Properties

Label 5.12.10.3
Base \(\Q_{5}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

\(x^{12} + 25 x^{6} + 200\)  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: $1$
$|\Gal(K/\Q_{ 5 })|$: $12$
This field is Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.3.2.1 x3, 5.4.2.2, 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:$\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{6} - 5 t^{3} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3:C_4$ (as 12T5)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} - 4 x^{11} + 26 x^{10} - 15 x^{9} + 25 x^{8} + 666 x^{7} - 1189 x^{6} + 4836 x^{5} + 2075 x^{4} - 7025 x^{3} + 54071 x^{2} - 35249 x + 68051$