Properties

Label 5.15.15.39
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group $(C_5^2 : C_4):C_3$ (as 15T19)

Related objects

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Defining polynomial

\( x^{15} + 10 x^{14} + 15 x^{12} + 20 x^{11} + 2 x^{10} + 5 x^{9} + 10 x^{8} + 15 x^{7} + 15 x^{6} + 12 x^{5} + 20 x^{4} + 10 x^{3} + 10 x^{2} + 20 x + 12 \)

Invariants

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

Invariants of the Galois closure

Galois group:$C_5^2:C_{12}$ (as 15T19)
Inertia group:Intransitive group isomorphic to $C_5^2:C_4$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$123/100$
Galois splitting model:$x^{15} - 95 x^{13} + 3365 x^{11} - 192 x^{10} - 57850 x^{9} + 4810 x^{8} + 511225 x^{7} - 11830 x^{6} - 2226068 x^{5} - 329550 x^{4} + 4009525 x^{3} + 1428050 x^{2} - 1142440 x - 228488$