Properties

Label 5.4.2.1
Base \(\Q_{5}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{5}$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $4$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:None

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 20 x + 5 \) $\ \in\Q_{5}(t)[x]$ 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:$C_2^2$ (as 4T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model: $x^{4} + 15 x^{2} + 100$ Copy content Toggle raw display