Properties

 Label 3.6.6.4 Base $$\Q_{3}$$ Degree $$6$$ e $$3$$ f $$2$$ c $$6$$ Galois group $D_{6}$ (as 6T3)

Related objects

Defining polynomial

 $$x^{6} + 3 x^{4} + 6 x^{3} + 9 x^{2} + 63 x + 9$$

Invariants

 Base field: $\Q_{3}$ Degree $d$ : $6$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $6$ Discriminant root field: $\Q_{3}(\sqrt{*})$ Root number: $1$ $|\Aut(K/\Q_{ 3 })|$: $2$ 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: $x^{3} + \left(6 t + 6\right) x^{2} + 6 x + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

 Galois group: $D_6$ (as 6T3) Inertia group: Intransitive group isomorphic to $S_3$ Unramified degree: $2$ Tame degree: $2$ Wild slopes: [3/2] Galois mean slope: $7/6$ Galois splitting model: $x^{6} - 3 x^{5} + 21 x^{4} - 31 x^{3} + 63 x^{2} - 45 x + 50$