Properties

Label 11.10.8.4
Base \(\Q_{11}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{10} - 165 x^{5} - 4356\) Copy content Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $10$
Ramification exponent $e$: $5$
Residue field degree $f$: $2$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{11}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 11 }) }$: $10$
This field is Galois and abelian over $\Q_{11}.$
Visible slopes:None

Intermediate fields

$\Q_{11}(\sqrt{2})$, 11.5.4.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{5} + 33 t + 33 \) $\ \in\Q_{11}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + 5z^{3} + 10z^{2} + 10z + 5$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{10} - x^{9} + 2 x^{8} - 356 x^{7} - 877 x^{6} + 3793 x^{5} + 65887 x^{4} + 386316 x^{3} + 977865 x^{2} + 1230866 x + 2368673$