Properties

Label 5.15.15.48
Base \(\Q_{5}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(15\)
Galois group 15T27

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{5}(\sqrt{5})$
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:\( x^{15} + 5 x^{14} - 10 x^{13} - 10 x^{12} + 5 x^{11} - 5 x^{10} - 10 x^{9} + 10 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{3} + 10 x^{2} - 5 x + 5 \)

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:$x^{15} + 15 x^{13} - 220 x^{12} - 2580 x^{11} + 14280 x^{10} - 16700 x^{9} - 454800 x^{8} + 326295 x^{7} + 2442440 x^{6} + 21389325 x^{5} + 62883660 x^{4} + 191239960 x^{3} + 129110160 x^{2} - 41569980 x + 565413680$