# Properties

 Label 2.12.12.2 Base $$\Q_{2}$$ Degree $$12$$ e $$2$$ f $$6$$ c $$12$$ Galois group 12T105

# Learn more about

## Defining polynomial

 $$x^{12} - 6 x^{10} - 13 x^{8} - 28 x^{6} + 15 x^{4} - 30 x^{2} - 3$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $2$ Residue field degree $f$ : $6$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{2}(\sqrt{*})$ Root number: $i$ $|\Aut(K/\Q_{ 2 })|$: $2$ 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: 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{6} - x + 1$$ Relative Eisenstein polynomial: $x^{2} + \left(2 t^{4} - 2 t^{3} - 2 t^{2} + 4 t + 4\right) x + 4 t^{5} + 4 t^{4} - 2 t - 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: 12T105 Inertia group: Intransitive group isomorphic to $C_2^5$ Unramified degree: $6$ Tame degree: $1$ Wild slopes: [2, 2, 2, 2, 2] Galois mean slope: $31/16$ Global splitting model: $$x^{12} - 26 x^{8} - 78 x^{6} - 78 x^{4} - 13 x^{2} + 13$$