Properties

Label 5.10.11.4
Base \(\Q_{5}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(11\)
Galois group $F_{5}\times C_2$ (as 10T5)

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Defining polynomial

\(x^{10} + 15 x^{2} + 5\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{5}(\sqrt{5})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $2$
This field is not Galois over $\Q_{5}.$
Visible slopes:$[5/4]$

Intermediate fields

$\Q_{5}(\sqrt{5})$, 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:$\Q_{5}$
Relative Eisenstein polynomial: \( x^{10} + 15 x^{2} + 5 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$2z^{2} + 4$,$z^{5} + 2$
Associated inertia:$2$,$1$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:$F_5$ (as 10T4)
Wild inertia group:$C_5$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:$[5/4]$
Galois mean slope:$23/20$
Galois splitting model:$x^{10} - 5 x^{9} + 15 x^{8} - 30 x^{7} + 45 x^{6} - 51 x^{5} + 5 x^{4} + 50 x^{3} - 35 x^{2} + 5 x - 1$