Properties

Label 2.14.0.1
Base \(\Q_{2}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

Learn more about

Defining polynomial

\( x^{14} - x^{5} - x^{3} - x + 1 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $1$
Residue field degree $f$: $14$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 2 })|$: $14$
This field is Galois and abelian over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.14.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{14} - x^{5} - x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x - 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:trivial
Unramified degree:$14$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} - 8701 x^{5} + 18151 x^{4} - 8246 x^{3} + 17920 x^{2} - 8148 x + 9409$