Properties

Label 11.10.5.1
Base \(\Q_{11}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(5\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\(x^{10} - 13310 x^{4} - 1449459\) 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: $\Q_{11}(\sqrt{11})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 11 }) }$: $10$
This field is Galois and abelian over $\Q_{11}.$
Visible slopes:None

Intermediate fields

$\Q_{11}(\sqrt{11})$, 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: \( x^{2} + 11 t \) $\ \in\Q_{11}(t)[x]$ 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:$x^{10} - x^{9} - 91 x^{8} - 77 x^{7} + 2223 x^{6} + 4289 x^{5} - 16697 x^{4} - 46785 x^{3} + 20988 x^{2} + 143200 x + 102325$