Properties

Label 17.12.10.3
Base \(\Q_{17}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

\(x^{12} + 136 x^{6} + 7803\)  Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{17}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 17 })|$: $12$
This field is Galois over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{3})$, 17.3.2.1 x3, 17.4.2.2, 17.6.4.1

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_3:C_4$ (as 12T5)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:Not computed