Base \(\Q_{17}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

\(x^{12} - 51 x^{9} + 867 x^{6} - 4913 x^{3} + 111166451\)  Toggle raw display


Base field: $\Q_{17}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{17}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 17 })|$: $12$
This field is Galois over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{3})$, x3,,

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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{4} - x + 11 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 17 t^{3} \)$\ \in\Q_{17}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3:C_4$ (as 12T5)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed