Properties

Label 2.10.16.1
Base \(\Q_{2}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(16\)
Galois group $(C_2^4 : C_5):C_4$ (as 10T24)

Related objects

Learn more about

Defining polynomial

\( x^{10} - 2 x^{7} - 4 x^{4} - 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $10$
Ramification exponent $e$ : $10$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $16$
Discriminant root field: $\Q_{2}$
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:\( x^{10} - 20 x^{9} + 40 x^{8} - 1450 x^{7} + 360 x^{6} + 23720 x^{5} - 97500 x^{4} + 164400 x^{3} - 174200 x^{2} + 96600 x - 22050 \)

Invariants of the Galois closure

Galois group:$C_2^4:C_5:C_4$ (as 10T24)
Inertia group:$C_2^4 : C_5$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:[12/5, 12/5, 12/5, 12/5]
Galois mean slope:$23/10$
Galois splitting model:$x^{10} - 5 x^{8} + 10 x^{6} - 10 x^{4} + 5 x^{2} - 9$