Properties

Label 11.12.6.1
Base \(\Q_{11}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\(x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $12$
Ramification exponent $e$: $2$
Residue field degree $f$: $6$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{11}$
Root number: $1$
$|\Gal(K/\Q_{ 11 })|$: $12$
This field is Galois and abelian over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$, 11.3.0.1, 11.4.2.1, 11.6.0.1, 11.6.3.1, 11.6.3.2

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

Unramified/totally ramified tower

Unramified subfield:11.6.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{6} + x^{2} - 2 x + 8 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 11 t^{2} \)$\ \in\Q_{11}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{12} - 8 x^{9} + 37 x^{6} - 216 x^{3} + 729$