Properties

Label 11.12.8.2
Base \(\Q_{11}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

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Defining polynomial

\(x^{12} - 1331 x^{3} + 29282\)  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: $1$
$|\Aut(K/\Q_{ 11 })|$: $6$
This field is not Galois over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{2})$, 11.4.0.1, 11.6.4.2

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

Unramified/totally ramified tower

Unramified subfield:11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 11 t \)$\ \in\Q_{11}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3\times C_3:C_4$ (as 12T19)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$12$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{12} - 3 x^{11} - 112 x^{10} + 135 x^{9} + 4708 x^{8} + 2405 x^{7} - 82903 x^{6} - 159417 x^{5} + 429601 x^{4} + 1548270 x^{3} + 1263093 x^{2} + 15394 x - 41348$  Toggle raw display