Base \(\Q_{11}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_6\times S_3$ (as 12T18)

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Defining polynomial

\(x^{12} - 198 x^{6} - 10043\) Copy content Toggle raw display


Base field: $\Q_{11}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{11}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 11 }) }$: $6$
This field is not Galois over $\Q_{11}.$
Visible slopes:None

Intermediate fields

$\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$,,

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 \( x^{2} + 7 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{6} + 44 t + 55 \) $\ \in\Q_{11}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{5} + 6z^{4} + 4z^{3} + 9z^{2} + 4z + 6$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_6\times S_3$ (as 12T18)
Inertia group:Intransitive group isomorphic to $C_6$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model: $x^{12} + 143 x^{6} + 5929$ Copy content Toggle raw display