Properties

Label 5.15.15.21
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group 15T38

Related objects

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Defining polynomial

\( x^{15} + 10 x^{14} + 5 x^{13} + 20 x^{12} + 22 x^{10} + 15 x^{6} + 17 x^{5} + 5 x^{3} + 15 x^{2} + 2 \)

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(15 t^{2} + 20 t + 5\right) x^{4} + \left(15 t^{2} + 20\right) x^{3} + 20 t x^{2} + \left(10 t^{2} + 15\right) x + 20 t^{2} + 5 t + 10 \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:$x^{15} - 10 x^{13} - 120 x^{12} + 455 x^{11} + 336 x^{10} + 150 x^{9} - 1220 x^{8} - 46430 x^{7} + 118980 x^{6} + 120168 x^{5} - 764000 x^{4} + 1115185 x^{3} + 992820 x^{2} - 4379360 x + 440896$