Properties

Label 2.14.21.40
Base \(\Q_{2}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(21\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\( x^{14} + 8 x^{13} - 4 x^{12} + 4 x^{11} + 5 x^{10} - 4 x^{9} - 4 x^{8} + 2 x^{7} - x^{6} + 6 x^{5} - 4 x^{4} + 6 x^{3} + 3 x^{2} + 6 x + 3 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $21$
Discriminant root field: $\Q_{2}(\sqrt{-2*})$
Root number: $i$
$|\Gal(K/\Q_{ 2 })|$: $14$
This field is Galois and abelian over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{-2*})$, 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:2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + \left(4 t^{5} + 4 t^{3} + 4 t^{2} + 4\right) x + 4 t^{6} + 4 t^{3} + 2 t^{2} + 4 t + 6 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$7$
Tame degree:$1$
Wild slopes:[3]
Galois mean slope:$3/2$
Galois splitting model:$x^{14} + 28 x^{12} - 42 x^{11} + 1043 x^{10} + 686 x^{9} + 27041 x^{8} + 33840 x^{7} + 508025 x^{6} + 804706 x^{5} + 6352920 x^{4} + 8567650 x^{3} + 48628468 x^{2} + 47642756 x + 178610849$