Properties

Label 11.15.12.4
Base \(\Q_{11}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(12\)
Galois group $C_{15}$ (as 15T1)

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Defining polynomial

\( x^{15} + 99 x^{10} + 3146 x^{5} + 35937 \)

Invariants

Base field: $\Q_{11}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{11}$
Root number: $1$
$|\Gal(K/\Q_{ 11 })|$: $15$
This field is Galois and abelian over $\Q_{11}.$

Intermediate fields

11.3.0.1, 11.5.4.3

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 \( x^{3} - x + 3 \)
Relative Eisenstein polynomial:$ x^{5} - 11 t^{3} \in\Q_{11}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{15}$ (as 15T1)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$3$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed