Base \(\Q_{17}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_6\times S_3$ (as 12T18)

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Defining polynomial

\( x^{12} + 85 x^{6} + 2601 \)


Base field: $\Q_{17}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{17}$
Root number: $-1$
$|\Aut(K/\Q_{ 17 })|$: $6$
This field is not Galois over $\Q_{17}$.

Intermediate fields

$\Q_{17}(\sqrt{*})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17*})$,,

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}(\sqrt{*})$ $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{6} - 17 t^{2} \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6\times S_3$ (as 12T18)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:Not computed