Properties

Label 5.15.15.3
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group 15T38

Related objects

Learn more about

Defining polynomial

\( x^{15} + 10 x^{14} + 20 x^{13} + 2 x^{10} + 20 x^{9} + 20 x^{8} + 15 x^{7} + 2 x^{5} + 10 x^{4} + 15 x^{3} + 10 x + 7 \)

Invariants

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

Intermediate fields

5.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{5} + \left(15 t^{2} + 5\right) x^{4} + \left(10 t^{2} + 10 t\right) x^{3} + \left(15 t^{2} + 10 t\right) x^{2} + \left(5 t^{2} + 10 t + 5\right) x + 5 t^{2} + 10 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:15T38
Inertia group:Intransitive group isomorphic to $C_5^2:D_5.C_2$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:[5/4, 5/4, 5/4]
Galois mean slope:$623/500$
Galois splitting model:$x^{15} + 10 x^{13} + 5 x^{11} - 18 x^{10} - 100 x^{9} + 90 x^{8} - 125 x^{7} - 70 x^{6} - 131 x^{5} - 850 x^{4} + 1185 x^{3} + 650 x^{2} - 1160 x + 344$