Properties

Label 5.10.12.18
Base \(\Q_{5}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(12\)
Galois group $(C_5^2 : C_8):C_2$ (as 10T28)

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Defining polynomial

\(x^{10} + 5 x^{3} + 10\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{5}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $1$
This field is not Galois over $\Q_{5}.$
Visible slopes:$[11/8]$

Intermediate fields

$\Q_{5}(\sqrt{5\cdot 2})$

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: \( x^{10} + 5 x^{3} + 10 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$2z + 1$,$z^{5} + 2$
Associated inertia:$1$,$1$
Indices of inseparability:$[3, 0]$

Invariants of the Galois closure

Galois group:$C_5^2:\OD_{16}$ (as 10T28)
Inertia group:$C_5^2:C_8$ (as 10T18)
Wild inertia group:$C_5^2$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:$[11/8, 11/8]$
Galois mean slope:$271/200$
Galois splitting model: $x^{10} - 5 x^{9} + 5 x^{7} + 140 x^{6} + 161 x^{5} + 220 x^{4} - 50 x^{3} + 325 x^{2} - 240 x + 29$ Copy content Toggle raw display