# Properties

 Label 3.15.20.65 Base $$\Q_{3}$$ Degree $$15$$ e $$3$$ f $$5$$ c $$20$$ Galois group $C_{15}$ (as 15T1)

# Related objects

## Defining polynomial

 $$x^{15} + 78 x^{14} + 39 x^{13} + 49 x^{12} + 24 x^{11} + 36 x^{10} + 2 x^{9} + 54 x^{8} + 69 x^{7} + 47 x^{6} + 18 x^{5} + 15 x^{4} + 36 x^{3} + 36 x^{2} + 63 x + 73$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$: $15$ Ramification exponent $e$: $3$ Residue field degree $f$: $5$ Discriminant exponent $c$: $20$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Gal(K/\Q_{ 3 })|$: $15$ This field is Galois and abelian 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: 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{5} - x + 1$$ Relative Eisenstein polynomial: $$x^{3} + \left(15 t^{4} + 24 t^{3} + 15 t^{2} + 9\right) x^{2} + \left(18 t^{4} + 18 t^{2} + 9 t + 18\right) x + 12 t^{3} + 6 t^{2} + 6$$$\ \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{15}$ (as 15T1) Inertia group: Intransitive group isomorphic to $C_3$ Unramified degree: $5$ Tame degree: $1$ Wild slopes: [2] Galois mean slope: $4/3$ Galois splitting model: $x^{15} - 3 x^{14} - 36 x^{13} + 151 x^{12} + 306 x^{11} - 2250 x^{10} + 1258 x^{9} + 10467 x^{8} - 18066 x^{7} - 7856 x^{6} + 39705 x^{5} - 23016 x^{4} - 8527 x^{3} + 5730 x^{2} + 2244 x + 199$