Properties

Label 5.15.15.34
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group $(C_5^2 : C_4):C_3$ (as 15T19)

Related objects

Learn more about

Defining polynomial

\(x^{15} + 15 x^{12} + 15 x^{11} + 12 x^{10} + 20 x^{8} + 22 x^{5} + 10 x^{4} + 20 x^{3} + 15 x + 17\)  Toggle raw display

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 \( x^{3} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{5} + \left(5 t^{2} + 10 t + 10\right) x^{4} + \left(10 t^{2} + 15 t + 10\right) x^{3} + \left(15 t^{2} + 15 t + 20\right) x^{2} + \left(15 t + 15\right) x + 20 t + 20 \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_5^2:C_{12}$ (as 15T19)
Inertia group:Intransitive group isomorphic to $C_5^2:C_4$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$123/100$
Galois splitting model:$x^{15} - 195 x^{13} + 15210 x^{11} - 1612 x^{10} - 604175 x^{9} + 209560 x^{8} + 12852450 x^{7} - 9534980 x^{6} - 138696779 x^{5} + 177078200 x^{4} + 568374885 x^{3} - 1151008300 x^{2} + 454691120 x + 78599872$  Toggle raw display