Properties

Label 7.9.6.3
Base \(\Q_{7}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $C_9$ (as 9T1)

Related objects

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Defining polynomial

\( x^{9} - 14 x^{6} + 49 x^{3} - 1372 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $9$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $9$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

7.3.0.1

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$3$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$