Properties

Label 2.14.14.34
Base \(\Q_{2}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(14\)
Galois group $C_2^3:F_8$ (as 14T21)

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Defining polynomial

\( x^{14} - x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{7} + 2 x^{4} + 2 x^{2} + 2 x + 1 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{2}$
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 \( x^{7} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + \left(4 t^{6} + 2 t^{5} + 4 t\right) x + 2 t^{5} + 4 t^{4} + 6 t^{3} + 6 t^{2} + 4 t + 4 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2^3:F_8$ (as 14T21)
Inertia group:Intransitive group isomorphic to $C_2^6$
Unramified degree:$7$
Tame degree:$1$
Wild slopes:[2, 2, 2, 2, 2, 2]
Galois mean slope:$63/32$
Galois splitting model:$x^{14} - 49 x^{10} + 441 x^{6} - 686 x^{4} + 343 x^{2} - 49$