} - 42 x^{12} + 126 x^{10} + 7938 x^{8} - 36855 x^{6} - 304479 x^{4} + 137781 x^{2} + 789507$  Toggle raw display