Properties

Label 5.15.24.88
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(24\)
Galois group $C_{15}$ (as 15T1)

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Defining polynomial

\(x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $24$
Discriminant root field: $\Q_{5}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $15$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:$[2]$

Intermediate fields

5.3.0.1, 5.5.8.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} + 3 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{5} + 20 x^{4} + 50 t + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_{15}$ (as 15T1)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_5$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$8/5$
Galois splitting model:$x^{15} - 5 x^{14} - 30 x^{13} + 150 x^{12} + 305 x^{11} - 1539 x^{10} - 1350 x^{9} + 6825 x^{8} + 3115 x^{7} - 13645 x^{6} - 4757 x^{5} + 11735 x^{4} + 3765 x^{3} - 3500 x^{2} - 770 x + 301$