Properties

Label 11.10.5.2
Base \(\Q_{11}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(5\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\(x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252\) Copy content Toggle raw display

Invariants

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

Intermediate fields

$\Q_{11}(\sqrt{11\cdot 2})$, 11.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:11.5.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{5} + 10 x^{2} + 9 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 11 \) $\ \in\Q_{11}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{10} - 5 x^{9} + 5 x^{8} + 90 x^{6} - 148 x^{5} + 610 x^{4} - 175 x^{3} + 2325 x^{2} - 1625 x + 5965$