Properties

Label 5.6.3.2
Base \(\Q_{5}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + 75 x^{2} - 375\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{5}(\sqrt{5\cdot 2})$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $6$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:None

Intermediate fields

$\Q_{5}(\sqrt{5\cdot 2})$, 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: \( x^{2} + 5 t \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$