Properties

Label 3.12.21.53
Base \(\Q_{3}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(21\)
Galois group 12T116

Related objects

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Defining polynomial

\( x^{12} - 12 x^{10} + 6 x^{9} - 9 x^{6} + 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x + 6 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $21$
Discriminant root field: $\Q_{3}(\sqrt{3*})$
Root number: $-i$
$|\Aut(K/\Q_{ 3 })|$: $3$
This field is not Galois over $\Q_{3}$.

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:\( x^{12} - 12 x^{10} + 6 x^{9} - 9 x^{6} + 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x + 6 \)

Invariants of the Galois closure

Galois group:12T116
Inertia group:12T72
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[3/2, 9/4, 9/4]
Galois Mean Slope:$77/36$
Global Splitting Model:\( x^{12} - 12 x^{10} + 54 x^{8} - 108 x^{6} + 567 x^{4} - 648 x^{3} + 324 x^{2} - 72 x + 6 \)