Properties

Label 5.15.10.1
Base \(\Q_{5}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(10\)
Galois group $S_3 \times C_5$ (as 15T4)

Related objects

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Defining polynomial

\( x^{15} - 625 x^{3} + 6250 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $3$
Residue field degree $f$: $5$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $5$
This field is not Galois over $\Q_{5}.$

Intermediate fields

5.3.2.1, 5.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.5.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{5} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} + 20 t + 10 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5\times S_3$ (as 15T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$10$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{15} - 2 x^{14} - 3 x^{13} - 6 x^{12} + 46 x^{11} - 96 x^{10} + 66 x^{9} - 271 x^{8} + 802 x^{7} - 1192 x^{6} + 2483 x^{5} - 2152 x^{4} + 5024 x^{3} - 605 x^{2} - 3875 x - 131$