Properties

Label 2.10.10.7
Base \(\Q_{2}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(10\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\( x^{10} - x^{8} - x^{6} - 3 x^{2} - 7 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $10$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{2}(\sqrt{-1})$
Root number: $i$
$|\Gal(K/\Q_{ 2 })|$: $10$
This field is Galois and abelian over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{-1})$, 2.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{5} + x^{2} + 1 \)
Relative Eisenstein polynomial:$ x^{2} + 2 x - 14 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$5$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$1$
Galois splitting model:$x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$