Properties

Label 5.12.10.1
Base \(\Q_{5}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $D_6$ (as 12T3)

Related objects

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Defining polynomial

\(x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44\)  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: $-1$
$|\Gal(K/\Q_{ 5 })|$: $12$
This field is Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.3.2.1 x3, 5.4.2.1, 5.6.4.1, 5.6.5.1 x3, 5.6.5.2 x3

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

Invariants of the Galois closure

Galois group:$D_6$ (as 12T3)
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:$x^{12} - 3 x^{11} + 4 x^{10} - 3 x^{7} - x^{6} + 3 x^{5} + 4 x^{2} + 3 x + 1$