# Properties

 Label 2.12.12.34 Base $$\Q_{2}$$ Degree $$12$$ e $$12$$ f $$1$$ c $$12$$ Galois group 12T254

# Related objects

## Defining polynomial

 $$x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2 x^{3} + 2 x + 2$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $12$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{2}(\sqrt{*})$ Root number: $i$ $|\Aut(K/\Q_{ 2 })|$: $1$ 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: $\Q_{2}$ Relative Eisenstein polynomial: $$x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2 x^{3} + 2 x + 2$$

## Invariants of the Galois closure

 Galois group: 12T254 Inertia group: 12T166 Unramified degree: $6$ Tame degree: $9$ Wild slopes: [10/9, 10/9, 10/9, 10/9, 10/9, 10/9] Galois mean slope: $319/288$ Galois splitting model: $x^{12} - 18 x^{10} - 186 x^{9} + 1179 x^{8} - 1818 x^{7} + 2874 x^{6} - 36360 x^{5} + 192951 x^{4} - 519944 x^{3} + 835722 x^{2} - 786834 x + 359163$