Properties

Label 17.6.3.2
Base \(\Q_{17}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} - 289 x^{2} + 14739 \)

Invariants

Base field: $\Q_{17}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{17}(\sqrt{17\cdot 3})$
Root number: $-1$
$|\Gal(K/\Q_{ 17 })|$: $6$
This field is Galois and abelian over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{17\cdot 3})$, 17.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:17.3.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{3} - x + 3 \)
Relative Eisenstein polynomial:$ x^{2} - 17 t \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed