Properties

Label 11.12.9.2
Base \(\Q_{11}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $D_4 \times C_3$ (as 12T14)

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Defining polynomial

\(x^{12} - 22 x^{8} + 121 x^{4} - 11979\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $12$
Ramification exponent $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{11}(\sqrt{11\cdot 2})$
Root number: $-i$
$|\Aut(K/\Q_{ 11 })|$: $6$
This field is not Galois over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{11})$, 11.3.0.1, 11.4.3.2, 11.6.3.1

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

Unramified/totally ramified tower

Unramified subfield:11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{3} - x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{4} - 11 t^{2} \)$\ \in\Q_{11}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$6$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{12} - 2 x^{11} - 24 x^{10} + 41 x^{9} + 168 x^{8} - 244 x^{7} - 467 x^{6} + 510 x^{5} + 597 x^{4} - 361 x^{3} - 200 x^{2} - 7 x - 41$  Toggle raw display