Properties

Label 3.12.18.62
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(18\)
Galois group 12T73

Related objects

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Defining polynomial

\( x^{12} + 36 x^{11} - 30 x^{10} - 15 x^{9} - 18 x^{8} + 18 x^{7} - 33 x^{6} + 27 x^{4} + 27 x^{3} - 27 x - 36 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $18$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $-1$
$|\Aut(K/\Q_{ 3 })|$: $3$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{6} + \left(-6 t - 12\right) x^{4} + \left(6 t - 6\right) x^{3} + \left(9 t - 9\right) x^{2} + \left(9 t + 9\right) x - 6 t - 9 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:12T73
Inertia group:Intransitive group isomorphic to $C_3\times C_3:S_3$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:[3/2, 3/2, 2]
Galois mean slope:$97/54$
Global splitting model:\( x^{12} - 84 x^{10} - 140 x^{9} + 4158 x^{8} + 252 x^{7} - 84154 x^{6} - 15876 x^{5} + 1566285 x^{4} - 3091312 x^{3} + 3313674 x^{2} - 30417828 x + 74869207 \)