Properties

Label 3.15.15.33
Base \(\Q_{3}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(15\)
Galois group $C_7^3:C_6$ (as 15T44)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{15} - 18 x^{14} + 129 x^{13} + 303 x^{12} - 5427 x^{11} + 14859 x^{10} + 179100 x^{9} + 421929 x^{8} + 433917 x^{7} + 265626 x^{6} + 127089 x^{5} + 1296 x^{4} - 17334 x^{3} + 13122 x^{2} - 2673 x + 243\) 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: \( x^{3} + \left(6 t^{4} + 3 t + 6\right) x^{2} + \left(6 t^{4} + 3 t^{3} + 3 t + 3\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t^{4} + 2t^{3} + 2t + 2$
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: $x^{15} - 3 x^{14} + 1860 x^{13} - 12784 x^{12} + 876054 x^{11} - 10174701 x^{10} + 66883923 x^{9} - 2430272985 x^{8} - 4534018347 x^{7} + 162286312994 x^{6} + 787737248991 x^{5} + 33342373879320 x^{4} - 177352964514079 x^{3} - 3133655109034716 x^{2} + 7214095667995362 x + 73208065322123071$ Copy content Toggle raw display