Properties

Label 5.12.8.1
Base \(\Q_{5}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

\(x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $12$
This field is Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.3.2.1 x3, 5.4.0.1, 5.6.4.1

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

Unramified/totally ramified tower

Unramified subfield:5.4.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{4} + x^{2} - 2 x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 5 t^{3} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3:C_4$ (as 12T5)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{12} - 3 x^{11} + 2 x^{9} + 43 x^{8} - 74 x^{7} - 71 x^{6} - 26 x^{5} + 271 x^{4} + 720 x^{3} - 406 x^{2} - 1633 x + 1699$