Properties

Label 2.10.12.3
Base \(\Q_{2}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(12\)
Galois group $(C_2^4 : C_5):C_4$ (as 10T25)

Related objects

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Defining polynomial

\( x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $10$
Ramification exponent $e$ : $10$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
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:\( x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2 \)

Invariants of the Galois closure

Galois group:$C_2^4:C_5:C_4$ (as 10T25)
Inertia group:$C_2^4 : C_5$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:[8/5, 8/5, 8/5, 8/5]
Galois mean slope:$31/20$
Galois splitting model:$x^{10} - x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} - 5$