Properties

Label 5.15.15.45
Base \(\Q_{5}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(15\)
Galois group 15T27

Related objects

Learn more about

Defining polynomial

\(x^{15} - 5 x^{14} - 5 x^{13} + 5 x^{12} - 10 x^{11} + 5 x^{10} - 5 x^{9} - 5 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} - 10 x^{4} + 10 x^{3} + 10 x^{2} - 10 x + 5\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
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

5.3.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{15} - 5 x^{14} - 5 x^{13} + 5 x^{12} - 10 x^{11} + 5 x^{10} - 5 x^{9} - 5 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} - 10 x^{4} + 10 x^{3} + 10 x^{2} - 10 x + 5 \)  Toggle raw display

Invariants of the Galois closure

Galois group:15T27
Inertia group:$(C_5^2 : C_4):C_3$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:[13/12, 13/12]
Galois mean slope:$323/300$
Galois splitting model:$x^{15} - 5 x^{14} - 1140 x^{13} + 3110 x^{12} + 470165 x^{11} + 101903 x^{10} - 84431845 x^{9} - 308853455 x^{8} + 5617820830 x^{7} + 41490873310 x^{6} + 19586232533 x^{5} - 440565472815 x^{4} - 628374042290 x^{3} + 1126034023020 x^{2} + 790212143220 x + 48157299081$  Toggle raw display