Properties

Label 5.15.15.32
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group 15T38

Related objects

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Defining polynomial

\( x^{15} + 15 x^{14} + 5 x^{13} + 20 x^{12} + 5 x^{11} + 12 x^{10} + 20 x^{9} + 15 x^{8} + 15 x^{6} + 2 x^{5} + 10 x^{4} + 10 x^{3} + 10 x + 12 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{5}(\sqrt{5\cdot 2})$
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 \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{5} + \left(10 t + 10\right) x^{4} + \left(10 t^{2} + 20 t\right) x^{3} + \left(5 t + 10\right) x^{2} + \left(5 t^{2} + 5\right) x + 5 t + 15 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:15T38
Inertia group:Intransitive group isomorphic to $C_5^2:D_5.C_2$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:[5/4, 5/4, 5/4]
Galois mean slope:$623/500$
Galois splitting model:$x^{15} + 20 x^{13} - 140 x^{12} + 130 x^{11} - 1468 x^{10} + 6850 x^{9} - 13180 x^{8} + 51325 x^{7} - 128840 x^{6} + 139274 x^{5} - 43300 x^{4} - 213375 x^{3} - 52800 x^{2} + 385040 x + 296512$