Base \(\Q_{11}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(12\)
Galois group $C_{11}:C_5$ (as 11T3)

Related objects

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Defining polynomial

\(x^{11} + 77 x^{2} + 11\)  Toggle raw display


Base field: $\Q_{11}$
Degree $d$: $11$
Ramification exponent $e$: $11$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{11}$
Root number: $1$
$|\Aut(K/\Q_{ 11 })|$: $1$
This field is not Galois over $\Q_{11}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{11} + 77 x^{2} + 11 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{11}:C_5$ (as 11T3)
Inertia group:$C_{11}:C_5$
Unramified degree:$1$
Tame degree:$5$
Wild slopes:[6/5]
Galois mean slope:$64/55$
Galois splitting model:Not computed