Properties

Label 2.14.14.39
Base \(\Q_{2}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)
Galois group $C_2\times F_8:C_3$ (as 14T18)

Related objects

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Defining polynomial

\( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{3} + 2 x + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
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:\( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{3} + 2 x + 2 \)

Invariants of the Galois closure

Galois group:$C_2\times F_8:C_3$ (as 14T18)
Inertia group:14T6
Unramified degree:$6$
Tame degree:$7$
Wild slopes:[8/7, 8/7, 8/7]
Galois mean slope:$31/28$
Galois splitting model:$x^{14} - 35 x^{12} - 42 x^{11} + 371 x^{10} + 1218 x^{9} - 1617 x^{8} - 10412 x^{7} + 2247 x^{6} + 38108 x^{5} - 469 x^{4} - 69034 x^{3} + 10885 x^{2} + 38402 x - 3623$