Properties

Label 23.11.10.1
Base \(\Q_{23}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(10\)
Galois group $C_{11}$ (as 11T1)

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Defining polynomial

\(x^{11} + 230\) Copy content Toggle raw display

Invariants

Base field: $\Q_{23}$
Degree $d$: $11$
Ramification exponent $e$: $11$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{23}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 23 }) }$: $11$
This field is Galois and abelian over $\Q_{23}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{23}$
Relative Eisenstein polynomial: \( x^{11} + 230 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{10} + 11z^{9} + 9z^{8} + 4z^{7} + 8z^{6} + 2z^{5} + 2z^{4} + 8z^{3} + 4z^{2} + 9z + 11$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{11}$ (as 11T1)
Inertia group:$C_{11}$ (as 11T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$11$
Wild slopes:None
Galois mean slope:$10/11$
Galois splitting model:Not computed