Properties

Label 13.15.10.2
Base \(\Q_{13}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(10\)
Galois group $C_{15}$ (as 15T1)

Related objects

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Defining polynomial

\( x^{15} - 57122 x^{3} + 2227758 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $15$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{13}$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $15$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

13.3.2.1, 13.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:13.5.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{5} - 2 x + 6 \)
Relative Eisenstein polynomial:$ x^{3} - 13 t \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{15}$ (as 15T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$5$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed