Properties

Label 3.15.20.2
Base \(\Q_{3}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(20\)
Galois group $C_7^3:C_6$ (as 15T44)

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Defining polynomial

\(x^{15} + 6 x^{14} + 189 x^{13} + 321 x^{12} - 1980 x^{11} + 9072 x^{10} - 55935 x^{9} - 77193 x^{8} + 33291 x^{7} + 582984 x^{6} + 438777 x^{5} + 3456918 x^{4} + 1635066 x^{3} + 3636738 x^{2} - 8018757\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $15$
Ramification exponent $e$: $3$
Residue field degree $f$: $5$
Discriminant exponent $c$: $20$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[2]$

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: \( x^{3} + \left(3 t^{4} + 6 t^{2} + 3 t + 6\right) x^{2} + 9 t^{4} + 9 t^{3} + 18 t + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_7^3:C_6$ (as 15T44)
Inertia group:Intransitive group isomorphic to $C_3^5$
Wild inertia group:$C_3^5$
Unramified degree:$10$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 2, 2]$
Galois mean slope:$484/243$
Galois splitting model: $x^{15} - 6 x^{13} - 28 x^{12} - 144 x^{11} - 156 x^{10} - 167 x^{9} - 36 x^{8} + 1806 x^{7} - 14 x^{6} + 801 x^{5} + 1224 x^{4} - 13460 x^{3} + 10944 x^{2} + 1248 x - 2144$ Copy content Toggle raw display