Properties

Label 5.15.15.12
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

\(x^{15} - 45 x^{12} + 45 x^{11} + 15 x^{10} + 750 x^{9} + 1950 x^{8} + 1425 x^{7} + 45075 x^{6} + 15825 x^{5} + 43125 x^{4} + 7125 x^{3} + 8250 x^{2} + 1125 x + 125\) 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: $\Q_{5}(\sqrt{5})$
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: \( x^{5} + \left(15 t^{2} + 5 t + 15\right) x^{2} + \left(20 t + 15\right) x + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t + 2$
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: $x^{15} + 10 x^{13} + 5 x^{11} - 18 x^{10} - 100 x^{9} - 190 x^{8} - 125 x^{7} - 630 x^{6} + 205 x^{5} - 850 x^{4} + 625 x^{3} - 750 x^{2} + 1080 x - 104$ Copy content Toggle raw display