Properties

Label 7.6.4.3
Base \(\Q_{7}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} + 56 x^{3} + 1323 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $6$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $6$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.3.2.2

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} + 5 x^{4} + 6 x^{2} + 1$