Properties

 Label 2.12.12.9 Base $$\Q_{2}$$ Degree $$12$$ e $$2$$ f $$6$$ c $$12$$ Galois group $C_2^2\wr C_2:C_3$ (as 12T58)

Learn more about

Defining polynomial

 $$x^{12} - 18 x^{10} + 7 x^{8} - 28 x^{6} - x^{4} - 18 x^{2} - 7$$

Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $2$ Residue field degree $f$ : $6$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{2}$ Root number: $-1$ $|\Aut(K/\Q_{ 2 })|$: $2$ 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: $x^{2} + \left(2 t^{5} + 2\right) x + 2 t^{4} \in\Q_{2}(t)[x]$

Invariants of the Galois closure

 Galois group: $C_2^2\wr C_2:C_3$ (as 12T58) Inertia group: Intransitive group isomorphic to $C_2^4$ Unramified degree: $6$ Tame degree: $1$ Wild slopes: [2, 2, 2, 2] Galois mean slope: $15/8$ Galois splitting model: $x^{12} - 2 x^{10} - 7 x^{8} + 6 x^{6} + 5 x^{4} - 5 x^{2} + 1$