Properties

Label 3.15.15.11
Base \(\Q_{3}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(15\)
Galois group 15T33

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Defining polynomial

\(x^{15} + 24 x^{13} + 24 x^{12} + 24 x^{11} + 25 x^{9} + 6 x^{8} + 12 x^{7} + 18 x^{6} + 15 x^{5} + 21 x^{4} + 19 x^{3} + 21 x^{2} + 6 x + 4\)  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: $\Q_{3}(\sqrt{3})$
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:\( x^{3} + \left(6 t^{4} + 6 t^{3} + 3 t + 3\right) x^{2} + \left(6 t^{4} + 3 t^{3} + 3 t + 3\right) x + 6 t^{4} + 6 t^{3} + 6 t^{2} + 6 t + 6 \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:15T33
Inertia group:Intransitive group isomorphic to $C_3:(C_3^3:C_2)$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:[3/2, 3/2, 3/2, 3/2]
Galois mean slope:$241/162$
Galois splitting model:$x^{15} - 9 x^{13} - 15 x^{12} - 27 x^{11} - 24 x^{10} + 99 x^{9} + 144 x^{8} + 57 x^{7} + 6 x^{6} - 279 x^{5} + 3 x^{4} + 365 x^{3} + 234 x^{2} - 135 x - 23$  Toggle raw display