Properties

Label 5.15.15.27
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group $C_5^3:C_{12}$ (as 15T38)

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Defining polynomial

\(x^{15} - 60 x^{12} + 45 x^{11} + 15 x^{10} + 975 x^{9} - 750 x^{8} + 150 x^{7} + 105575 x^{6} + 16575 x^{5} + 7500 x^{4} + 375 x^{3} + 2250 x^{2} + 1125 x + 125\) Copy content Toggle raw display

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: $1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $1$
This field is not Galois over $\Q_{5}.$
Visible slopes:$[5/4]$

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 \( x^{3} + 3 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{5} + \left(20 t^{2} + 5 t + 20\right) x^{2} + \left(5 t + 15\right) x + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 4t + 2$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_5^3:C_{12}$ (as 15T38)
Inertia group:Intransitive group isomorphic to $C_5^3:C_4$
Wild inertia group:$C_5^3$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:$[5/4, 5/4, 5/4]$
Galois mean slope:$623/500$
Galois splitting model: $x^{15} + 40 x^{13} - 60 x^{12} + 680 x^{11} + 60 x^{10} + 5550 x^{9} + 9020 x^{8} + 53525 x^{7} + 31840 x^{6} + 316614 x^{5} + 631900 x^{4} + 1236015 x^{3} + 1166480 x^{2} - 41040 x - 235712$ Copy content Toggle raw display