Properties

Label 5.12.11.1
Base \(\Q_{5}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $S_3 \times C_4$ (as 12T11)

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Defining polynomial

\(x^{12} - 5\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{5}(\sqrt{5})$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $4$
This field is not Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{5})$, 5.3.2.1, 5.4.3.1, 5.6.5.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{12} - 5 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_4\times S_3$ (as 12T11)
Inertia group:$C_{12}$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:$x^{12} - 2 x^{11} + x^{10} + 5 x^{9} - 12 x^{7} - x^{6} + 18 x^{5} - 20 x^{4} - 5 x^{3} + 16 x^{2} - 7 x + 1$