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

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Defining polynomial

\( x^{10} - 781 x^{5} + 290521 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $10$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{11}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 11 })|$: $10$
This field is Galois and abelian over $\Q_{11}$.

Intermediate fields

$\Q_{11}(\sqrt{*})$, 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:$\Q_{11}(\sqrt{*})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} - x + 7 \)
Relative Eisenstein polynomial:$ x^{5} - 11 t^{4} \in\Q_{11}(t)[x]$

Invariants of the Galois closure

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