Properties

Label 11.9.6.1
Base \(\Q_{11}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $S_3\times C_3$ (as 9T4)

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Defining polynomial

\(x^{9} - 121 x^{3} + 3993\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
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

11.3.2.1, 11.3.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{3} - x + 3 \)  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 9T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{9} - 4 x^{8} + 32 x^{7} - 97 x^{6} + 331 x^{5} - 647 x^{4} + 965 x^{3} - 926 x^{2} + 345 x - 41$