Properties

Label 5.12.9.1
Base \(\Q_{5}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\(x^{12} - 10 x^{8} - 375 x^{4} - 2000\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $12$
Ramification exponent $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{5}(\sqrt{5})$
Root number: $-1$
$|\Gal(K/\Q_{ 5 })|$: $12$
This field is Galois and abelian over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{5})$, 5.3.0.1, 5.4.3.1, 5.6.3.1

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 \( x^{3} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{4} - 5 t^{4} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{12} - x^{11} - 12 x^{10} + 11 x^{9} + 54 x^{8} - 43 x^{7} - 113 x^{6} + 71 x^{5} + 110 x^{4} - 46 x^{3} - 40 x^{2} + 8 x + 1$