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

\( x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275 \)

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: $-i$
$|\Gal(K/\Q_{ 11 })|$: $10$
This field is Galois and abelian over $\Q_{11}$.

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 \( x^{5} + x^{2} - x + 5 \)
Relative Eisenstein polynomial:$ x^{2} - 11 t^{2} \in\Q_{11}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{10} - x^{9} - 91 x^{8} - 77 x^{7} + 2223 x^{6} + 4289 x^{5} - 16697 x^{4} - 46785 x^{3} + 20988 x^{2} + 143200 x + 102325$