Properties

Label 2.14.26.60
Base \(\Q_{2}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(26\)
Galois group 14T35

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$ : $14$
Ramification exponent $e$ : $14$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $26$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.7.6.1

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:\( x^{14} - 2 x^{13} + 2 x^{12} - 2 x^{10} + 4 x^{9} + 2 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2 \)

Invariants of the Galois closure

Galois group:14T35
Inertia group:14T21
Unramified degree:$3$
Tame degree:$7$
Wild slopes:[18/7, 18/7, 18/7, 20/7, 20/7, 20/7]
Galois mean slope:$313/112$
Galois splitting model:$x^{14} - 35 x^{12} + 35 x^{10} + 427 x^{8} - 357 x^{6} - 945 x^{4} + 945 x^{2} - 81$