Properties

Label 2.14.18.4
Base \(\Q_{2}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(18\)
Galois group 14T44

Related objects

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Defining polynomial

\( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{5} + 2 x^{2} + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
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

2.7.6.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{5} + 2 x^{2} + 2 \)

Invariants of the Galois closure

Galois group:14T44
Inertia group:14T21
Unramified degree:$6$
Tame degree:$7$
Wild slopes:[8/7, 8/7, 8/7, 12/7, 12/7, 12/7]
Galois mean slope:$367/224$
Galois splitting model:$x^{14} - 14 x^{12} - 14 x^{11} + 14 x^{10} + 140 x^{9} + 378 x^{8} - 236 x^{7} - 770 x^{6} - 1918 x^{5} - 1568 x^{4} + 4032 x^{3} + 3878 x^{2} + 7476 x - 470$