Properties

Label 2.10.10.8
Base \(\Q_{2}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(10\)
Galois group $C_2 \times (C_2^4 : C_5)$ (as 10T14)

Related objects

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Defining polynomial

\( x^{10} + x^{8} - 2 x^{6} - 2 x^{4} + x^{2} + 33 \)

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: $-i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

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 - 2 t^{3} \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_2^4:C_5$ (as 10T14)
Inertia group:Intransitive group isomorphic to $C_2^5$
Unramified degree:$5$
Tame degree:$1$
Wild slopes:[2, 2, 2, 2, 2]
Galois mean slope:$31/16$
Galois splitting model:$x^{10} + 4 x^{8} + 2 x^{6} - 5 x^{4} - 2 x^{2} + 1$