Properties

Label 5.6.5.2
Base \(\Q_{5}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $D_{6}$ (as 6T3)

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Defining polynomial

\(x^{6} + 10\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{5}(\sqrt{5\cdot 2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $2$
This field is not Galois over $\Q_{5}.$
Visible slopes:None

Intermediate fields

$\Q_{5}(\sqrt{5\cdot 2})$, 5.3.2.1

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: \( x^{6} + 10 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$D_6$ (as 6T3)
Inertia group:$C_6$ (as 6T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{6} + 10$