Properties

Label 2.10.10.4
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

Downloads

Learn more

Defining polynomial

\(x^{10} - 2 x^{8} + 16 x^{7} - 8 x^{6} - 432 x^{5} - 176 x^{4} - 960 x^{3} - 1776 x^{2} - 448 x - 32\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $10$
Ramification exponent $e$: $2$
Residue field degree $f$: $5$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2]$

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 \( x^{5} + x^{2} + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 2 t x + 4 t^{3} + 4 t^{2} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_2\wr C_5$ (as 10T14)
Inertia group:Intransitive group isomorphic to $C_2^4$
Wild inertia group:$C_2^4$
Unramified degree:$10$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 2]$
Galois mean slope:$15/8$
Galois splitting model:$x^{10} - 11 x^{6} - 11 x^{4} + 11 x^{2} + 11$