Properties

Label 5.12.6.1
Base \(\Q_{5}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\(x^{12} + 500 x^{6} - 3125 x^{2} + 62500\)  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: $\Q_{5}$
Root number: $-1$
$|\Gal(K/\Q_{ 5 })|$: $12$
This field is Galois and abelian over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.3.0.1, 5.4.2.1, 5.6.0.1, 5.6.3.1, 5.6.3.2

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:\( x^{2} - 5 t^{2} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
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:$x^{12} - x^{11} + 2 x^{10} - 3 x^{9} + 5 x^{8} - 8 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1$