Properties

Label 11.4.2.2
Base \(\Q_{11}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_4$ (as 4T1)

Related objects

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Defining polynomial

\(x^{4} - 11 x^{2} + 847\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{11}(\sqrt{2})$
Root number: $-1$
$|\Gal(K/\Q_{ 11 })|$: $4$
This field is Galois and abelian over $\Q_{11}.$

Intermediate fields

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

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 \( x^{2} - x + 7 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 11 t \)$\ \in\Q_{11}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_4$ (as 4T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{4} + 44 x^{2} + 242$