Properties

Label 11.6.4.2
Base \(\Q_{11}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $S_3\times C_3$ (as 6T5)

Related objects

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Defining polynomial

\(x^{6} - 11 x^{3} + 847\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{11}(\sqrt{2})$
Root number: $1$
$|\Aut(K/\Q_{ 11 })|$: $3$
This field is not Galois 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:\( x^{3} - 11 t \)$\ \in\Q_{11}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 6T5)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} - 11 x^{3} + 847$  Toggle raw display