Properties

Label 89.15.14.1
Base \(\Q_{89}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(14\)
Galois group $D_{15}$ (as 15T2)

Related objects

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Defining polynomial

\(x^{15} - 89\)  Toggle raw display

Invariants

Base field: $\Q_{89}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{89}(\sqrt{3})$
Root number: $1$
$|\Aut(K/\Q_{ 89 })|$: $1$
This field is not Galois over $\Q_{89}.$

Intermediate fields

89.3.2.1, 89.5.4.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{89}$
Relative Eisenstein polynomial:\( x^{15} - 89 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$D_{15}$ (as 15T2)
Inertia group:$C_{15}$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$15$
Wild slopes:None
Galois mean slope:$14/15$
Galois splitting model:Not computed