Properties

Label 2.10.16.10
Base \(\Q_{2}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(16\)
Galois group $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29)

Related objects

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Defining polynomial

\( x^{10} - 2 x^{9} + 2 x^{7} + 2 x^{6} - 6 x^{4} - 4 x^{3} - 6 x^{2} - 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $10$
Ramification exponent $e$ : $10$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $16$
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

2.5.4.1

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:\( x^{10} - 2 x^{9} + 2 x^{7} + 2 x^{6} - 6 x^{4} - 4 x^{3} - 6 x^{2} - 2 \)

Invariants of the Galois closure

Galois group:$((C_2^4 : C_5):C_4)\times C_2$ (as 10T29)
Inertia group:$C_2 \times (C_2^4 : C_5)$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:[2, 12/5, 12/5, 12/5, 12/5]
Galois mean slope:$187/80$
Galois splitting model:$x^{10} + 2 x^{8} - 4 x^{6} + 8 x^{2} + 4$