Properties

Label 5.15.15.22
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group 15T38

Related objects

Learn more about

Defining polynomial

\( x^{15} + 15 x^{13} + 5 x^{12} + 15 x^{11} + 12 x^{10} + 20 x^{9} + 15 x^{8} + 15 x^{6} + 22 x^{5} + 20 x^{4} + 10 x^{3} + 2 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{5}(\sqrt{5\cdot 2})$
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} + 10 t^{2} x^{4} + \left(20 t^{2} + 20 t + 15\right) x^{3} + \left(5 t^{2} + 15\right) x^{2} + \left(10 t^{2} + 10 t + 15\right) x + 20 t + 5 \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:$x^{15} + 20 x^{13} - 20 x^{12} - 10 x^{11} - 788 x^{10} - 1650 x^{9} + 1740 x^{8} - 15875 x^{7} + 41320 x^{6} + 188522 x^{5} + 203300 x^{4} - 535425 x^{3} - 2105880 x^{2} - 388560 x - 32192$