Properties

Label 11.12.8.1
Base \(\Q_{11}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

\(x^{12} - 33 x^{9} + 363 x^{6} - 1331 x^{3} + 117128\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{11}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 11 })|$: $12$
This field is Galois over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{2})$, 11.3.2.1 x3, 11.4.0.1, 11.6.4.1

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

Unramified/totally ramified tower

Unramified subfield:11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 11 t^{3} \)$\ \in\Q_{11}(t)[x]$  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:$x^{12} - 3 x^{11} - 24 x^{10} + 69 x^{9} + 176 x^{8} - 499 x^{7} - 326 x^{6} + 1172 x^{5} - 411 x^{4} - 211 x^{3} + 40 x^{2} + 16 x + 1$