Properties

Label 7.15.10.2
Base \(\Q_{7}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(10\)
Galois group $C_{15}$ (as 15T1)

Related objects

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Defining polynomial

\( x^{15} - 2401 x^{3} + 67228 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $15$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $15$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

7.3.2.1, 7.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.5.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{5} - x + 4 \)
Relative Eisenstein polynomial:$ x^{3} - 7 t \in\Q_{7}(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