Properties

Label 5.12.6.2
Base \(\Q_{5}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\(x^{12} - 3125 x^{2} + 31250\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $12$
Ramification exponent $e$: $2$
Residue field degree $f$: $6$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $12$
This field is Galois and abelian over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.3.0.1, 5.4.2.2, 5.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.6.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{6} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 5 t \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{12} - 4 x^{11} - 62 x^{10} + 224 x^{9} + 1319 x^{8} - 4132 x^{7} - 12278 x^{6} + 31124 x^{5} + 50090 x^{4} - 89004 x^{3} - 69720 x^{2} + 54712 x - 4319$