Properties

Label 5.12.10.2
Base \(\Q_{5}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_6\times S_3$ (as 12T18)

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Defining polynomial

\(x^{12} + 15 x^{6} + 100\)  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: $\Q_{5}$
Root number: $-1$
$|\Aut(K/\Q_{ 5 })|$: $6$
This field is not Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.4.2.1, 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:\( x^{6} - 5 t^{2} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_6\times S_3$ (as 12T18)
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:$x^{12} - 5 x^{11} + 13 x^{10} - 40 x^{9} + 100 x^{8} - 195 x^{7} + 370 x^{6} - 590 x^{5} + 855 x^{4} - 870 x^{3} + 908 x^{2} - 400 x + 64$