Properties

Label 7.15.12.1
Base \(\Q_{7}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(12\)
Galois group $F_5\times C_3$ (as 15T8)

Related objects

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Defining polynomial

\( x^{15} - 49 x^{5} + 686 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $15$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $3$
This field is not Galois over $\Q_{7}$.

Intermediate fields

7.3.0.1, 7.5.4.1

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

Unramified/totally ramified tower

Unramified subfield:7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{5} - 7 t \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$12$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed