Properties

Label 3.15.15.24
Base \(\Q_{3}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(15\)
Galois group $C_7^3:C_6$ (as 15T44)

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Defining polynomial

\(x^{15} + 150 x^{13} - 2235 x^{12} + 11961 x^{11} - 32922 x^{10} + 45342 x^{9} - 17577 x^{8} - 6723 x^{7} - 34884 x^{6} + 193671 x^{5} + 181197 x^{4} + 39285 x^{3} + 5103 x^{2} + 486 x + 243\) Copy content 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: $i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[3/2]$

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

Ramification polygon

Residual polynomials:$z + 2t^{4} + t^{3} + 2t + 1$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_7^3:C_6$ (as 15T44)
Inertia group:Intransitive group isomorphic to $C_3^4:S_3$
Wild inertia group:$C_3^5$
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: $x^{15} - 40 x^{12} - 75 x^{9} + 2250 x^{6} - 6250 x^{3} + 3125$ Copy content Toggle raw display