Properties

Label 5.12.10.4
Base \(\Q_{5}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

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Defining polynomial

\(x^{12} - 5 x^{6} + 50\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $6$
This field is not Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.4.2.2, 5.6.4.2

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 \( x^{2} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{6} - 5 t \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3\times C_3:C_4$ (as 12T19)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} - 480 x^{10} - 1260 x^{9} + 80100 x^{8} + 413700 x^{7} - 5076750 x^{6} - 40519500 x^{5} + 59710125 x^{4} + 1214507000 x^{3} + 2584503750 x^{2} - 3523957500 x - 12020550625$  Toggle raw display