Properties

Label 3.12.13.3
Base \(\Q_{3}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(13\)
Galois group $S_3^2:C_6$ (as 12T121)

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Defining polynomial

\(x^{12} + 3 x^{3} + 3 x^{2} + 6\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $3$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[5/4]$

Intermediate fields

$\Q_{3}(\sqrt{3})$, 3.4.3.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{12} + 3 x^{3} + 3 x^{2} + 6 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$S_3^2:C_6$ (as 12T121)
Inertia group:$C_3^2:C_4$ (as 12T17)
Wild inertia group:$C_3^2$
Unramified degree:$6$
Tame degree:$4$
Wild slopes:$[5/4, 5/4]$
Galois mean slope:$43/36$
Galois splitting model: $x^{12} - 8 x^{9} - 3 x^{8} - 36 x^{7} - 45 x^{6} + 96 x^{5} + 36 x^{4} - 80 x^{3} - 15 x^{2} + 6 x + 1$ Copy content Toggle raw display