Properties

Label 3.15.15.35
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

\(x^{15} + 51 x^{13} - 975 x^{12} + 2907 x^{11} + 2043 x^{10} - 9684 x^{9} + 9315 x^{8} + 34830 x^{7} + 11853 x^{6} - 13122 x^{5} - 3969 x^{4} + 9639 x^{3} + 6318 x^{2} + 1215 x + 243\) 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: $\Q_{3}(\sqrt{3})$
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: \( x^{3} + \left(3 t^{2} + 6 t\right) x^{2} + \left(6 t^{2} + 3 t + 3\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t^{2} + 2t + 2$
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: $x^{15} - 3 x^{14} - 810 x^{13} + 18989 x^{12} - 346005 x^{11} + 3263676 x^{10} + 56238989 x^{9} - 1599823419 x^{8} + 17281209177 x^{7} - 118982301841 x^{6} - 963684815676 x^{5} + 4405617135960 x^{4} + 84905048383392 x^{3} - 1086443883724617 x^{2} - 672847854760317 x + 6867637514341207$ Copy content Toggle raw display