Properties

Label 2.14.18.3
Base \(\Q_{2}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(18\)
Galois group $C_2^3:F_8:C_3$ (as 14T35)

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Defining polynomial

\(x^{14} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[12/7]$

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: \( x^{14} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{12} + z^{10} + z^{8} + z^{6} + z^{4} + z^{2} + 1$
Associated inertia:$1$,$3$
Indices of inseparability:$[5, 0]$

Invariants of the Galois closure

Galois group:$C_2^3:F_8:C_3$ (as 14T35)
Inertia group:$C_2^3:F_8$ (as 14T21)
Wild inertia group:$C_2^6$
Unramified degree:$3$
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: $x^{14} - 7 x^{12} - 35 x^{10} + 161 x^{8} + 399 x^{6} + 35 x^{4} - 133 x^{2} - 1$ Copy content Toggle raw display