Properties

Label 11.12.10.4
Base \(\Q_{11}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

Related objects

Learn more about

Defining polynomial

\(x^{12} - 11 x^{6} + 847\)  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: $-1$
$|\Aut(K/\Q_{ 11 })|$: $6$
This field is not Galois over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{2})$, 11.4.2.2, 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:$\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} - x + 7 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{6} - 11 t \)$\ \in\Q_{11}(t)[x]$  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 $C_6$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} - 3 x^{11} + 41 x^{10} - 273 x^{9} + 5320 x^{8} - 1216 x^{7} + 185714 x^{6} - 462051 x^{5} + 6328363 x^{4} + 19619049 x^{3} + 296778921 x^{2} + 695676814 x + 2813125384$  Toggle raw display