Properties

Label 3.15.15.25
Base \(\Q_{3}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(15\)
Galois group 15T44

Related objects

Learn more about

Defining polynomial

\(x^{15} + 6 x^{14} + 3 x^{13} + 18 x^{12} + 24 x^{11} + 18 x^{10} + 7 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{6} + 9 x^{5} + 3 x^{4} + 4 x^{3} + 15 x^{2} + 12 x + 25\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $15$
Ramification exponent $e$: $3$
Residue field degree $f$: $5$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$
Root number: $i$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}.$

Intermediate fields

3.5.0.1

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 \( x^{5} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} + 3 t^{2} x^{2} + \left(6 t^{3} + 3 t^{2} + 6 t + 6\right) x + 3 t^{4} + 6 t^{3} + 3 t^{2} + 6 t + 6 \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:15T44
Inertia group:Intransitive group isomorphic to $C_3^2:(C_3^3:C_2)$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:[3/2, 3/2, 3/2, 3/2, 3/2]
Galois mean slope:$727/486$
Galois splitting model:$x^{15} - 15 x^{12} - 75 x^{9} + 1875 x^{6} - 6250 x^{3} - 3125$  Toggle raw display