Properties

 Label 3.12.12.25 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$12$$ Galois group 12T215

Related objects

Defining polynomial

 $$x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{9} + 9 x^{8} - 3 x^{7} + 3 x^{6} - 9 x^{5} + 9 x^{2} - 9$$

Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}$ 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: $\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{6} + \left(-3 t + 3\right) x^{5} - 3 t x^{4} + 3 x^{3} + \left(-3 t - 3\right) x^{2} - 3 t x - 3 t + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

 Galois group: 12T215 Inertia group: Intransitive group isomorphic to $C_3^2:(C_3:S_3.C_2)$ Unramified degree: $4$ Tame degree: $4$ Wild slopes: [5/4, 5/4, 5/4, 5/4] Galois mean slope: $403/324$ Galois splitting model: $x^{12} - 3 x^{11} + 9 x^{10} - 7 x^{9} - 9 x^{8} + 30 x^{7} - 75 x^{6} + 90 x^{5} - 105 x^{4} + 95 x^{3} - 60 x^{2} + 30 x - 5$