# Properties

 Label 3.12.12.30 Base $$\Q_{3}$$ Degree $$12$$ e $$12$$ f $$1$$ c $$12$$ Galois group $PSU(3,2):C_2$ (as 12T84)

# Related objects

## Defining polynomial

 $$x^{12} - 3 x^{10} + 3 x^{7} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} - 3 x - 3$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $12$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Aut(K/\Q_{ 3 })|$: $1$ 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: $\Q_{3}$ Relative Eisenstein polynomial: $$x^{12} - 3 x^{10} + 3 x^{7} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} - 3 x - 3$$

## Invariants of the Galois closure

 Galois group: $PSU(3,2):C_2$ (as 12T84) Inertia group: 12T46 Unramified degree: $2$ Tame degree: $8$ Wild slopes: [9/8, 9/8] Galois mean slope: $79/72$ Galois splitting model: $x^{12} - 6 x^{10} - 4 x^{9} + 6 x^{8} + 24 x^{5} + 21 x^{4} + 8 x^{3} + 18 x^{2} + 12 x - 2$