Properties

Label 5.15.15.38
Base \(\Q_{5}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(15\)
Galois group $F_5\times C_3$ (as 15T8)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{15} - 60 x^{12} + 45 x^{11} + 15 x^{10} + 4425 x^{9} - 1800 x^{8} + 75 x^{7} + 17575 x^{6} + 66450 x^{5} + 8625 x^{4} - 5625 x^{3} + 1875 x^{2} + 1125 x + 125\) 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: $1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $3$
This field is not Galois over $\Q_{5}.$
Visible slopes:$[5/4]$

Intermediate fields

5.3.0.1, 5.5.5.3

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: \( x^{5} + \left(15 t^{2} + 20 t + 10\right) x^{2} + 15 x + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:Intransitive group isomorphic to $F_5$
Wild inertia group:$C_5$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:$[5/4]$
Galois mean slope:$23/20$
Galois splitting model: $x^{15} - 5 x^{14} - 30 x^{13} + 105 x^{12} + 500 x^{11} - 976 x^{10} - 3110 x^{9} + 690 x^{8} + 16045 x^{7} + 565 x^{6} - 4932 x^{5} - 65245 x^{4} + 18535 x^{3} + 23695 x^{2} + 7350 x - 105209$ Copy content Toggle raw display