Properties

Label 2.14.12.1
Base \(\Q_{2}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(12\)
Galois group $(C_7:C_3) \times C_2$ (as 14T5)

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Defining polynomial

\( x^{14} - 2 x^{7} + 4 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $7$
Residue field degree $f$: $2$
Discriminant exponent $c$: $12$
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

$\Q_{2}(\sqrt{5})$, 2.7.6.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{7} + 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_7:C_3$ (as 14T5)
Inertia group:Intransitive group isomorphic to $C_7$
Unramified degree:$6$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:$x^{14} + 14 x^{12} + 140 x^{10} + 672 x^{8} - 22 x^{7} + 2352 x^{6} + 308 x^{5} + 3136 x^{4} + 2464 x^{3} + 3136 x^{2} + 1232 x + 484$