Properties

Label 2.10.14.8
Base \(\Q_{2}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(14\)
Galois group $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29)

Related objects

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Defining polynomial

\( x^{10} - 2 x^{9} + 2 x^{5} - 2 x^{4} + 2 x^{2} - 6 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $10$
Ramification exponent $e$ : $10$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{2}(\sqrt{-*})$
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:\( x^{10} - 10 x^{9} - 1120 x^{8} - 4780 x^{7} + 224840 x^{6} + 1086138 x^{5} - 22976430 x^{4} - 20509260 x^{3} + 65579870 x^{2} - 274635860 x - 649182262 \)

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:[8/5, 8/5, 8/5, 8/5, 2]
Galois mean slope:$71/40$
Galois splitting model:$x^{10} + x^{8} + 10 x^{6} - 10 x^{4} + 5 x^{2} + 5$