Properties

 Label 3.12.12.8 Base $$\Q_{3}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$12$$ Galois group $S_3\times C_3:S_3.C_2$ (as 12T119)

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Defining polynomial

 $$x^{12} + 21 x^{11} + 81 x^{10} - 78 x^{9} - 63 x^{8} - 99 x^{7} + 117 x^{6} - 54 x^{5} - 54 x^{4} + 81 x^{2} - 81 x + 81$$

Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $4$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}(\sqrt{*})$ Root number: $-1$ $|\Aut(K/\Q_{ 3 })|$: $1$ This field is not Galois over $\Q_{3}$.

Intermediate fields

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

Unramified/totally ramified tower

 Unramified subfield: 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{4} - x + 2$$ Relative Eisenstein polynomial: $x^{3} + \left(3 t^{3} + 3\right) x^{2} + \left(-3 t^{3} + 3 t^{2} + 3 t - 3\right) x - 3 t^{3} - 3 t^{2} - 3 t - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

 Galois group: $S_3\times C_3:S_3.C_2$ (as 12T119) Inertia group: Intransitive group isomorphic to $C_3^3:C_2$ Unramified degree: $4$ Tame degree: $2$ Wild slopes: [3/2, 3/2, 3/2] Galois mean slope: $79/54$ Galois splitting model: $x^{12} - 4 x^{9} + x^{6} + 6 x^{3} + 1$