Properties

Label 11.11.15.8
Base \(\Q_{11}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(15\)
Galois group $F_{11}$ (as 11T4)

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Defining polynomial

\(x^{11} + 33 x^{5} + 11\) Copy content Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $11$
Ramification exponent $e$: $11$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{11}(\sqrt{11\cdot 2})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 11 }) }$: $1$
This field is not Galois over $\Q_{11}.$
Visible slopes:$[3/2]$

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: \( x^{11} + 33 x^{5} + 11 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{5} + 7$
Associated inertia:$5$
Indices of inseparability:$[5, 0]$

Invariants of the Galois closure

Galois group:$F_{11}$ (as 11T4)
Inertia group:$D_{11}$ (as 11T2)
Wild inertia group:$C_{11}$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:$[3/2]$
Galois mean slope:$31/22$
Galois splitting model:Not computed