Properties

 Label 2.12.12.28 Base $$\Q_{2}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$12$$ Galois group $S_4$ (as 12T9)

Related objects

Defining polynomial

 $$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$

Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{2}$ Root number: $-i$ $|\Aut(K/\Q_{ 2 })|$: $4$ This field is not Galois over $\Q_{2}$.

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: $\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{2} - x + 1$$ Relative Eisenstein polynomial: $x^{6} + 2 x^{4} + \left(2 t + 2\right) x^{3} + \left(2 t + 2\right) x + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

 Galois group: $S_4$ (as 12T9) Inertia group: Intransitive group isomorphic to $A_4$ Unramified degree: $2$ Tame degree: $3$ Wild slopes: [4/3, 4/3] Galois mean slope: $7/6$ Galois splitting model: $$x^{12} - 4 x^{11} + x^{10} + 14 x^{9} - 14 x^{8} - 10 x^{7} + 25 x^{6} - 10 x^{5} - 14 x^{4} + 14 x^{3} + x^{2} - 4 x + 1$$