Properties

Label 5.15.15.46
Base \(\Q_{5}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(15\)
Galois group 15T27

Related objects

Learn more about

Defining polynomial

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

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

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:$x^{15} - 5 x^{14} - 1305 x^{13} + 14425 x^{12} + 443435 x^{11} - 7738479 x^{10} - 17722590 x^{9} + 1066146055 x^{8} - 6587684480 x^{7} - 1964223865 x^{6} + 136632641122 x^{5} - 355500926340 x^{4} - 80127675915 x^{3} + 966211026480 x^{2} - 300974243220 x - 541837582767$