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

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Defining polynomial

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

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

3.3.4.2, 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:\( x^{3} + \left(15 t^{4} + 24 t^{3} + 15 t^{2} + 9\right) x^{2} + \left(18 t^{4} + 18 t^{2} + 9 t + 18\right) x + 12 t^{3} + 6 t^{2} + 6 \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

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:$x^{15} - 3 x^{14} - 36 x^{13} + 151 x^{12} + 306 x^{11} - 2250 x^{10} + 1258 x^{9} + 10467 x^{8} - 18066 x^{7} - 7856 x^{6} + 39705 x^{5} - 23016 x^{4} - 8527 x^{3} + 5730 x^{2} + 2244 x + 199$  Toggle raw display