Properties

Label 2.14.16.4
Base \(\Q_{2}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(16\)
Galois group $F_8:C_3$ (as 14T11)

Related objects

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Defining polynomial

\( x^{14} + 2 x^{13} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
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:\( x^{14} + 2 x^{13} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 \)

Invariants of the Galois closure

Galois group:$F_8:C_3$ (as 14T11)
Inertia group:14T6
Unramified degree:$3$
Tame degree:$7$
Wild slopes:[10/7, 10/7, 10/7]
Galois mean slope:$19/14$
Galois splitting model:$x^{14} - 7 x^{12} - 91 x^{10} + 497 x^{8} + 2331 x^{6} - 6993 x^{4} - 22869 x^{2} - 6561$