Properties

Label 11.12.9.1
Base \(\Q_{11}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $D_4 \times C_3$ (as 12T14)

Related objects

Learn more about

Defining polynomial

\(x^{12} - 121 x^{4} + 3993\)  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: $\Q_{11}(\sqrt{11})$
Root number: $i$
$|\Aut(K/\Q_{ 11 })|$: $6$
This field is not Galois over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{11\cdot 2})$, 11.3.0.1, 11.4.3.1, 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:11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{3} - x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{4} - 11 t \)$\ \in\Q_{11}(t)[x]$  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:$x^{12} - x^{11} - 7 x^{10} + 10 x^{9} + 77 x^{8} - 8 x^{7} - 171 x^{6} + 22 x^{5} + 476 x^{4} - 192 x^{3} - 112 x^{2} + 64 x + 64$  Toggle raw display