Properties

Label 11.12.10.3
Base \(\Q_{11}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

\(x^{12} + 220 x^{6} + 41503\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
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, 11.4.2.2, 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:$\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} - x + 7 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{6} - 11 t^{3} \)$\ \in\Q_{11}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3:C_4$ (as 12T5)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} - 5 x^{11} + 41 x^{10} - 81 x^{9} + 278 x^{8} - 316 x^{7} + 2315 x^{6} - 7280 x^{5} + 17286 x^{4} + 34538 x^{3} + 86932 x^{2} + 51353 x + 332929$